Unit 3: Components, Cliques, & Key Actors

ECI 589 Social Network Analysis and Education

1. PREPARE

The primary goal in this case study is to examine the use of different algorithms to analyze a network’s groups and individual actors. Using an open educational dataset prepared by Kellogg and Edelman Kellogg and Edelmann (2015), we’ll explore both “top-down” and “bottom-up” approaches to identifying groups within our network, as well as node-level measures of centrality to identify key actors. Specifically, the Unit 3 case study will cover the following topics:

1. Prepare: Prior to analysis, we’ll take a look at the context from which our data came, formulate simple research questions to guide our analysis, and load some by now familiar R packages for network analysis.

2. Wrangle: Wrangling data entails the work of manipulating, cleaning, transforming, and merging data. In section 2 we focus on importing network data, and converting edge and node lists into a network object that can be analyzed and graphed.

3. Explore: In section 3, we look at components and cliques within our network, and learn how to add add summarize important node-level measures of centrality.

4. Model: While we won’t dig into approaches for modeling network data until Unit 4, we will take a quick look at some approaches used in the research article guiding this study.

5. Communicate: Finally, you’ll prepare a simple “data product” consisting of a data visualization and/or table that highlights some key findings from our analysis.

1a. Review the Research

In Social Network Analysis and Education: Theory, Methods & Applications, Carolan (2013) notes that:

the social network perspective is one concerned with the structure of relations and the implication this structure has on individual or group behavior and attitudes

More specifically, Carolan cites the following four features used by Freeman (2004) to define the social network perspective:

1. Social network analysis is motivated by a relational intuition based on ties connecting social actors.

2. It is firmly grounded in systematic empirical data.

3. It makes use of graphic imagery to represent actors and their relations with one another.

4. It relies on mathematical and/or computational models to succinctly represent the complexity of social life.

For Unit 3, our case study is guided by previous research and evaluation work conducted by the Friday Institute for Educational Innovation as part of the Massively Open Online Courses for Educators (MOOC-Ed) initiative. The study introduced next and our hands-on analysis with R will help to illustrate these four defining features of the social network perspective.

A Social Network Perspective in MOOC-Eds

Kellogg, S., Booth, S., & Oliver, K. (2014). A social network perspective on peer supported learning in MOOCs for educators. International Review of Research in Open and Distributed Learning, 15(5), 263-289.

Research Context

In the spring of 2013, The Friday Institute launched the MOOC-Ed Initiative to explore the potential of delivering personalized, high-quality professional development to educators at scale (Kleiman et al., 2013). In collaboration with the Alliance for Excellent Education, the Friday Institute launched this initiative with a 6-week pilot course called Planning for the Digital Learning Transition in K-12 Schools (DLT 1), which was offered again in September 2013 (DLT 2). This course was designed to help school and district leaders plan and implement K-12 digital learning initiatives.

Academics, as well as pundits from traditional and new media, have raised a number of concerns about MOOCs, including the lack of instructional and social supports. Among the core design principles of MOOC-Eds are collaboration and peer-supported learning. It is an assumption of this study that challenges arising form this problem of scale can be addressed by leveraging these massive numbers to develop robust online learning communities.

This mixed-methods case study used both SNA and qualitative methods to better understand peer support in MOOC-Eds through an examination of the characteristics, mechanisms, and outcomes of peer networks. Findings from this study demonstrate that even with technology as basic as a discussion forum, MOOCs can be leveraged to foster these networks and facilitate peer-supported learning. Although this study was limited to two unique cases along the wide spectrum of MOOCs, the methods applied provide other researchers with an approach for better understanding the dynamic process of peer supported learning in MOOCs.

Data Sources

MOOC-Ed registration form. All participants completed a registration form for each MOOC-Ed course. The registration form consists of self-reported demographic data, including information related to their professional role and work setting, years of experience in education, and personal learning goals.

MOOC-Ed discussion forums. All peer interaction, including peer discussion, feedback, and reactions (e.g., likes), take place within the forum area of MOOC-Eds, which are powered by Vanilla Forums. Because of the specific focus on peer supported learning, postings to or from course facilitators and staff were removed from the data set. Finally, analyses described below exclude more passive forms of interactions (i.e., read and reaction logs), and include only postings among peers.

For our Unit 3 case study, we’ll take a look at data from the original Digital Learning Transition in K-12 Schools (DLT 1) that was not included in this study. For your independent analysis next week, you may want to consider working with the DLT 2 data to see if you can replicate some of the findings from this paper!

Note: In the data we’re using, instructors have not yet been removed and only direct replies to forum posts have been included, though “weaker” ties like reactions with emoticons and even views of posts were captured in this study.

👉 Your Turn ⤵

Take a quick look at the Description of the Dataset section from the Massively Open Online Course for Educators (MOOC-Ed) network dataset BJET article and the accompanying data sets stored on Harvard Dataverse that we’ll be using for this case study.

In the space below, type a brief response to the following questions:

1. What were some of the steps necessary to construct the edges in this dataset?

   • The edge data was constructed from a reply by one participant in the discussion forum to the post of another participant. Started by I and then participant(s) may reply to I and or a different participant may respond to the other participants without directly responding to I.

2. What “node attributes” are included in this dataset? How might they be used for analysis?

   • UID
   • Instructor
   • Role
   • experience
   • Grades
   • location
   • gender
   • group
   • expert
   • connect

3. What else do you notice/wonder about this dataset?

   • I wonder if the participants ended up clustering by grades taught, or role or experience. Who is the dominant player in the cluster?

1b. Identify a Question(s)

A Social Network Perspective on Peer Supported Learning was framed by three primary research questions related to peer supported learning:

1. What are the patterns of peer interaction and the structure of peer networks that emerge over the course of a MOOC-Ed?

2. To what extent do participant and network attributes (e.g., homophily, reciprocity, transitivity) account for the structure of these networks?

3. To what extent do these networks result in the co-construction of new knowledge?

For Unit 3, our exploratory analysis is guided by RQ1 from the original study with an additional emphasis on examining groups and key actors that emerge in the discussion forums for the first and second implementation of the Digital Learning Transition MOOC-Eds. Specifically, for our case we’re interested in the following questions examining groups and key actors:

1. How many network components and cliques are in first and second iteration of DLT 1 and DLT 2?
2. Which actors are central to each of these networks and what attributes, if any, do they share?

👉 Your Turn ⤵

Based on what you know about networks and the context so far, what other research question(s) might ask we ask in this context that a social network perspective might be able to answer?

• Does the network have strong ties and are homophilous? Or is the network heterogeneous?

1c. Load Libraries

As highlighted in Chapter 6 of Data Science in Education Using R (DSIEUR):

Packages are shareable collections of R code that can contain functions, data, and/or documentation. Packages increase the functionality of R by providing access to additional functions to suit a variety of needs.

You should see installed some familiar tidytext packages from our Getting Started Walkthrough like {dplyr} and {readr} which we’ll be using again shortly. You should also see an important package call {igraph} that we will rely on heavily for our network analyses in this course.

If you are working in RStudio Desktop, or notice that the packages have not been installed and/or loaded, run the following install.packages() function code to install the {tidyverse} and {igraph} packages:

install.packages("tidyverse")
install.packages("igraph") 

👉 Your Turn ⤵

Use the library() function and the code chunk below and load the following packages:

• tidyverse
• igraph
• tidygraph
• ggraph
• skmir
• janitor

library(tidyverse)

## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --

## v ggplot2 3.3.5     v purrr   0.3.4
## v tibble  3.1.6     v dplyr   1.0.7
## v tidyr   1.1.4     v stringr 1.4.0
## v readr   2.1.1     v forcats 0.5.1

## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(skimr)
library(igraph)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:dplyr':
## 
##     as_data_frame, groups, union

## The following objects are masked from 'package:purrr':
## 
##     compose, simplify

## The following object is masked from 'package:tidyr':
## 
##     crossing

## The following object is masked from 'package:tibble':
## 
##     as_data_frame

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

library(tidygraph)

## 
## Attaching package: 'tidygraph'

## The following object is masked from 'package:igraph':
## 
##     groups

## The following object is masked from 'package:stats':
## 
##     filter

library(ggraph)
library(janitor)

## 
## Attaching package: 'janitor'

## The following objects are masked from 'package:stats':
## 
##     chisq.test, fisher.test

library(RColorBrewer)

RStudio Tip: You can always check to see which packages have already been installed and loaded into RStudio Cloud by looking at the the Files, Plots, & Packages Pane in the lower right hand corner of RStudio as shown in the following screenshot:

---

2. WRANGLE

In general, data wrangling involves some combination of cleaning, reshaping, transforming, and merging data (Wickham and Grolemund 2016). The importance of data wrangling is difficult to overstate, as it involves the initial steps of going from the raw data to a dataset that can be explored and modeled (Krumm, Means, and Bienkowski 2018).

For our data wrangling this week, we’re keeping it simple since working with network data is a bit of a departure from our working with rectangular data frames. Our primary goals for Section 2 are learning how to:

1. Import Lists. An obvious and also important first step, we need to “read” our data into R and learn about formatting for edge-lists and node attribute files.

2. Create a Graph Object. Before performing network analyses, we’ll need to convert our data frames into a graph object for working with relational data.

2a. Import Data

The Edge-List Format

To get started, we need to import, or “read,” our data into R. The function used to import your data will depend on the file format of the data you are trying to import, but R is pretty adept at working with many files types.

Take a look in the /data folder in your Files pane. You should see the following .csv files:

• dlt1-edges.csv

• dlt1-nodes.csv

The first file dlt1-edges.csv is an edge-list that contains information about each tie, or relation between two actors in a network. In this context, a “tie” is a reply by one participant in the discussion forum to the post of another participant – or in some cases to their own post! These ties between a single actor are called “self-loops” and as we’ll see later, {tidygraph} has a special function to remove these self loops from a sociogram, or network visualization.

The edge-list format is slightly different than other formats you have likely worked with before in that the values in the first two columns each row represent a dyad, or tie between two nodes in a network. An edge-list can also contain other information regarding the strength, duration, or frequency of the relationship, sometime called “weight,” in addition to other “edge attributes.”

In addition to our Sender and Reciever dyad pairs, our DLT 1 dataset contains the following edge attributes:

• Sender = Unique identifier of author of comment

• Receiver = Unique identifier of identified recipient of comment

• Timestamp = Time post or reply was posted

• Parent = Primary category or topic of thread

• Category = Subcategory or subtopic of thread

• Thread_id = Unique identifier of a thread

• Comment_id = Unique identifier of a comment\

Let’s use the read_csv() function from the {readr} and {janitor} packages introduced in previous units to read in our edge-list and clean up the variables names:

dlt1_ties <- read_csv("data/dlt1-edges.csv", 
                      col_types = cols(Sender = col_character(), 
                                       Receiver = col_character(), 
                                       `Category Text` = col_skip(), 
                                       `Comment ID` = col_character(), 
                                       `Discussion ID` = col_character())) |>
  clean_names()

dlt1_ties

## # A tibble: 2,529 x 9
##    sender receiver timestamp  discussion_title  discussion_cate~ parent_category
##    <chr>  <chr>    <chr>      <chr>             <chr>            <chr>          
##  1 360    444      4/4/13 16~ Most important c~ Group N          Units 1-3 Disc~
##  2 356    444      4/4/13 18~ Most important c~ Group D-L        Units 1-3 Disc~
##  3 356    444      4/4/13 18~ DLT Resources—Co~ Group D-L        Units 1-3 Disc~
##  4 344    444      4/4/13 18~ Most important c~ Group O-T        Units 1-3 Disc~
##  5 392    444      4/4/13 19~ Most important c~ Group U-Z        Units 1-3 Disc~
##  6 219    444      4/4/13 19~ Most important c~ Group M          Units 1-3 Disc~
##  7 318    444      4/4/13 19~ Most important c~ Group M          Units 1-3 Disc~
##  8 4      444      4/4/13 19~ Most important c~ Group N          Units 1-3 Disc~
##  9 355    356      4/4/13 20~ DLT Resources—Co~ Group D-L        Units 1-3 Disc~
## 10 355    444      4/4/13 20~ Most important c~ Group D-L        Units 1-3 Disc~
## # ... with 2,519 more rows, and 3 more variables: discussion_identifier <chr>,
## #   comment_id <chr>, discussion_id <chr>

Note the addition of the col_types = argument for changing the column types to character strings since the numbers for those particular columns indicate actors (Sender and Reciever) and attributes (Comment_ID and Discussion_Id). We also skipped the Category Text since this was left blank for deidentification purposes.

👉 Your Turn ⤵

Consider the example pictured below of a discussion thread from the Planning for the Digital Learning Transition in K-12 Schools (DLT 1) where our data orginated. This thread was initiated by participant I, so the comments by J and N are considered to be directed at I. The comment of B, however, is a direct response to the comment by N as signaled by the use of the quote-feature as well as the explicit mentioning of N’s name within B’s comment.

Now answer the following questions as they relate to the DLT 1 edge-list we just read into R.

1. Which actors in this thread are the Sender and the Reciever? Which actor is both?

   • I is the sender. J,N and B are receivers, and N is both

2. How many dyads are in this thread? Which pairs of actors are dyads?

   • 3 dyads: N to I , J to I , N to B

Sidebar: Unfortunately, these types of nuances in discussion forum data as illustrated by this simple example are rarely captured through automated approaches to constructing networks. Fortunately, the dataset you are working with was carefully reviewed to try and capture more accurately the intended recipients of each reply.

Node Attributes

The second file we’ll be using contains all the nodes or actors (i.e., participants who posted to the discussion forum) as well as some of their attributes such as gender and years of experience in education.

Carolan (2013) notes that most social network analyses include variables that describe attributes of actors, ones that are either categorical (e.g., sex, race, etc.) or continuous in nature (e.g., test scores, number of times absent, etc.). These attributes that can be incorporated into a network graph or model, making it more informative and can aid in testing or generating hypotheses.

These attribute variables are typically included in a rectangular array, or dataframe, that mimics the actor-by-attribute that is the dominant convention in social science, i.e. rows represent cases, columns represent variables, and cells consist of values on those variables.

As an aside, Carolan also refers to this historical preference by researchers for “actor-by-attribute” data, in the absence of relational data in which the actor has been removed their social context, as the “sociological meatgrinder” in action. Specifically, this historical approach assumes that the actor does not interact with anyone else in the study and that outcomes are solely dependent of the characteristics of the individual.

👉 Your Turn ⤵

Use the code chunk below to import the dlt1-nodes.csv attribute file and be sure to set the following variables as character data types: UID, Facilitator, expert, connect.

dlt1_actors <- read_csv("data/dlt1-nodes.csv", 
                      col_types = cols(UID = col_character(), 
                                       Facilitator = col_character(), 
                                       `expert` = col_character(), 
                                       `connect` = col_character())) |>
  clean_names()

dlt1_actors

## # A tibble: 445 x 13
##    uid   facilitator role1 experience experience2 grades location region country
##    <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
##  1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
##  2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
##  3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
##  4 4     0           clas~          2 11 to 20    middle NC       South  US     
##  5 5     0           othe~          3 20+         gener~ AL       South  US     
##  6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
##  7 7     0           inst~          2 11 to 20    gener~ SD       Midwe~ US     
##  8 8     0           spec~          1 6 to 10     secon~ BE       Inter~ BE     
##  9 9     0           clas~          1 6 to 10     middle NC       South  US     
## 10 10    0           scho~          2 11 to 20    middle NC       South  US     
## # ... with 435 more rows, and 4 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>

Now use the code chunk below to inspect the data you imported and complete the matching exercise that follows:

dlt1_actors

## # A tibble: 445 x 13
##    uid   facilitator role1 experience experience2 grades location region country
##    <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
##  1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
##  2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
##  3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
##  4 4     0           clas~          2 11 to 20    middle NC       South  US     
##  5 5     0           othe~          3 20+         gener~ AL       South  US     
##  6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
##  7 7     0           inst~          2 11 to 20    gener~ SD       Midwe~ US     
##  8 8     0           spec~          1 6 to 10     secon~ BE       Inter~ BE     
##  9 9     0           clas~          1 6 to 10     middle NC       South  US     
## 10 10    0           scho~          2 11 to 20    middle NC       South  US     
## # ... with 435 more rows, and 4 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>

Match up the attributes included in the node file with the following codebook descriptors. The first one has been done as an example.

• Facilitator = Identification of course facilitator (1 = instructor)
• UID = Dummy variable for whether participants listed networking and collaboration with others as one of their course goals on the registration form
• expert = Identifier of “expert panelists” invited to course to share experience through recorded Q&A
• Facilitator = Identification of course facilitator (1 = instructor)
• role1 = Professional role (eg, teacher, librarian, administrator)
• experience2 = Years of experience as an educator
• grades = Works with elementary, middle, and/or high school students
• geoup = Initial assignment of discussion group

RStudio Tip: Importing data and dealing with data types can be a bit tricky, especially for beginners. Fortunately, RStudio has an “Import Dataset” feature in the Environment Pane that can help you use the {readr} package and associated functions to greatly facilitate this process.

2b. Create Network Object

As demonstrated in our previous case study, we first need to convert the data frames that we imported into an network object before we can begin using many of the functions from our network packages for summarizing and visualizing our DLT 1 network.

Convert to Graph Object

To do that, we will use the tbl_graph() function from the {tidygraph} package. Note that I included the eval=FALSE argument in the code block below to prevent this code from running when we knit our final document. Otherwise it will produce an error since we can’t include help documentation in our knitted HTML file.

Run the following code to take a look at the help documentation for this function:

?tbl_graph

You probably saw that this particular function takes the following three arguments, two of which are data frames:

• edges = A data.frame containing information about the edges in the graph. The terminal nodes of each edge must either be encoded in a to and from column, or be in the two first columns.

• nodes = a node list that starts with a column of node IDs. Any following columns are interpreted as node attributes.

• node_key = The name of the column in nodes that character represented to and from columns should be matched against.

• directed = determines whether or not to create a directed graph.

Run the following code to specify our ties data frame as the edges of our network, our actors data frame for the vertices of our network and their attributes, and indicate that this is indeed a directed network.

dlt1_network <- tbl_graph(edges = dlt1_ties,
                          nodes = dlt1_actors,
                          node_key = "uid",
                          directed = TRUE)

# create dlt2

dlt2_ties <- read_csv("data/dlt2-edges.csv", 
                      col_types = cols(Sender = col_character(), 
                                       Reciever = col_character(), 
                                       `Category` = col_skip(), 
                                       `CommentID` = col_character(), 
                                       `DiscussionID` = col_character())) |>
  clean_names()

dlt2_actors <- read_csv("data/dlt2-nodes.csv", 
                      col_types = cols(uid = col_character(), 
                                       facilitator = col_character(), 
                                       `expert` = col_character(), 
                                       `connect` = col_character())) |>
  clean_names()

dlt2_network <- tbl_graph(edges = dlt2_ties,
                          nodes = dlt2_actors,
                          node_key = "uid",
                          directed = TRUE)

👉 Your Turn ⤵

Take a look at the output for our dlt1_network and answer the questions that follow:

dlt1_network

## # A tbl_graph: 445 nodes and 2529 edges
## #
## # A directed multigraph with 4 components
## #
## # Node Data: 445 x 13 (active)
##   uid   facilitator role1 experience experience2 grades location region country
##   <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
## 1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
## 2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
## 3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
## 4 4     0           clas~          2 11 to 20    middle NC       South  US     
## 5 5     0           othe~          3 20+         gener~ AL       South  US     
## 6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
## # ... with 439 more rows, and 4 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>
## #
## # Edge Data: 2,529 x 9
##    from    to timestamp discussion_title discussion_cate~ parent_category
##   <int> <int> <chr>     <chr>            <chr>            <chr>          
## 1   360   444 4/4/13 1~ Most important ~ Group N          Units 1-3 Disc~
## 2   356   444 4/4/13 1~ Most important ~ Group D-L        Units 1-3 Disc~
## 3   356   444 4/4/13 1~ DLT Resources—C~ Group D-L        Units 1-3 Disc~
## # ... with 2,526 more rows, and 3 more variables: discussion_identifier <chr>,
## #   comment_id <chr>, discussion_id <chr>

dlt2_network

## # A tbl_graph: 492 nodes and 2584 edges
## #
## # A directed multigraph with 69 components
## #
## # Node Data: 492 x 13 (active)
##   uid   facilitator role  experience2 experience grades location region country
##   <chr> <chr>       <chr>       <dbl>      <dbl> <chr>  <chr>    <chr>  <chr>  
## 1 1     0           curr~           2         17 gener~ IN       Midwe~ US     
## 2 2     0           other           1          3 prima~ NC       South  US     
## 3 3     0           inst~           2         20 gener~ US       South  US     
## 4 4     0           inst~           2         12 middle TX       South  US     
## 5 5     0           other           1          0 gener~ CAN      Inter~ CA     
## 6 6     0           inst~           1          7 gener~ CAN      Inter~ CA     
## # ... with 486 more rows, and 4 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>
## #
## # Edge Data: 2,584 x 8
##    from    to timestamp title   parent  description     comment_id discussion_id
##   <int> <int> <chr>     <chr>   <chr>   <chr>           <chr>      <chr>        
## 1    37   461 9/27/13   Greeti~ Group ~ Please introdu~ 3          1            
## 2   434   434 9/27/13   Sherry~ Group ~ Please introdu~ 4          2            
## 3   302   434 9/27/13   Sherry~ Group ~ Please introdu~ 5          3            
## # ... with 2,581 more rows

1. How does the number of node and edges in our DLT 1 network compare to the totals reported for the DLT 2, the second iteration of this MOOC-Ed, reported on in our guiding study?

   • Dlt 1 network has 445 nodes and 2529 edges, Dlt 2 has 492 nodes and 2584 edges. From first glance, it would appear that Dlt 2 network has less nodes and edges. However, DLT 1 actually has more edges and nodes. However, DLT 1 has less component than DLT 2.,

2. Do the number of nodes and edges are in our network match the number of observations in our node and edge list .csv files? Hint: Check the Environment pane.

   • Yes, the observations and the number of nodes and edges match.

3. Our output notes that our network is a directed multigraph, indicating that some dyads may have multiple edges. Why might that be?

   • Assymetrical ties may not be reciprocated. If you are asking who someone goes to for advice one person in a dyad may have multiple arcs in a directed graph. Or a tie may have been send to and from itself.

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3. EXPLORE

As noted in the previous units, exploratory data analysis involves the processes of describing your network through mathematical measures and/or data visualization. In Section 3, we learn some new functions for identifying groups and subgroups and creating node-level measures to identify central actors in our network. Specifically, in this section we’ll focus on: a. Group Analysis. We take a look at both a top-down and bottom-up approach to identifying groups within a network by examining the weak and strong components in our network as well as existing cliques.

2. Egocentric Analysis. We also examine the size of an ego’s “local neighborhood” as well as key centrality measures for each node such as degree, closeness, and betweenness.

3a. Analyze Groups: Components & Cliques

Components

As noted in Social Network Analysis and Education (Carolan 2014), one of the most basic ways in which network researchers first characterize a network’s substructure is to identify its components.

A component is a connected subgraph in which there is a path between all pairs of nodes.

Recall from our output above that our “multigraph” had 4 components. Let’s take a quick look at our network and see if we can spot the 4 components:

autograph(dlt1_network)

As you can see, three of our components are isolates in our network, meaning they have no connections, or forum replies to or from, anyone else in the network. That is, they posted to a discussion forum and never received a reply, or replied to the posts of others.

Directed graphs, such as the DLT 1 network, have two different kinds of components: weak and strong. A weak component, like ours above, ignores the direction of a tie; strong components do not. Rather,

Strong components consist of nodes that are connected to one another via both directions along the path that connects them.

This concept of a component is simple and very useful, especially when analyzing a large network with many components. Often, researchers extract each component and analyze it as a separate network. We won’t be doing that here, but it would be very reasonable to remove the isolates from our network prior to further analysis.

The {igraph} package has a simple function for identifying the number of components in a network, the size of each component, and which actors belong to each. Let’s first take a quick look at the summaries for “weak” components in our network:

components(dlt1_network, mode = c("weak"))

## $membership
##   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [149] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [223] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [260] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [297] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [334] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [371] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [408] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 1
## [445] 1
## 
## $csize
## [1] 442   1   1   1
## 
## $no
## [1] 4

While not incredibly intuitive to interpret, the output indicates that there are 4 components as we discovered above, 1 component has 442 members and three (our isolates) have only 1 member.

One issue with this output that the {tidygraph} package is great at solving is identifying and saving as a new variable who belongs to which component. We can sort of see from above that the first 400+ actors in our network belong to component 1, while the three isolates towards the end belong to components 2, 3, and 4 respectively; and the last two nodes in our network also belong to component 1.

How do you think you might find the “strong” components in our network using the component() function? Use the following code chunk to test out your theory:

components(dlt1_network, mode = c("strong"))

## $membership
##   [1] 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 193
##  [19] 194 194 203 194 209 194 194 194 194 192 194 194 191 194 194 194 194 194
##  [37] 194 194 194 190 194 194 194 194 194 194 189 194 194 194 194 194 194 194
##  [55] 188 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194
##  [73] 187 194 194 194 194 194 186 194 194 194 194 208 194 185 194 194 184 194
##  [91] 194 194 183 194 181 180 178 194 194 194 194 177 194 194 194 194 194 176
## [109] 194 194 175 194 194 194 194 194 194 194 194 174 194 194 169 168 167 166
## [127] 165 194 194 164 194 194 194 163 162 194 194 194 161 159 194 194 158 194
## [145] 173 182 157 156 155 172 171 194 206 194 194 194 194 194 194 154 194 194
## [163] 194 153 194 152 194 151 150 194 149 194 194 148 202 194 194 194 147 146
## [181] 194 201 194 194 194 145 195 194 198 194 194 194 194 194 194 144 194 194
## [199] 194 194 194 143 194 142 194 194 194 194 194 141 194 194 140 139 138 194
## [217] 194 194 194 137 194 194 194 136 135 194 194 134 133 131 130 129 128 204
## [235] 194 127 126 125 160 124 123 122 194 121 119 194 194 194 194 194 194 194
## [253] 194 194 118 194 194 179 117 115 116 194 120 114 194 194 113 194 112 194
## [271] 194 194 199 111 194 194 194 110 194 109 194 108 106 107 194 200 194 105
## [289] 104 103 102 194 194 101 194 100  99  98  97 194 194 194 194  96 194 197
## [307] 194 207  95 194  94  93  92  91  90  89 194 194 194  88 194 194 194 205
## [325] 194  87  86  85 194 170 194  84  83 196 194 194 194 194 194 194 194 194
## [343] 194 132 194 194 194  82  81 194 194  80  79  78 194 194  77 194  76  75
## [361] 194  74  73  72  71  70  69  68  67  66  65  64  63  62  61  60  59  58
## [379]  57  56  55  54  53  52  51  50  49  48  47  46  45  44  43  42  41  40
## [397]  39  38  37  36  35  34  33  32  31  30  29  28  27  26  25  24 194  23
## [415]  22  21  20  19  18  17  16 194  15  14  13  12  11  10   9   8   7 194
## [433] 194 194   6   5 194 194 194   4   3   2   1 194 194
## 
## $csize
##   [1]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##  [19]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##  [37]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##  [55]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##  [73]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
##  [91]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
## [109]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
## [127]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
## [145]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
## [163]   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
## [181]   1   1   1   1   1   1   1   1   1   1   1   1   1 237   1   1   1   1
## [199]   1   1   1   1   1   1   1   1   1   1   1
## 
## $no
## [1] 209

Wow! If your code was correct, you’ll see that we have a whopping 209 distinct components!

As demonstrated in the previous case study, the {tidygraph} package has a very useful activate() function for working with our nodes and edges as if they were standard “tibbles,” or tidy data tables.

Let’s use the activate() function to single out the node list in our network and use the familiar mutate() and group_components() functions to create a new strong_component variable that indicates and saves the strong components to which each node belongs:

dlt1_network <- dlt1_network |>
  activate(nodes) |>
  mutate(strong_component = group_components(type = "strong"))

Let’s take a look at the nodes in our new dlt1_network, which should now contain our new variable. To do so, we’ll use another handy function from the {tidygraph} package, as_tibble(), that will temporarily convert our node list to a standard table that will allow us to view every row in our node list:

as_tibble(dlt1_network)

## # A tibble: 445 x 14
##    uid   facilitator role1 experience experience2 grades location region country
##    <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
##  1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
##  2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
##  3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
##  4 4     0           clas~          2 11 to 20    middle NC       South  US     
##  5 5     0           othe~          3 20+         gener~ AL       South  US     
##  6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
##  7 7     0           inst~          2 11 to 20    gener~ SD       Midwe~ US     
##  8 8     0           spec~          1 6 to 10     secon~ BE       Inter~ BE     
##  9 9     0           clas~          1 6 to 10     middle NC       South  US     
## 10 10    0           scho~          2 11 to 20    middle NC       South  US     
## # ... with 435 more rows, and 5 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>, strong_component <int>

We could even extend this if we liked to create some standard table summaries using the summarise() function. Run the code below to create a count of the number of nodes in each strong component:

dlt1_network |>
  as_tibble() |>
  group_by(strong_component) |>
  summarise(count = n()) |>
  arrange(desc(count))

## # A tibble: 209 x 2
##    strong_component count
##               <int> <int>
##  1                1   237
##  2                2     1
##  3                3     1
##  4                4     1
##  5                5     1
##  6                6     1
##  7                7     1
##  8                8     1
##  9                9     1
## 10               10     1
## # ... with 199 more rows

Similar to our graph of weak components, we see this network has a strong component with many members (n=237), and the remaining components are all isolated nodes.

If we wanted to illustrate this with a sociogram, we could create a new edge variable using the same activate() and mutate() functions and filter() our edges so our graph only contains reciprocated ties, like so:

dlt1_network |>
  activate(edges) |>
  mutate( reciprocated = edge_is_mutual()) |> 
  filter(reciprocated == TRUE) |>
  autograph()

Or we could filter out all isolates in our strong component network entirely using the same activate() and filter() functions:

dlt1_network |>
  activate(nodes) |>
  filter(strong_component == 1) |>
  autograph()

Obviously, this graph would need some polishing before sharing with others, but you get the point. The activate() function from the {tidygraph} package is a powerful tool that let’s you leverage the entire suite of {tidyverse} packages to analyze nodes and edges in a network! We’ll be using the activate function again in section 3b. Egocentric Analysis: Size & Centrality to calculate node-level measures.

Cliques

Whereas top-down approaches to group analysis focus on the complete network and considers parts of it that distinct from the rest of the network, the bottom-up approach focuses is built up from simple dyads and triads that extend into dense clusters that give the network its “clumpiness.” Similar to complete network measures, there are several important properties for which precise definitions and algorithms have been developed, including cliques, clans, plexes, and cores (Carolan 2014).

Examining cliques is one bottom-up approach that reveals how groups are distributed in the network and which actors belong to which groups.

A clique is a maximally connected subgraph of nodes (> 2) in which all nodes are connected to each other.

Similar to our component analysis, the {igraph} function has a simple clique_num() function for identifying number of completely connected subgroups in a network:

clique_num(dlt1_network)

## Warning in clique_num(dlt1_network): At cliques.c:1125 :directionality of edges
## is ignored for directed graphs

## [1] 8

As you probably saw above, the clique_num() function does not take into account directionality of our ties. It looks like we have 8 dense clusters of nodes that all have ties to one another.

The {igraph} function also has a simple cliques() function for identifying members who belong to the same group. In addition to specifying the network you want to examine, this function also allows you to set the minimum and maximum number of members to included in a clique.

Let’s see if there are any cliques that contain a minimum of 8 nodes?

cliques(dlt1_network, min = 8, max = NULL)

## Warning in cliques(dlt1_network, min = 8, max = NULL): At
## igraph_cliquer.c:57 :Edge directions are ignored for clique calculations

## [[1]]
## + 8/445 vertices, from 4749393:
## [1]  11  19  24  30  44  60 444 445

Unfortunately, the {tidygraph} package does not have a clustering function for cliques. It does, however, include a range popular clustering functions provided that use range of functions that group nodes and edges based on “community structure.” Some of these community detection algorithms are designed for directed graphs (e.g. our strong component analysis above), while others are for undirected graphs.

One function similar to our clique analysis above is the group_edge_betweenness() function, which group densely connected nodes together. The betweenness centrality measures is something we will look at more closely in the next section.

Because this function can only be used with undirected networks, we will need to pipe |> our dlt_network through the following functions in sequence:

1. morph() with the to_undirected argument will temporarily change our directed network to an undirected network, or “symmetrize” our network as discussed in Carolan Carolan (2014);
2. activate() will select just our nodes list;
3. mutate() will created a new subgroup variable using the group_edgebetweenness() function;
4. unmorph() will change our undirected network back to a directed network.

Run the following code to group our nodes based on their edge betweenness and print our updated dlt1_network object and take a quick look. Because this is a bit computationally intensive, it make take a minute or so to run.

dlt1_network <- dlt1_network |>
  morph(to_undirected) |>
  activate(nodes) |>
  mutate(sub_group = group_edge_betweenness()) |>
  unmorph()

dlt1_network |>
  as_tibble()

## # A tibble: 445 x 15
##    uid   facilitator role1 experience experience2 grades location region country
##    <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
##  1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
##  2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
##  3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
##  4 4     0           clas~          2 11 to 20    middle NC       South  US     
##  5 5     0           othe~          3 20+         gener~ AL       South  US     
##  6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
##  7 7     0           inst~          2 11 to 20    gener~ SD       Midwe~ US     
##  8 8     0           spec~          1 6 to 10     secon~ BE       Inter~ BE     
##  9 9     0           clas~          1 6 to 10     middle NC       South  US     
## 10 10    0           scho~          2 11 to 20    middle NC       South  US     
## # ... with 435 more rows, and 6 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>, strong_component <int>, sub_group <int>

As you scroll through the nodes tibble produced, you should now see at the far end a new subgroup variable that includes an ID number indicating to which densely connected cluster each node belongs.

Finally, let get a count of Run the following code to group our nodes and print our new cccss_network_groups object and take a quick look:

dlt1_network |>
  activate(nodes) |>
  as_tibble() |>
  group_by(sub_group) |>
  summarise(count = n()) |>
  arrange(desc(count))

## # A tibble: 326 x 2
##    sub_group count
##        <int> <int>
##  1         1    72
##  2         2     6
##  3         3     5
##  4         4     4
##  5         5     3
##  6         6     3
##  7         7     3
##  8         8     3
##  9         9     2
## 10        10     2
## # ... with 316 more rows

This out is very telling and suggestive that our network may have a typical core-periphery structure: a group of well-connected actors at the center of the network with a set of actors residing on the periphery whose only connections are typically to those actors affiliated with this core group. At the core of our network might reside these actors in subgroup 1, with the remaining actors residing on the periphery.

Note, that if we had wanted to permanently convert our network to an undirected network, we could use the to_undirected argument as a stand-alone function.

dlt1_undirected <-  to_undirected(dlt1_network)

dlt1_undirected

## # A tbl_graph: 445 nodes and 2529 edges
## #
## # An undirected multigraph with 4 components
## #
## # Node Data: 445 x 15 (active)
##   uid   facilitator role1 experience experience2 grades location region country
##   <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
## 1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
## 2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
## 3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
## 4 4     0           clas~          2 11 to 20    middle NC       South  US     
## 5 5     0           othe~          3 20+         gener~ AL       South  US     
## 6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
## # ... with 439 more rows, and 6 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>, strong_component <int>, sub_group <int>
## #
## # Edge Data: 2,529 x 9
##    from    to timestamp discussion_title discussion_cate~ parent_category
##   <int> <int> <chr>     <chr>            <chr>            <chr>          
## 1   360   444 4/4/13 1~ Most important ~ Group N          Units 1-3 Disc~
## 2   356   444 4/4/13 1~ Most important ~ Group D-L        Units 1-3 Disc~
## 3   356   444 4/4/13 1~ DLT Resources—C~ Group D-L        Units 1-3 Disc~
## # ... with 2,526 more rows, and 3 more variables: discussion_identifier <chr>,
## #   comment_id <chr>, discussion_id <chr>

The {tidygraph} package has a wide range of functions to generate alternate representations of graphs. They can be used as stand alone functions to permanently alter your network, or as an argument with the morph() function to temporarily change your network depending on the analyses you need to perform.

👉 Your Turn ⤵

We’ve only scratched the surface of the range of top-down and bottom-up group analysis functions available in the {igraph} and {tidygraph} packages. In the code chunk below, try out one of the other grouping functions available in the tidygraph or igraph packages to examine groups withing the dlt1_network:

dlt1_network2 <- dlt1_network |>
  morph(to_subgraph, seq_len(graph_order()) %% 2 == 0) |>
  mutate(neighbour_count = centrality_degree())|>
  unmorph()

## Subsetting by nodes

lay <- layout_with_kk(dlt1_network2)
plot(dlt1_network2, layout = lay, 
     vertex.label = NA)

3b. Egocentric Analysis: Size & Centrality

In this section, we shift out analytical lens from groups to egos and the pattern of relations in which each individual is embedded. Implicit in egocentric analysis, and also very intuitive to understand, is that:

An individual’s (ego) connections with others (alters) provides access to some instrumental (e.g., advice) or expressive (e.g., support) resource that may, in turn, be beneficial.

As summarized by Carolan Carolan (2014), the structure and content of these relations between an ego and a set of alters is the focus of egocentric network analysis. Since our research question aims to identify “key actors” in our network, we will look at two characteristics of an ego’s connections that are indicative their importance: size and centrality.

Size

Recall from our previous case study focused on complete network analysis, one simple and often ignored structural property of a social network is its size. This also applies to egocentric analysis.

Size is simply the number of alters that are directly connected to ego.

This characteristic is also sometimes referred to as an ego’s local neighborhood, and as Carolan points out, size matters, because it indicates the amount of potential resources available in one’s network. For example, a student with a small friendship network may be at a disadvantage when it comes seeking support on schoolwork or advice on course or college enrollment.

We can use the same {tidygraph} process that we used for identifying groups and individual membership to calculate the size for each node. Specifically, we will need to activate() the nodes in our network and create a new variable which we’ll call size that is calculated using the local_size() function.

And since it’s much easier to inspect our data as a tibble than using the graph output, we’ll also convert our node to a table and arrange() in descending order by size to make it easier to see the range in values of our network:

dlt1_network <- dlt1_network |>
  activate(nodes) |>
  mutate(size = local_size())

dlt1_network |> 
  as_tibble() |>
  arrange(desc(size)) |> 
  select(uid, facilitator, size)

## # A tibble: 445 x 3
##    uid   facilitator  size
##    <chr> <chr>       <dbl>
##  1 444   1             295
##  2 445   1             160
##  3 44    0              61
##  4 11    0              51
##  5 7     0              42
##  6 30    0              42
##  7 19    0              39
##  8 60    0              39
##  9 36    0              37
## 10 432   0              36
## # ... with 435 more rows

Not surprisingly, the egos with the most alters are the course facilitators who played a very active role in this course and therefore have an outsize influence on the structure of this network.

Note: In the code immediately above or earlier in this case study, it’s important to take note when we have permanently altered our dlt1_network object and when we have only made temporary changes to our network. For example, when creating a new variable for local neighborhood size, we assigned the changes made to dlt1_network to an object of the same name using the <- assignment operator. This effectively overwrites the old object with the new object because was wanted to save the new variable created. However, when converting network to a tibble consisting of nodes and their facilitor and size attributes, we did not assign to dlt1_network because we did not want to permanently change to our network object to a tibble.

Centrality

As we learned in our previous case study and readings, a key structural property of networks is the concept of centralization. A network that is highly centralized is one in which relations are focused on a small number of actors or even a single actor in a network, whereas ties in a decentralized network are diffuse and spread over a number of actors. As we saw above, the facilitators in our network play an outsize role in the MOOC-Ed discussions!

Degree

One of the most common descriptives reported in network studies and a primary measure of centralization is degree.

Degree is the number of ties to and from an ego. In a directed network, in-degree is the number of ties received, whereas out-degree is the number of ties sent.

The {tidygraph} package has an aptly named function centrality_degree() for calculating degree, in-degree, and out-degree for all actors in a network.

Run the following create two new variables for our nodes: in_dgree and out_degree. We’ll set the mode = argument in centrality_degree() function to "in" and "out" respectively.

dlt1_network <- dlt1_network |>
  activate(nodes) |>
  mutate(in_degree = centrality_degree(mode = "in"),
         out_degree = centrality_degree(mode = "out"))
  
dlt1_network |> 
  as_tibble()

## # A tibble: 445 x 18
##    uid   facilitator role1 experience experience2 grades location region country
##    <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
##  1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
##  2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
##  3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
##  4 4     0           clas~          2 11 to 20    middle NC       South  US     
##  5 5     0           othe~          3 20+         gener~ AL       South  US     
##  6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
##  7 7     0           inst~          2 11 to 20    gener~ SD       Midwe~ US     
##  8 8     0           spec~          1 6 to 10     secon~ BE       Inter~ BE     
##  9 9     0           clas~          1 6 to 10     middle NC       South  US     
## 10 10    0           scho~          2 11 to 20    middle NC       South  US     
## # ... with 435 more rows, and 9 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>, strong_component <int>, sub_group <int>,
## #   size <dbl>, in_degree <dbl>, out_degree <dbl>

A Quick Note about Multigraphs and Tie Strength

You may have noticed in this network that indegree exceeds the total number of nodes in our network and are wondering how that is possible. Recall that our network is currently treated at a “multigraph,” indicating that some dyads have multiple edges. Therefore, some edges between dyads are counted more than once.

Alternatively, we could “simplify” our network by collapsing multiple edges between the same actors and assigning a value to edges between actors indicating tie strength. In this case it would be a measure of frequency of replies between actors. However, in the process of simplifying our network, we would lose important contextual information about the context of each tie, such as the when (timestamp) and where (forum) each tie was created.

Let’s dust off our ggplot() skills and take a look at the distribution of out_degree, or the number of replies to other posts, by using in geom_histogram() function for creating histograms.

dlt1_network |> 
  as_tibble() |>
  ggplot() +
  geom_histogram(aes(x = out_degree))

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

We can see that most egos in the network sent very few replies to alters in the course, while a handful of actors in this network have sent 30 or more replies.

Compositional and Variance Measures

Carolan Carolan (2014) notes that there are two types of measures that can emerge from egocentric data: compositional and variance. Compositional measures are those created by counting or taking the average of egocentric network variables. Variance measures, on the other hand, are simply those that are derived by calculating the variance or standard deviation of the egocentric network variables.

While we could calculate invdividually the mean and variance for each egocentric measure created above using the summarise() function demonstrated earlier, R has a package and function to save us the effort.

To quickly calculate summary statistics for our nodes, including compositional and variance measures for our egocentric measures, we can use the skim() function from the {skimr} package to take a quick look at the variables in our node list:

dlt1_network |> 
  as_tibble() |>
  skim()

Data summary

Name	as_tibble(dlt1_network)
Number of rows	445
Number of columns	18
_______________________
Column type frequency:
character	12
numeric	6
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
uid	0	1	1	3	0	445	0
facilitator	0	1	1	1	0	2	0
role1	0	1	4	18	0	13	0
experience2	0	1	3	8	0	6	0
grades	0	1	4	10	0	8	0
location	0	1	2	4	0	62	0
region	0	1	4	13	0	6	0
country	0	1	2	4	0	21	0
group	0	1	1	4	0	7	0
gender	0	1	4	6	0	3	0
expert	0	1	1	4	0	3	0
connect	0	1	1	4	0	3	0

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
experience	0	1	2.13	0.80	1	1	2	3	3	▆▁▇▁▇
strong_component	0	1	49.84	66.43	1	1	1	98	209	▇▁▁▁▁
sub_group	0	1	121.55	106.59	1	15	104	215	326	▇▃▃▃▃
size	0	1	8.75	17.51	1	3	4	10	295	▇▁▁▁▁
in_degree	0	1	5.68	26.74	0	0	1	5	475	▇▁▁▁▁
out_degree	0	1	5.68	9.84	0	1	2	6	106	▇▁▁▁▁

We can see, for example, that egos in our network are connected with on average 8.75 alters with a standards deviation of 17.5. Notice also that our mean for in-degree and out-degree are identical. This will always be the case since for every tie that goes “in” there is always one going “out.”

Finally, the {skimr} package like our {tidgraph} package plays nicely with other {tidyverse} packages for data wrangling and analysis. For example, let’s select only MOOC-Ed participants who are located in the United States and calculate compositional and variance measures for size by educator’s role:

dlt1_network |> 
  as_tibble() |>
  filter(country == "US") |>
  group_by(role1) |>
  select(size) |>
  skim()

## Adding missing grouping variables: `role1`

Data summary

Name	select(…)
Number of rows	412
Number of columns	2
_______________________
Column type frequency:
numeric	1
________________________
Group variables	role1

Variable type: numeric

skim_variable	role1	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
size	classteaching	0	1	8.97	8.22	2	3.00	6.0	12.00	39	▇▂▁▁▁
size	curriculum	0	1	7.70	7.22	2	2.50	5.0	8.50	24	▇▁▁▁▂
size	districtadmin	0	1	5.50	4.74	1	2.00	3.5	7.00	22	▇▂▁▁▁
size	instructionaltech	0	1	7.46	9.38	1	2.00	4.0	7.75	61	▇▁▁▁▁
size	libmedia	0	1	9.91	9.91	1	2.00	6.0	15.00	33	▇▂▁▁▁
size	Operations	0	1	3.00	NA	3	3.00	3.0	3.00	3	▁▁▇▁▁
size	other	0	1	8.25	8.10	2	2.00	6.0	12.25	19	▇▁▃▁▃
size	otheredprof	0	1	18.95	53.48	2	3.00	4.0	9.00	295	▇▁▁▁▁
size	profdev	0	1	5.71	4.44	2	2.00	4.0	7.50	21	▇▂▂▁▁
size	schooladmin	0	1	9.40	9.10	2	3.25	5.5	13.50	42	▇▂▁▁▁
size	specialed	0	1	12.00	6.63	2	11.00	11.0	17.00	19	▃▁▇▁▇
size	techinfrastructure	0	1	5.44	3.22	2	3.00	4.5	7.00	14	▇▂▃▁▁

As illustrated by the output above, we can see that “otheredprof” (i.e., other educational professionals including our facilitators) were connected with the most individuals on average (18.9), while educators in “Operations” connected with the fewest individuals on average (3).

Closeness & Betweenness

Two other centrality measures for identifying key actors in a network are closeness and betweenness. Whereas degree centrality is a local measure and can be calculated without needing information about the overall pattern of relations among ego and alters, closeness and betweenness address this shortcoming by taking into account indirect ties among all the alters in an ego’s network.

Closeness centrality captures the average distance an actor is from all other actors in the network and is a function of an actor’s geodesic distance to others, which equals the length of shortest path connecting a pair of actors. It’s an intuitively appealing measure in that being “close” to others may provide an advantage by giving you early access to new information or by indicating how quickly an actor can exchange something with others (Knoke & Yang, 2008).

Betweenness centrality measures the degree to which other actors lie on the shortest geodesic path between pairs of actors in the network. Betweenness may be indicative of how actors control or mediate the relations between pairs of actors that are not directly connected. Therefore, this measure is an important indicator of control over information exchange or resource flows within a network (Knoke & Yang, 2008). The more any given actor is located on the path between numerous dyads, the higher that actor’s potential to control network interactions. This concept and its associated measure is very appealing, as it captures the degree to which an actor occupies a strategically important position.

👉 Your Turn ⤵

Use the code chunk below to add a closeness and betweenness variable to the nodes in our network and answer the questions that follow. Hint: you may want to look at the functions to calculate node and edge centrality on the {tidygraph} package website.

set.seed(123)
dlt1_stats <- dlt1_undirected %>% 
  activate(nodes)%>%
  mutate(community = as.factor(group_louvain())) %>%
  mutate(degree_c = centrality_degree()) %>%
  mutate(betweenness_c = centrality_betweenness(directed = FALSE ,normalized = TRUE)) %>%
  mutate(closeness_c = centrality_closeness(normalized = TRUE))

## Warning in closeness(graph = graph, vids = V(graph), mode = mode, weights =
## weights, : At centrality.c:2874 :closeness centrality is not well-defined for
## disconnected graphs

dlt1_stats

## # A tbl_graph: 445 nodes and 2529 edges
## #
## # An undirected multigraph with 4 components
## #
## # Node Data: 445 x 19 (active)
##   uid   facilitator role1 experience experience2 grades location region country
##   <chr> <chr>       <chr>      <dbl> <chr>       <chr>  <chr>    <chr>  <chr>  
## 1 1     0           libm~          1 6 to 10     secon~ VA       South  US     
## 2 2     0           clas~          1 6 to 10     secon~ FL       South  US     
## 3 3     0           dist~          2 11 to 20    gener~ PA       North~ US     
## 4 4     0           clas~          2 11 to 20    middle NC       South  US     
## 5 5     0           othe~          3 20+         gener~ AL       South  US     
## 6 6     0           clas~          1 4 to 5      gener~ AL       South  US     
## # ... with 439 more rows, and 10 more variables: group <chr>, gender <chr>,
## #   expert <chr>, connect <chr>, strong_component <int>, sub_group <int>,
## #   community <fct>, degree_c <dbl>, betweenness_c <dbl>, closeness_c <dbl>
## #
## # Edge Data: 2,529 x 9
##    from    to timestamp discussion_title discussion_cate~ parent_category
##   <int> <int> <chr>     <chr>            <chr>            <chr>          
## 1   360   444 4/4/13 1~ Most important ~ Group N          Units 1-3 Disc~
## 2   356   444 4/4/13 1~ Most important ~ Group D-L        Units 1-3 Disc~
## 3   356   444 4/4/13 1~ DLT Resources—C~ Group D-L        Units 1-3 Disc~
## # ... with 2,526 more rows, and 3 more variables: discussion_identifier <chr>,
## #   comment_id <chr>, discussion_id <chr>

dltl_stats_ego <- as.data.frame(dlt1_stats)  
dltl_stats_ego

##     uid facilitator              role1 experience experience2     grades
## 1     1           0           libmedia          1     6 to 10  secondary
## 2     2           0      classteaching          1     6 to 10  secondary
## 3     3           0      districtadmin          2    11 to 20 generalist
## 4     4           0      classteaching          2    11 to 20     middle
## 5     5           0        otheredprof          3         20+ generalist
## 6     6           0      classteaching          1      4 to 5 generalist
## 7     7           0  instructionaltech          2    11 to 20 generalist
## 8     8           0          specialed          1     6 to 10  secondary
## 9     9           0      classteaching          1     6 to 10     middle
## 10   10           0        schooladmin          2    11 to 20     middle
## 11   11           0              other          3         20+ generalist
## 12   12           0      classteaching          3         20+  secondary
## 13   13           0      classteaching          2    11 to 20     middle
## 14   14           0              other          1      0 to 3     middle
## 15   15           0        schooladmin          3         20+  secondary
## 16   16           0        otheredprof          1      0 to 3     middle
## 17   17           0           libmedia          1      0 to 3    primary
## 18   18           0          specialed          1      4 to 5     middle
## 19   19           0        otheredprof          3         20+    college
## 20   20           0      districtadmin          1      0 to 3 generalist
## 21   21           0        otheredprof          1      0 to 3 generalist
## 22   22           0        schooladmin          1     6 to 10  secondary
## 23   23           0           libmedia          3         20+     middle
## 24   24           0  instructionaltech          2    11 to 20 generalist
## 25   25           0  instructionaltech          2    11 to 20 generalist
## 26   26           0        otheredprof          3         20+    primary
## 27   27           0        otheredprof          1      0 to 3 generalist
## 28   28           0            profdev          3         20+     middle
## 29   29           0        schooladmin          2    11 to 20 generalist
## 30   30           0        schooladmin          3         20+  secondary
## 31   31           0      classteaching          1      4 to 5 generalist
## 32   32           0            profdev          1     6 to 10    primary
## 33   33           0  instructionaltech          3         20+ generalist
## 34   34           0         curriculum          3         20+    primary
## 35   35           0         curriculum          2    11 to 20    college
## 36   36           0      classteaching          2    11 to 20  secondary
## 37   37           0         curriculum          2    11 to 20  secondary
## 38   38           0      classteaching          1     6 to 10  secondary
## 39   39           0      classteaching          2    11 to 20  secondary
## 40   40           0      districtadmin          3         20+ generalist
## 41   41           0  instructionaltech          1     6 to 10 generalist
## 42   42           0         curriculum          3         20+ generalist
## 43   43           0            profdev          3         20+  secondary
## 44   44           0  instructionaltech          2    11 to 20     middle
## 45   45           0      districtadmin          2    11 to 20 generalist
## 46   46           0      classteaching          2    11 to 20    primary
## 47   47           0            profdev          3         20+ generalist
## 48   48           0        schooladmin          3         20+     middle
## 49   49           0           libmedia          3         20+ generalist
## 50   50           0        schooladmin          3         20+    primary
## 51   51           0        schooladmin          3         20+    primary
## 52   52           0        schooladmin          3         20+    primary
## 53   53           0  instructionaltech          3         20+    primary
## 54   54           0        otheredprof          2    11 to 20    primary
## 55   55           0  instructionaltech          1      4 to 5 generalist
## 56   56           0      districtadmin          3         20+ generalist
## 57   57           0         curriculum          3         20+ generalist
## 58   58           0  instructionaltech          2    11 to 20 generalist
## 59   59           0           libmedia          1      0 to 3    primary
## 60   60           0      classteaching          2    11 to 20     middle
## 61   61           0        schooladmin          3         20+    primary
## 62   62           0      classteaching          1      0 to 3 generalist
## 63   63           0          specialed          1     6 to 10  secondary
## 64   64           0  instructionaltech          2    11 to 20 generalist
## 65   65           0           libmedia          2    11 to 20  secondary
## 66   66           0  instructionaltech          2    11 to 20  secondary
## 67   67           0           libmedia          3         20+    primary
## 68   68           0           libmedia          3         20+ generalist
## 69   69           0            profdev          3         20+ generalist
## 70   70           0        schooladmin          1     6 to 10     middle
## 71   71           0      districtadmin          3         20+ generalist
## 72   72           0  instructionaltech          2    11 to 20    primary
## 73   73           0      districtadmin          3         20+    primary
## 74   74           0        schooladmin          3         20+ generalist
## 75   75           0      classteaching          3         20+  secondary
## 76   76           0         curriculum          3         20+ generalist
## 77   77           0            profdev          2    11 to 20 generalist
## 78   78           0        schooladmin          3         20+  secondary
## 79   79           0            profdev          1      4 to 5 generalist
## 80   80           0      classteaching          2    11 to 20     middle
## 81   81           0      classteaching          3         20+    primary
## 82   82           0      classteaching          1     6 to 10     middle
## 83   83           0      classteaching          3         20+    primary
## 84   84           0         curriculum          1     6 to 10 generalist
## 85   85           0      classteaching          1     6 to 10 generalist
## 86   86           0        schooladmin          3         20+     middle
## 87   87           0           libmedia          2    11 to 20 generalist
## 88   88           0      classteaching          1      0 to 3  secondary
## 89   89           0            profdev          3         20+ generalist
## 90   90           0        otheredprof          3         20+ generalist
## 91   91           0  instructionaltech          2    11 to 20  secondary
## 92   92           0         curriculum          3         20+    primary
## 93   93           0      classteaching          1     6 to 10  secondary
## 94   94           0      classteaching          1     6 to 10     middle
## 95   95           0  instructionaltech          1     6 to 10  secondary
## 96   96           0  instructionaltech          2    11 to 20  secondary
## 97   97           0           libmedia          1     6 to 10    primary
## 98   98           0      classteaching          2    11 to 20    primary
## 99   99           0  instructionaltech          3         20+ generalist
## 100 100           0  instructionaltech          2    11 to 20 generalist
## 101 101           0  instructionaltech          1      4 to 5    primary
## 102 102           0      classteaching          3         20+ generalist
## 103 103           0      classteaching          2    11 to 20     middle
## 104 104           0      classteaching          3         20+     middle
## 105 105           0 techinfrastructure          2    11 to 20    primary
## 106 106           0      districtadmin          3         20+ generalist
## 107 107           0  instructionaltech          2    11 to 20 generalist
## 108 108           0      classteaching          1      4 to 5  secondary
## 109 109           0  instructionaltech          3         20+    primary
## 110 110           0  instructionaltech          1      4 to 5     middle
## 111 111           0      districtadmin          2    11 to 20 generalist
## 112 112           0      districtadmin          2    11 to 20 generalist
## 113 113           0        schooladmin          2    11 to 20  secondary
## 114 114           0            profdev          2    11 to 20  secondary
## 115 115           0        schooladmin          2    11 to 20     middle
## 116 116           0      classteaching          3         20+    college
## 117 117           0         curriculum          1     6 to 10 generalist
## 118 118           0      classteaching          1     6 to 10    college
## 119 119           0           libmedia          2    11 to 20     middle
## 120 120           0  instructionaltech          3         20+ generalist
## 121 121           0  instructionaltech          2    11 to 20 generalist
## 122 122           0        otheredprof          3         20+ generalist
## 123 123           0         curriculum          2    11 to 20 generalist
## 124 124           0  instructionaltech          2    11 to 20 generalist
## 125 125           0  instructionaltech          3         20+ generalist
## 126 126           0 techinfrastructure          2    11 to 20 generalist
## 127 127           0 techinfrastructure          2    11 to 20 generalist
## 128 128           0         curriculum          1     6 to 10    college
## 129 129           0  instructionaltech          2    11 to 20    primary
## 130 130           0        otheredprof          3         20+ generalist
## 131 131           0            profdev          2    11 to 20 generalist
## 132 132           0  instructionaltech          1     6 to 10 generalist
## 133 133           0 techinfrastructure          3         20+  secondary
## 134 134           0      classteaching          1     6 to 10  secondary
## 135 135           0        otheredprof          1      0 to 3 generalist
## 136 136           0      districtadmin          3         20+ generalist
## 137 137           0      districtadmin          3         20+ generalist
## 138 138           0            profdev          2    11 to 20 generalist
## 139 139           0  instructionaltech          2    11 to 20 generalist
## 140 140           0        otheredprof          3         20+ generalist
## 141 141           0  instructionaltech          3         20+  secondary
## 142 142           0            profdev          2    11 to 20 generalist
## 143 143           0  instructionaltech          2    11 to 20 generalist
## 144 144           0      districtadmin          2    11 to 20 generalist
## 145 145           0      classteaching          1     6 to 10     middle
## 146 146           0        otheredprof          1     6 to 10  secondary
## 147 147           0      classteaching          2    11 to 20  secondary
## 148 148           0      districtadmin          2         20+ generalist
## 149 149           0  instructionaltech          1     6 to 10 generalist
## 150 150           0            profdev          1     6 to 10 generalist
## 151 151           0 techinfrastructure          2    11 to 20 generalist
## 152 152           0        otheredprof          2    11 to 20 generalist
## 153 153           0        schooladmin          1     6 to 10  secondary
## 154 154           0  instructionaltech          3         20+  secondary
## 155 155           0  instructionaltech          3         20+ generalist
## 156 156           0        otheredprof          3         20+    primary
## 157 157           0      classteaching          1      4 to 5     middle
## 158 158           0           libmedia          3         20+     middle
## 159 159           0  instructionaltech          2    11 to 20  secondary
## 160 160           0         curriculum          3         20+ generalist
## 161 161           0  instructionaltech          3         20+    primary
## 162 162           0        schooladmin          3         20+ generalist
## 163 163           0              other          1      0 to 3 generalist
## 164 164           0        schooladmin          2    11 to 20  secondary
## 165 165           0            profdev          3         20+ generalist
## 166 166           0 techinfrastructure          1      0 to 3 generalist
## 167 167           0  instructionaltech          3         20+ generalist
## 168 168           0  instructionaltech          3         20+ generalist
## 169 169           0  instructionaltech          3         20+ generalist
## 170 170           0 techinfrastructure          1     6 to 10 generalist
## 171 171           0      classteaching          2    11 to 20 generalist
## 172 172           0        otheredprof          2    11 to 20    college
## 173 173           0        schooladmin          2    11 to 20  secondary
## 174 174           0        schooladmin          2    11 to 20  secondary
## 175 175           0         curriculum          3         20+ generalist
## 176 176           0 techinfrastructure          2    11 to 20 generalist
## 177 177           0      districtadmin          2    11 to 20 generalist
## 178 178           0      districtadmin          2    11 to 20  secondary
## 179 179           0  instructionaltech          2    11 to 20 generalist
## 180 180           0        otheredprof          3         20+    college
## 181 181           0 techinfrastructure          1     6 to 10 generalist
## 182 182           0        otheredprof          1     6 to 10 generalist
## 183 183           0      districtadmin          2    11 to 20 generalist
## 184 184           0      classteaching          1     6 to 10  secondary
## 185 185           0      classteaching          2    11 to 20  secondary
## 186 186           0 techinfrastructure          1      4 to 5  secondary
## 187 187           0 techinfrastructure          2    11 to 20 generalist
## 188 188           0      classteaching          3         20+  secondary
## 189 189           0        schooladmin          3         20+     middle
## 190 190           0           libmedia          1      4 to 5     middle
## 191 191           0      classteaching          3         20+     middle
## 192 192           0      classteaching          3         20+     middle
## 193 193           0  instructionaltech          2    11 to 20 generalist
## 194 194           0  instructionaltech          3         20+ generalist
## 195 195           0      districtadmin          3         20+ generalist
## 196 196           0        otheredprof          1      0 to 3  secondary
## 197 197           0      districtadmin          3         20+ generalist
## 198 198           0           libmedia          2    11 to 20    primary
## 199 199           0        schooladmin          3         20+     middle
## 200 200           0        schooladmin          2    11 to 20     middle
## 201 201           0      classteaching          1     6 to 10     middle
## 202 202           0  instructionaltech          2    11 to 20    primary
## 203 203           0            profdev          3         20+ generalist
## 204 204           0  instructionaltech          2    11 to 20 generalist
## 205 205           0        otheredprof          1      0 to 3 generalist
## 206 206           0         curriculum          2    11 to 20 generalist
## 207 207           0  instructionaltech          1      0 to 3  secondary
## 208 208           0        otheredprof          2    11 to 20 generalist
## 209 209           0        schooladmin          3         20+  secondary
## 210 210           0         Operations          2     6 to 10 generalist
## 211 211           0        otheredprof          2    11 to 20 generalist
## 212 212           0        schooladmin          3         20+  secondary
## 213 213           0  instructionaltech          2    11 to 20 generalist
## 214 214           0            profdev          1     6 to 10     middle
## 215 215           0      classteaching          1     6 to 10  secondary
## 216 216           0        schooladmin          3         20+     middle
## 217 217           0        otheredprof          3         20+ generalist
## 218 218           0        otheredprof          1     6 to 10 generalist
## 219 219           0         curriculum          3         20+  secondary
## 220 220           0            profdev          3         20+ generalist
## 221 221           0          specialed          1     6 to 10  secondary
## 222 222           0        otheredprof          3         20+  secondary
## 223 223           0         curriculum          2    11 to 20 generalist
## 224 224           0  instructionaltech          3         20+ generalist
## 225 225           0        otheredprof          3         20+ generalist
## 226 226           0 techinfrastructure          3         20+ generalist
## 227 227           0      classteaching          2    11 to 20     middle
## 228 228           0        otheredprof          1      0 to 3 generalist
## 229 229           0  instructionaltech          2    11 to 20    primary
## 230 230           0      districtadmin          3         20+ generalist
## 231 231           0      districtadmin          3         20+ generalist
## 232 232           0           libmedia          3         20+  secondary
## 233 233           0        otheredprof          2    11 to 20 generalist
## 234 234           0  instructionaltech          3         20+ generalist
## 235 235           0            profdev          2    11 to 20 generalist
## 236 236           0      classteaching          1     6 to 10    primary
## 237 237           0  instructionaltech          2    11 to 20     middle
## 238 238           0  instructionaltech          1      4 to 5 generalist
## 239 239           0  instructionaltech          1     6 to 10 generalist
## 240 240           0        schooladmin          1     6 to 10  secondary
## 241 241           0      classteaching          1      4 to 5 generalist
## 242 242           0        otheredprof          1      4 to 5    college
## 243 243           0  instructionaltech          1     6 to 10  secondary
## 244 244           0            profdev          3         20+ generalist
## 245 245           0            profdev          1      0 to 3 generalist
## 246 246           0  instructionaltech          3         20+    primary
## 247 247           0      classteaching          2    11 to 20  secondary
## 248 248           0 techinfrastructure          1      0 to 3 generalist
## 249 249           0  instructionaltech          2    11 to 20 generalist
## 250 250           0  instructionaltech          2    11 to 20 generalist
## 251 251           0  instructionaltech          2    11 to 20     middle
## 252 252           0        schooladmin          2    11 to 20  secondary
## 253 253           0  instructionaltech          3         20+ generalist
## 254 254           0      classteaching          3         20+    primary
## 255 255           0  instructionaltech          2    11 to 20 generalist
## 256 256           0  instructionaltech          3         20+ generalist
## 257 257           0        otheredprof          3         20+ generalist
## 258 258           0        schooladmin          3         20+     middle
## 259 259           0      classteaching          3         20+    primary
## 260 260           0      classteaching          3         20+    primary
## 261 261           0  instructionaltech          2    11 to 20  secondary
## 262 262           0      classteaching          3         20+    primary
## 263 263           0              other          2    11 to 20  secondary
## 264 264           0            profdev          2    11 to 20    primary
## 265 265           0      districtadmin          3         20+ generalist
## 266 266           0      classteaching          1     6 to 10  secondary
## 267 267           0      districtadmin          3         20+ generalist
## 268 268           0  instructionaltech          1     6 to 10 generalist
## 269 269           0  instructionaltech          3         20+     middle
## 270 270           0      classteaching          1     6 to 10  secondary
## 271 271           0        schooladmin          2    11 to 20    primary
## 272 272           0      classteaching          3         20+  secondary
## 273 273           0  instructionaltech          3         20+ generalist
## 274 274           0  instructionaltech          3         20+    primary
## 275 275           0 techinfrastructure          1      0 to 3 generalist
## 276 276           0 techinfrastructure          3         20+ generalist
## 277 277           0  instructionaltech          3         20+ generalist
## 278 278           0        otheredprof          2    11 to 20  secondary
## 279 279           0          specialed          3         20+     middle
## 280 280           0      districtadmin          3         20+ generalist
## 281 281           0        schooladmin          3         20+  secondary
## 282 282           0      classteaching          1      4 to 5    primary
## 283 283           0           libmedia          2    11 to 20     middle
## 284 284           0  instructionaltech          1     6 to 10 generalist
## 285 285           0      classteaching          1      0 to 3  secondary
## 286 286           0      districtadmin          3         20+ generalist
## 287 287           0      districtadmin          1     6 to 10 generalist
## 288 288           0      classteaching          1      4 to 5  secondary
## 289 289           0        schooladmin          2    11 to 20  secondary
## 290 290           0  instructionaltech          3         20+ generalist
## 291 291           0  instructionaltech          1     6 to 10  secondary
## 292 292           0      classteaching          1      0 to 3    primary
## 293 293           0  instructionaltech          3         20+  secondary
## 294 294           0              other          3         20+  secondary
## 295 295           0      classteaching          1     6 to 10  secondary
## 296 296           0  instructionaltech          3         20+    college
## 297 297           0        schooladmin          3         20+    primary
## 298 298           0  instructionaltech          3         20+    primary
## 299 299           0      districtadmin          3         20+ generalist
## 300 300           0        schooladmin          2    11 to 20  secondary
## 301 301           0           libmedia          2    11 to 20 generalist
## 302 302           0      classteaching          3         20+  secondary
## 303 303           0      classteaching          3         20+  secondary
## 304 304           0  instructionaltech          1     6 to 10    college
## 305 305           0  instructionaltech          3         20+ generalist
## 306 306           0  instructionaltech          1     6 to 10 generalist
## 307 307           0         curriculum          2    11 to 20 generalist
## 308 308           0      classteaching          1      4 to 5  secondary
## 309 309           0      classteaching          1      4 to 5  secondary
## 310 310           0  instructionaltech          2    11 to 20    college
## 311 311           0      districtadmin          2    11 to 20 generalist
## 312 312           0      districtadmin          2    11 to 20 generalist
## 313 313           0      classteaching          2    11 to 20  secondary
## 314 314           0  instructionaltech          2    11 to 20    primary
## 315 315           0           libmedia          3         20+  secondary
## 316 316           0      classteaching          3         20+  secondary
## 317 317           0        otheredprof          3         20+ generalist
## 318 318           0      classteaching          2    11 to 20  secondary
## 319 319           0      districtadmin          2    11 to 20 generalist
## 320 320           0      districtadmin          3         20+ generalist
## 321 321           0        schooladmin          2    11 to 20    primary
## 322 322           0        schooladmin          1     6 to 10 generalist
## 323 323           0      classteaching          1     6 to 10  secondary
## 324 324           0        schooladmin          3         20+  secondary
## 325 325           0      districtadmin          2    11 to 20 generalist
## 326 326           0        otheredprof          3         20+  secondary
## 327 327           0      classteaching          1     6 to 10    primary
## 328 328           0  instructionaltech          3         20+ generalist
## 329 329           0 techinfrastructure          2    11 to 20 generalist
## 330 330           0        otheredprof          3         20+    college
## 331 331           0         curriculum          2    11 to 20 generalist
## 332 332           0  instructionaltech          1     6 to 10 generalist
## 333 333           0            profdev          1     6 to 10    primary
## 334 334           0        schooladmin          3         20+  secondary
## 335 335           0  instructionaltech          3         20+  secondary
## 336 336           0  instructionaltech          2    11 to 20    primary
## 337 337           0            profdev          3         20+  secondary
## 338 338           0        otheredprof          3         20+ generalist
## 339 339           0      classteaching          3         20+  secondary
## 340 340           0      districtadmin          3         20+ generalist
## 341 341           0      classteaching          3         20+  secondary
## 342 342           0        schooladmin          3         20+ generalist
## 343 343           0      classteaching          1      4 to 5    primary
## 344 344           0  instructionaltech          3         20+ generalist
## 345 345           0  instructionaltech          3         20+    primary
## 346 346           0  instructionaltech          1     6 to 10  secondary
## 347 347           0  instructionaltech          2    11 to 20     middle
## 348 348           0        schooladmin          3         20+  secondary
## 349 349           0  instructionaltech          1      0 to 3    primary
## 350 350           0        schooladmin          2    11 to 20     middle
## 351 351           0  instructionaltech          2    11 to 20 generalist
## 352 352           0      districtadmin          1     6 to 10 generalist
## 353 353           0           libmedia          1     6 to 10     middle
## 354 354           0      classteaching          2    11 to 20  secondary
## 355 355           0        schooladmin          2    11 to 20  secondary
## 356 356           0  instructionaltech          2    11 to 20 generalist
## 357 357           0      classteaching          3         20+  secondary
## 358 358           0        schooladmin          3         20+    primary
## 359 359           0  instructionaltech          3         20+ generalist
## 360 360           0  instructionaltech          1      0 to 3 generalist
## 361 361           0            profdev          3         20+ generalist
## 362 362           0  instructionaltech          2    11 to 20  secondary
## 363 363           0         curriculum          3         20+  secondary
## 364 364           0        otheredprof          3         20+ generalist
## 365 365           0           libmedia          2    11 to 20    college
## 366 366           0            profdev          2    11 to 20 generalist
## 367 367           0            profdev          1      0 to 3    college
## 368 368           0  instructionaltech          2    11 to 20  secondary
## 369 369           0        otheredprof          3         20+    college
## 370 370           0        otheredprof          3         20+ generalist
## 371 371           0  instructionaltech          2    11 to 20 generalist
## 372 372           0  instructionaltech          3         20+ generalist
## 373 373           0  instructionaltech          1     6 to 10    primary
## 374 374           0      classteaching          2    11 to 20  secondary
## 375 375           0  instructionaltech          3         20+       NULL
## 376 376           0      classteaching          2    11 to 20  secondary
## 377 377           0  instructionaltech          2    11 to 20 generalist
## 378 378           0      districtadmin          1     6 to 10 generalist
## 379 379           0        otheredprof          2    11 to 20  secondary
## 380 380           0      districtadmin          3         20+  secondary
## 381 381           0        schooladmin          2    11 to 20 generalist
## 382 382           0         curriculum          2    11 to 20  secondary
## 383 383           0      districtadmin          3         20+ generalist
## 384 384           0  instructionaltech          2    11 to 20 generalist
## 385 385           0      districtadmin          3         20+ generalist
## 386 386           0  instructionaltech          2    11 to 20 generalist
## 387 387           0  instructionaltech          3         20+ generalist
## 388 388           0  instructionaltech          2    11 to 20 generalist
## 389 389           0         curriculum          3         20+ generalist
## 390 390           0           libmedia          1     6 to 10    primary
## 391 391           0         curriculum          3         20+    primary
## 392 392           0 techinfrastructure          3         20+ generalist
## 393 393           0  instructionaltech          2    11 to 20     middle
## 394 394           0  instructionaltech          1     6 to 10 generalist
## 395 395           0      districtadmin          2    11 to 20 generalist
## 396 396           0         curriculum          2    11 to 20 generalist
## 397 397           0        otheredprof          1      4 to 5 generalist
## 398 398           0         curriculum          1     6 to 10     middle
## 399 399           0         curriculum          2    11 to 20 generalist
## 400 400           0      classteaching          3         20+    college
## 401 401           0 techinfrastructure          2    11 to 20  secondary
## 402 402           0        schooladmin          2    11 to 20 generalist
## 403 403           0  instructionaltech          3         20+ generalist
## 404 404           0        otheredprof          1     6 to 10    primary
## 405 405           0      districtadmin          1      4 to 5 generalist
## 406 406           0  instructionaltech          2    11 to 20 generalist
## 407 407           0  instructionaltech          3         20+  secondary
## 408 408           0 techinfrastructure          3         20+ generalist
## 409 409           0  instructionaltech          1     6 to 10 generalist
## 410 410           0         curriculum          1      4 to 5 generalist
## 411 411           0  instructionaltech          3         20+ generalist
## 412 412           0          specialed          1     6 to 10 generalist
## 413 413           0  instructionaltech          3         20+ generalist
## 414 414           0  instructionaltech          2    11 to 20 generalist
## 415 415           0            profdev          3         20+ generalist
## 416 416           0           libmedia          2    11 to 20  secondary
## 417 417           0      classteaching          2    11 to 20 generalist
## 418 418           0  instructionaltech          2    11 to 20 generalist
## 419 419           0      classteaching          3         20+  secondary
## 420 420           0      districtadmin          2    11 to 20 generalist
## 421 421           0  instructionaltech          2    11 to 20 generalist
## 422 422           0            profdev          2    11 to 20 generalist
## 423 423           0         curriculum          3         20+ generalist
## 424 424           0           libmedia          2    11 to 20    primary
## 425 425           0      districtadmin          3         20+ generalist
## 426 426           0      districtadmin          1     6 to 10 generalist
## 427 427           0           libmedia          3         20+ generalist
## 428 428           0        otheredprof          3         20+    primary
## 429 429           0  instructionaltech          1      0 to 3 generalist
## 430 430           0  instructionaltech          2    11 to 20 generalist
## 431 431           0      classteaching          2    11 to 20     middle
## 432 432           0  instructionaltech          1     6 to 10  secondary
## 433 433           0         curriculum          3         20+ curriculum
## 434 434           0  instructionaltech          1         20+ generalist
## 435 435           0  instructionaltech          1         20+       high
## 436 436           0            profdev          2    11 to 20 generalist
## 437 437           0  instructionaltech          1         20+ generalist
## 438 438           0  instructionaltech          1    11 to 20 generalist
## 439 439           0               NULL          1        NULL       NULL
## 440 440           0            profdev          2    11 to 20  secondary
## 441 441           0           libmedia          3         20+     middle
## 442 442           0      districtadmin          3         20+ generalist
## 443 443           0  instructionaltech          2    11 to 20 generalist
## 444 444           1        otheredprof          3         20+ generalist
## 445 445           1        otheredprof          3         20+ generalist
##     location        region country group gender expert connect strong_component
## 1         VA         South      US    UZ female      0       1                1
## 2         FL         South      US    DL female      0       0                1
## 3         PA     Northeast      US    OT female      0       1                1
## 4         NC         South      US     N female      0       0                1
## 5         AL         South      US    AC female      0       0                1
## 6         AL         South      US    AC female      0       1                1
## 7         SD       Midwest      US    OT female      0       0                1
## 8         BE International      BE    AC female      0       0                1
## 9         NC         South      US     N female      0       0                1
## 10        NC         South      US     N   male      0       0                1
## 11        KG International      KG    DL female      0       1                1
## 12        AL         South      US    AC female      0       0                1
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## 14        NC         South      US     N female      0       0                1
## 15        WV         South      US    UZ female      0       0                1
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## 17        NC         South      US     N female      0       0                1
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## 19        AU International      AU    AC   male      0       1                1
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## 21        NC         South      US     N female      0       0              203
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## 24        IN       Midwest      US    DL female      0       1                1
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## 28        ME     Northeast      US     M female      1       0              193
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## 30        NC         South      US     N female      0       1                1
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## 33        CA          West      US    AC   male      0       0                1
## 34        AL         South      US    AC female      0       0                1
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## 36        UT          West      US    UZ female      0       1                1
## 37        CA          West      US    AC   male      0       0                1
## 38        ME     Northeast      US     M   male      0       1                1
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## 41        PA     Northeast      US    OT   male      1       1                1
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## 43        MA     Northeast      US     M female      1       1                1
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## 47        MA     Northeast      US     M female      1       0              190
## 48        KS       Midwest      US    DL   male      1       0                1
## 49        ME     Northeast      US     M female      0       0                1
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## 56        PA     Northeast      US    OT female      1       1                1
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## 66        NC         South      US     N female      0       0                1
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## 106       NC         South      US     N   male      0       0                1
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## 115       PA     Northeast      US    OT female      0       0                1
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## 130       RI     Northeast      US    OT female      0       1              165
## 131       NC         South      US     N female      0       0                1
## 132       MI       Midwest      US     M female      0       1                1
## 133       NC         South      US     N   male      0       1                1
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## 138       VT     Northeast      US    UZ female      0       0                1
## 139       VA         South      US    UZ female      0       0              162
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## 142       AL         South      US    AC female      0       0                1
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## 144       VA         South      US    UZ   male      0       0                1
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## 149       OH       Midwest      US    OT   male      0       1              156
## 150       TX         South      US    OT   male      0       0              173
## 151       RI     Northeast      US    OT   male      0       1              172
## 152       VT     Northeast      US    UZ   male      0       0                1
## 153       WY          West      US    UZ   male      0       0              206
## 154       MN       Midwest      US     M female      0       0                1
## 155       VA         South      US    UZ female      0       0                1
## 156       ME     Northeast      US     M female      0       0                1
## 157       AL         South      US    AC   male      0       0                1
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## 159       VT     Northeast      US    UZ   male      0       0                1
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## 165       RI     Northeast      US    OT female      0       0                1
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## 183       AL         South      US    AC   male      0       0                1
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## 185       VA         South      US    UZ female      0       0                1
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## 188       PA     Northeast      US    OT   male      0       0                1
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## 191       BR International      BR    AC female      0       0                1
## 192       ME     Northeast      US     M female      0       0                1
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## 194       NC         South      US     N female      0       1                1
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## 199       NJ     Northeast      US     N female      0       0                1
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## 201       ME     Northeast      US     M female      0       0                1
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## 203       IA       Midwest      US    DL female      0       0                1
## 204       WY          West      US    UZ female      0       0              143
## 205       AL         South      US    AC female      0       0                1
## 206       MT          West      US     M   male      0       1                1
## 207       NC         South      US     N   male      0       1                1
## 208       NJ     Northeast      US     N   male      0       0                1
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## 210       RI     Northeast      US    OT   male      1       0              142
## 211       DC         South      US    DL   male      1       1                1
## 212       NH     Northeast      US     N   male      0       0                1
## 213       NY     Northeast      US     N female      0       0              141
## 214       ME     Northeast      US     M female      0       0              140
## 215       ME     Northeast      US     M female      0       1              139
## 216       NC         South      US     N female      0       0                1
## 217       CA          West      US    AC   male      0       1                1
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## 219       AL         South      US    AC female      0       0                1
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## 221       NY     Northeast      US     N female      0       0                1
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## 224       OK       Midwest      US    OT female      0       0              137
## 225       AL         South      US    AC female      0       1              136
## 226       ME     Northeast      US     M   male      0       0                1
## 227       ME     Northeast      US     M female      0       0                1
## 228       AL         South      US    AC   male      1       0              135
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## 232       WV         South      US    UZ female      0       0              130
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## 256       ME     Northeast      US     M female      0       0                1
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## 270       PA     Northeast      US    OT female      0       0                1
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## 295      COL International      CO    AC   male      0       0                1
## 296      CAN International      CA    AC   male      0       0              101
## 297       NC         South      US     N   male      0       1              100
## 298       NC         South      US     N female      0       0               99
## 299       PA     Northeast      US    OT   male      0       0               98
## 300       WI       Midwest      US    UZ   male      0       0                1
## 301       WI       Midwest      US    UZ female      0       0                1
## 302       NC         South      US     N female      0       0                1
## 303       NC         South      US     N female      0       0                1
## 304       NJ     Northeast      US     N female      0       0               97
## 305       WI       Midwest      US    UZ female      0       0                1
## 306       NC         South      US     N   male      0       0              197
## 307       WI       Midwest      US    UZ female      0       0                1
## 308       NY     Northeast      US     N female      0       0              207
## 309       GB International      GB    DL female      0       0               96
## 310       GB International      GB    DL   male      0       0                1
## 311       CT     Northeast      US    AC   male      0       0               95
## 312       CT     Northeast      US    AC female      0       0               94
## 313       CA          West      US    AC female      0       0               93
## 314       AZ          West      US    AC female      0       0               92
## 315       CA          West      US    AC female      0       0               91
## 316      CAN International      CA    AC female      0       0               90
## 317       CA          West      US    AC   male      0       0                1
## 318       MO       Midwest      US     M female      0       0                1
## 319       WI       Midwest      US    UZ   male      0       0                1
## 320       VA         South      US    UZ female      0       0               89
## 321       NC         South      US     N   male      0       0                1
## 322       VA         South      US    UZ   male      0       0                1
## 323       CT     Northeast      US    AC female      0       0                1
## 324       VA         South      US    UZ female      0       0              205
## 325       NC         South      US     N   male      1       0                1
## 326       NC         South      US     N female      0       1               88
## 327       PA     Northeast      US    OT   male      0       0               87
## 328       MO       Midwest      US     M   male      0       0               86
## 329       MO       Midwest      US     M   male      0       0                1
## 330       OR          West      US    OT female      0       0              171
## 331       CA          West      US    AC   male      0       0                1
## 332       TN         South      US    OT female      0       1               85
## 333       AL         South      US    AC female      0       0               84
## 334       NC         South      US     N female      0       0              196
## 335       VA         South      US    UZ female      0       0                1
## 336       NC         South      US     N female      0       0                1
## 337       NY     Northeast      US     N   male      0       1                1
## 338       AL         South      US    AC female      0       0                1
## 339       VA         South      US    UZ female      0       0                1
## 340       SC         South      US    OT female      0       0                1
## 341       NY     Northeast      US     N female      0       1                1
## 342      CAN International      CA    AC   male      0       0                1
## 343       TX         South      US    OT female      0       0                1
## 344       OH       Midwest      US    OT female      0       0              133
## 345       VA         South      US    UZ female      0       0                1
## 346       AL         South      US    AC   male      0       0                1
## 347       NC         South      US     N   male      0       1                1
## 348       NC         South      US     N female      0       0               83
## 349       NY     Northeast      US     N female      0       0               82
## 350       ME     Northeast      US     M   male      0       0                1
## 351       WA          West      US    UZ   male      0       0                1
## 352       OH       Midwest      US    OT female      0       0               81
## 353       NH     Northeast      US     N female      0       0               80
## 354       NC         South      US     N female      0       0               79
## 355       KS       Midwest      US    DL female      0       0                1
## 356       IL       Midwest      US    DL female      0       0                1
## 357       NC         South      US     N female      0       0               78
## 358       NY     Northeast      US     N female      0       0                1
## 359       MA     Northeast      US     M female      0       0               77
## 360       NC         South      US     N female      0       0               76
## 361       NC         South      US     N female      0       1                1
## 362       KS       Midwest      US    DL female      0       0               75
## 363       IL       Midwest      US    DL   male      0       0               74
## 364       IN International      IN    DL   male      0       0               73
## 365       NC         South      US     N female      0       0               72
## 366       NZ International      NZ     N female      0       0               71
## 367       NY     Northeast      US     N female      0       0               70
## 368       NH     Northeast      US     N female      0       0               69
## 369       PA     Northeast      US    OT female      0       0               68
## 370       NC         South      US     N female      0       0               67
## 371       NM          West      US     N female      0       0               66
## 372       NC         South      US     N female      0       0               65
## 373       NY     Northeast      US     N female      0       0               64
## 374       NC         South      US     N female      0       0               63
## 375       AL         South      US    AC female      0       0               62
## 376       NH     Northeast      US     N female      0       0               61
## 377       NC         South      US     N   male      0       0               60
## 378       NC         South      US     N   male      0       0               59
## 379       NC         South      US     N female      0       0               58
## 380       NJ     Northeast      US     N female      0       0               57
## 381       NJ     Northeast      US     N female      0       0               56
## 382       NC         South      US     N female      0       0               55
## 383       NC         South      US     N female      0       0               54
## 384       GA         South      US    DL female      0       1               53
## 385       TX         South      US    OT female      0       0               52
## 386       TH International      TH    OT female      0       0               51
## 387       OH       Midwest      US    OT female      0       1               50
## 388       TX         South      US    OT   male      0       0               49
## 389       PA     Northeast      US    OT female      0       0               48
## 390       RI     Northeast      US    OT female      0       0               47
## 391       TX         South      US    OT female      0       0               46
## 392       AL         South      US    AC   male      0       0               45
## 393       WY          West      US    UZ female      0       0               44
## 394       WI       Midwest      US    UZ   male      0       0               43
## 395       WY          West      US    UZ female      0       0               42
## 396       WI       Midwest      US    UZ   male      0       0               41
## 397       WI       Midwest      US    UZ female      0       0               40
## 398       MD         South      US     M female      0       0               39
## 399       MA     Northeast      US     M female      0       0               38
## 400       MX International      MX     M   male      0       1               37
## 401       MA International      MA     M female      0       0               36
## 402       MN       Midwest      US     M   male      0       0               35
## 403       MI       Midwest      US     M female      0       0               34
## 404       MO       Midwest      US     M female      0       0               33
## 405       MA     Northeast      US     M   male      0       0               32
## 406       MI       Midwest      US     M female      0       0               31
## 407       ME     Northeast      US     M female      0       0               30
## 408       MO       Midwest      US     M   male      0       0               29
## 409       NC         South      US     N female      0       0               28
## 410       NM          West      US     N female      0       1               27
## 411       OR          West      US    OT female      0       1               26
## 412       VA         South      US    UZ female      0       0               25
## 413       SC         South      US    OT female      0       1                1
## 414       AZ          West      US    AC female      0       0               24
## 415       AR         South      US    AC   male      0       0               23
## 416       CA          West      US    AC female      0       0               22
## 417       NJ     Northeast      US     N female      0       0               21
## 418       NH     Northeast      US     N female      0       0               20
## 419       NC         South      US     N female      0       0               19
## 420       MA     Northeast      US     M female      0       0               18
## 421       MN       Midwest      US     M female      0       0               17
## 422       NC         South      US     N female      0       1                1
## 423       VT     Northeast      US    UZ   male      0       0               16
## 424       CO          West      US    AC female      0       0               15
## 425       PA     Northeast      US    OT   male      0       1               14
## 426       OH       Midwest      US    OT female      0       0               13
## 427       WI       Midwest      US    UZ female      0       0               12
## 428       MA     Northeast      US     M female      0       0               11
## 429       NM          West      US     N   male      0       1               10
## 430       NM          West      US     N female      0       0                9
## 431       NH     Northeast      US     N female      0       0                8
## 432       NC         South      US     N female      0       0                1
## 433       AL         South      US    AC female      0       0                1
## 434       MO       Midwest      US     M female      1       0                1
## 435       MA     Northeast      US     M   male      1       0                7
## 436       NC         South      US     N   male      0       0                6
## 437       MO       Midwest      US     M female      1       0                1
## 438       OR          West      US    OT   male      1       0                1
## 439     NULL          NULL    NULL  NULL   NULL   NULL    NULL                1
## 440       NC         South      US     N female      0       0                5
## 441       TX         South      US    OT female      0       0                4
## 442       SC         South      US    OT female      0       0                3
## 443       NY     Northeast      US     N female      0       0                2
## 444       NC         South      US     N   male      0       0                1
## 445       NC         South      US     N female      0       0                1
##     sub_group community degree_c betweenness_c closeness_c
## 1           1         4       53  9.997258e-03 0.197508897
## 2          37         5        7  9.338289e-05 0.172628305
## 3           9         5        6  1.332240e-04 0.171627368
## 4           1         7       16  2.173936e-04 0.191709845
## 5           1         2       33  4.915016e-03 0.194907814
## 6           1         3       33  4.110439e-03 0.196113074
## 7           1         2       58  1.514472e-02 0.198835647
## 8           1         2       31  1.092029e-03 0.194907814
## 9           1         1       14  6.887348e-05 0.193295603
## 10          1         7       20  4.643416e-03 0.194736842
## 11          1         3      121  1.357036e-02 0.200814111
## 12          1         2       24  4.039876e-04 0.181669394
## 13          1         3       35  3.087664e-03 0.194140796
## 14          1         3       25  2.642207e-04 0.194993412
## 15          1         7       31  1.926153e-03 0.196460177
## 16         38         7        3  1.508410e-05 0.167610419
## 17          1         3       24  6.282821e-04 0.195250660
## 18          1         3       19  5.392684e-04 0.193971166
## 19          1         5       79  3.362383e-02 0.198835647
## 20         39         5        3  6.686092e-06 0.166791886
## 21         40         3        2  1.977146e-06 0.166666667
## 22          1         1       23  1.796529e-03 0.194140796
## 23         41         3        6  8.503385e-06 0.170441459
## 24          1         3       57  6.177946e-03 0.198746643
## 25         42         3       12  2.348410e-04 0.189339019
## 26          1         3       34  1.848973e-03 0.195422535
## 27          1         1       31  6.929039e-03 0.193548387
## 28         43         7        1  0.000000e+00 0.163235294
## 29          1         7       35  9.491100e-03 0.196721311
## 30          1         7       90  1.645036e-02 0.199192463
## 31         44         7        1  0.000000e+00 0.164627364
## 32         45         7       15  5.051407e-04 0.189662537
## 33         46         8       23  6.929501e-04 0.191875540
## 34          1         3       39  1.092729e-03 0.196113074
## 35          1         7       34  5.316433e-03 0.193464052
## 36          1         7       47  7.218586e-03 0.198657718
## 37         10         7       11  1.623720e-04 0.173099415
## 38         47         8        9  1.016427e-04 0.190721649
## 39          1         4       18  4.665541e-04 0.194140796
## 40         48         4        1  0.000000e+00 0.151535836
## 41          1         4       35  5.602716e-03 0.178313253
## 42          1         7       19  4.711667e-04 0.174734357
## 43         49         4        8  3.957020e-05 0.170900693
## 44          1         4      117  2.750977e-02 0.201360544
## 45         50         3        3  3.627871e-05 0.189339019
## 46         51         4       11  4.804831e-04 0.179757085
## 47         52         4        5  1.592974e-05 0.177316294
## 48         53         7        6  9.722768e-05 0.189097104
## 49          1         4       42  7.914720e-03 0.196460177
## 50          1         2       30  9.747479e-04 0.195079086
## 51          1         2       19  1.788484e-04 0.193295603
## 52          1         3       17  5.245653e-04 0.181002854
## 53          1         3       35  1.838537e-03 0.184538653
## 54          1         2       42  8.083196e-03 0.196634190
## 55         54         2        4  1.094847e-04 0.175632911
## 56          1         2       24  6.434155e-04 0.182415776
## 57         55         3       11  8.681808e-05 0.189258312
## 58          1         3       39  9.939424e-04 0.196547145
## 59         56         6       21  5.947597e-03 0.195250660
## 60          1         3       65  1.478094e-02 0.198037467
## 61          1         2       53  1.966297e-02 0.196808511
## 62          1         3       33  4.608381e-04 0.195767196
## 63          1         3       36  1.528650e-03 0.194395797
## 64          1         3       45  4.634750e-03 0.195680917
## 65         57         3        3  0.000000e+00 0.188535032
## 66         58         1        8  1.031376e-04 0.193043478
## 67          1         5       30  6.168175e-03 0.196460177
## 68          1         3       61  4.240547e-03 0.196113074
## 69         59         4       10  4.555436e-03 0.192792010
## 70         60         3        4  4.041984e-05 0.188775510
## 71          1         2       13  1.165292e-03 0.192124621
## 72         61         8        6  1.957734e-03 0.189177674
## 73         62         8        2  2.371625e-06 0.164261931
## 74         63         3       16  7.699256e-04 0.179393939
## 75         64         7       10  4.696823e-04 0.192959583
## 76         65         3        5  8.353946e-06 0.188615123
## 77         66         2       11  1.603071e-04 0.192457737
## 78         11         8        6  1.030969e-03 0.189177674
## 79         11         8        4  1.316145e-05 0.164871890
## 80         67         3        7  6.675907e-05 0.173844949
## 81         68         2        9  2.023521e-04 0.189824711
## 82         69         1        7  1.565938e-04 0.192374350
## 83          1         4       14  2.723061e-03 0.194310722
## 84         12         4        5  4.482609e-03 0.166666667
## 85         12         4        7  9.703950e-04 0.192792010
## 86         70         4        1  0.000000e+00 0.143041237
## 87         71         4       12  2.490986e-03 0.193971166
## 88          1         8       30  7.380835e-03 0.194480946
## 89         72         8        1  0.000000e+00 0.163055454
## 90         73         8        4  2.870534e-06 0.188375053
## 91         13         8       18  9.879306e-04 0.193211488
## 92          1         3       33  3.693240e-03 0.195853551
## 93         74         8        3  2.428104e-05 0.164505372
## 94         75         1        7  1.838970e-03 0.191792657
## 95          4         1        7  4.493683e-03 0.188375053
## 96          1         1       12  4.745181e-04 0.192875760
## 97         14         1        5  4.446413e-05 0.176190476
## 98          1         3       28  4.422862e-04 0.192792010
## 99         76         8        6  1.074300e-04 0.192792010
## 100         1         4       29  2.226758e-03 0.195939982
## 101        15         4       12  1.491776e-03 0.193211488
## 102        77         2        6  0.000000e+00 0.192291035
## 103        78         6       14  5.205746e-03 0.192792010
## 104        16         8       21  9.561525e-04 0.182640889
## 105        79         8        5  9.977457e-05 0.189662537
## 106        80         4        6  1.553601e-05 0.172293364
## 107        81         4       19  5.186231e-04 0.190967742
## 108        82         4        4  3.381618e-06 0.168949772
## 109         1         4       22  7.966098e-03 0.192124621
## 110        17         5        4  4.479720e-03 0.188775510
## 111        17         5        1  0.000000e+00 0.159025788
## 112        83         4        5  8.488116e-05 0.188855806
## 113        84         5       10  1.765519e-04 0.189743590
## 114        85         2       11  9.969978e-05 0.180855397
## 115        18         6       26  2.333754e-02 0.193127447
## 116        18         6       35  1.524048e-02 0.197070573
## 117        86         3        6  2.994202e-05 0.171627368
## 118        87         6        6  5.342236e-05 0.166917293
## 119        88         6        7  5.582580e-04 0.169659916
## 120        19         6        3  4.474000e-03 0.188615123
## 121        89         7       11  1.416391e-04 0.193464052
## 122        90         6        3  2.903839e-05 0.166729253
## 123         2         6        5  7.004260e-04 0.168118137
## 124        91         6        2  0.000000e+00 0.188455008
## 125        92         6        1  0.000000e+00 0.164871890
## 126        93         1        3  0.000000e+00 0.188455008
## 127        94         6        2  0.000000e+00 0.188455008
## 128        95         6       24  1.690773e-03 0.192291035
## 129        20         6       14  1.254664e-03 0.190149893
## 130        96         6        1  0.000000e+00 0.164871890
## 131        97         1        5  1.821601e-05 0.188936170
## 132        98         1        6  3.640050e-05 0.171230235
## 133        99         1        9  6.186174e-04 0.189905902
## 134       100         1        2  1.138480e-04 0.187896741
## 135         5         6        2  2.412686e-05 0.160927872
## 136         1         6       25  1.148455e-03 0.193717277
## 137         1         6       30  3.241522e-03 0.195767196
## 138       101         2        9  1.111568e-04 0.192291035
## 139       102         7        4  1.847047e-05 0.188455008
## 140        21         1        3  2.542045e-06 0.161807580
## 141       103         1        5  1.773828e-03 0.188535032
## 142       104         1       14  2.481858e-04 0.193379791
## 143       105         6        6  1.068402e-06 0.189016603
## 144         1         3       21  2.019987e-04 0.190803610
## 145         2         6        2  1.866003e-05 0.164566345
## 146         4         1        3  4.023374e-05 0.191461837
## 147       106         8       13  1.800397e-03 0.193211488
## 148        22         2        8  3.947696e-05 0.177955912
## 149         2         1        1  0.000000e+00 0.158571429
## 150         2         1        4  8.477025e-03 0.188135593
## 151         2         6        2  2.440364e-05 0.160000000
## 152       107         1        5  5.905531e-04 0.189016603
## 153       108         1        3  7.111750e-06 0.163415532
## 154       109         1       10  2.301671e-03 0.192541197
## 155       110         8       18  2.595965e-03 0.193971166
## 156       111         8        3  2.979074e-06 0.169465649
## 157         1         2       11  1.112162e-03 0.192708333
## 158         1         1       15  1.195712e-03 0.190312902
## 159       112         7        5  2.855870e-05 0.176610979
## 160        23         1        3  7.270113e-04 0.188135593
## 161         1         5       18  2.865065e-04 0.192041522
## 162       113         6        7  3.611767e-05 0.191875540
## 163         1         7       12  7.773809e-05 0.193127447
## 164       114         6        1  0.000000e+00 0.161866569
## 165         9         6        8  2.495069e-04 0.190803610
## 166       115         2        6  1.530205e-04 0.191958495
## 167       116         1       14  3.070046e-03 0.190394511
## 168       117         1        2  0.000000e+00 0.161807580
## 169         3         1        5  0.000000e+00 0.188455008
## 170         3         1        9  2.442540e-04 0.189177674
## 171       118         1        7  4.474000e-03 0.188615123
## 172       119         1        8  1.063241e-04 0.189500640
## 173       120         5        9  1.893158e-03 0.189743590
## 174        24         5        2  3.389394e-06 0.162756598
## 175       121         2        5  1.207379e-05 0.166917293
## 176         1         3       14  4.787245e-03 0.192875760
## 177         1         3       15  5.059019e-04 0.182116489
## 178       122         8        8  8.215182e-04 0.189016603
## 179         4         1        6  2.089695e-04 0.191875540
## 180         4         1        1  0.000000e+00 0.158741509
## 181       123         1        3  0.000000e+00 0.187896741
## 182       124         2       10  8.758506e-05 0.172964550
## 183       125         2       12  6.552675e-04 0.180781759
## 184       126         3       12  1.835474e-03 0.193211488
## 185       127         2       14  8.593368e-04 0.193295603
## 186       128         1        2  0.000000e+00 0.187817259
## 187       129         1        3  9.243802e-07 0.187896741
## 188       130         1        4  0.000000e+00 0.188535032
## 189       131         5        3  5.180931e-06 0.187976291
## 190       132         2        5  6.130620e-05 0.191709845
## 191       133         7        4  3.038756e-05 0.169078446
## 192         1         7       18  8.311724e-03 0.192041522
## 193         1         3       22  4.425342e-04 0.193379791
## 194         6         4        6  1.290771e-04 0.189419795
## 195       134         4       12  1.090249e-04 0.194651469
## 196         6         4        5  1.783289e-05 0.189419795
## 197        25         2        7  6.288008e-04 0.178743961
## 198         1         2       36  7.005840e-04 0.195336560
## 199        16         8       14  2.846943e-04 0.193295603
## 200       135         1        5  1.945641e-04 0.189097104
## 201       136         4       22  6.532441e-04 0.193886463
## 202         1         3       13  8.398502e-05 0.194310722
## 203       137         4       12  2.806158e-04 0.190721649
## 204       138         4        2  0.000000e+00 0.187976291
## 205        13         8       17  9.459393e-04 0.190312902
## 206       139         1        3  0.000000e+00 0.188135593
## 207       140         5       16  3.426864e-04 0.183017312
## 208       141         3        7  9.279767e-05 0.170180146
## 209       142         4        3  4.400729e-05 0.175841584
## 210       143         3        2  0.000000e+00 0.175355450
## 211       144         3       10  3.336131e-04 0.192708333
## 212         1         3       12  9.547411e-05 0.192457737
## 213       145         3        1  0.000000e+00 0.165548098
## 214       146         4        2  0.000000e+00 0.188215345
## 215       147         2        5  1.321273e-05 0.177316294
## 216       148         5        7  7.396383e-05 0.170049789
## 217        26         5       13  2.370083e-03 0.190394511
## 218        27         7        5  2.627770e-04 0.189419795
## 219       149         8       30  3.178656e-03 0.192624729
## 220       150         5        1  0.000000e+00 0.166105499
## 221       151         4       10  6.195117e-04 0.191461837
## 222       152         5        2  0.000000e+00 0.166105499
## 223         1         5       39  8.530463e-03 0.196026490
## 224       153         5        2  0.000000e+00 0.167673716
## 225       154         5        1  0.000000e+00 0.166105499
## 226         1         2       15  3.017241e-04 0.194310722
## 227       155         5        2  0.000000e+00 0.166105499
## 228       156         5        2  0.000000e+00 0.177387135
## 229       157         7        1  0.000000e+00 0.161337209
## 230         7         6        3  2.736394e-05 0.161689731
## 231       158         6        7  8.912657e-05 0.170966500
## 232       159         7        3  1.664355e-05 0.191709845
## 233       160         1        1  0.000000e+00 0.161689731
## 234         3         1       37  2.242049e-02 0.192541197
## 235       161         1        4  4.327795e-05 0.191875540
## 236       162         1        3  0.000000e+00 0.188295165
## 237       163         1        1  0.000000e+00 0.161689731
## 238         3         1        2  0.000000e+00 0.161689731
## 239        21         1        3  8.818300e-04 0.188375053
## 240       164         1        2  0.000000e+00 0.188295165
## 241         3         1        3  0.000000e+00 0.188295165
## 242       165         1        4  0.000000e+00 0.188375053
## 243         5         1        8  8.748607e-03 0.189662537
## 244       166         1        1  0.000000e+00 0.161689731
## 245        28         1        3  4.474000e-03 0.188455008
## 246       167         1        6  9.804484e-06 0.188936170
## 247       168         7        8  4.513245e-03 0.192541197
## 248       169         4        9  4.524555e-03 0.190639760
## 249         1         8       12  3.758989e-03 0.192541197
## 250       170         5        4  1.124611e-04 0.191709845
## 251        10         7        5  2.123824e-04 0.188535032
## 252       171         1        3  4.994438e-07 0.188295165
## 253       172         1        3  0.000000e+00 0.188295165
## 254       173         8        2  1.520385e-06 0.161984677
## 255        29         8        2  0.000000e+00 0.187896741
## 256        29         8        5  4.152514e-05 0.188695283
## 257         6         8        4  7.012946e-05 0.188936170
## 258        14         1        5  2.263293e-04 0.191709845
## 259       174         3        3  7.710743e-05 0.188695283
## 260        30         1        1  0.000000e+00 0.158401712
## 261        30         1        2  4.474000e-03 0.187896741
## 262       175         2        4  7.810434e-05 0.176120587
## 263        28         1        1  0.000000e+00 0.158798283
## 264       176         7        2  1.475825e-05 0.175632911
## 265       177         2        4  4.980090e-04 0.188535032
## 266        31         2        5  5.145883e-05 0.164627364
## 267       178         2        2  0.000000e+00 0.188775510
## 268       179         3        9  1.532313e-04 0.173844949
## 269       180         6        3  2.866235e-06 0.167673716
## 270       181         6        2  0.000000e+00 0.162102957
## 271       182         1        4  2.285280e-04 0.188135593
## 272       183         5        2  0.000000e+00 0.164140481
## 273       184         3        4  1.041269e-05 0.168373151
## 274       185         4        1  0.000000e+00 0.167863894
## 275       186         2        5  1.959549e-06 0.177528988
## 276       187         1        5  0.000000e+00 0.192124621
## 277       188         2        6  3.919098e-06 0.177528988
## 278        24         5        5  2.665584e-03 0.192457737
## 279       189         5       12  1.773613e-03 0.191296855
## 280       190         1        3  0.000000e+00 0.188295165
## 281       191         8        8  8.324394e-05 0.192124621
## 282         5         1        1  0.000000e+00 0.159654800
## 283        32         1        1  0.000000e+00 0.158401712
## 284        32         1        2  4.474000e-03 0.187896741
## 285        15         4        6  1.193956e-05 0.168693009
## 286        23         3        2  2.944515e-06 0.162102957
## 287       192         6        4  3.862214e-05 0.169271826
## 288        33         6        2  1.171883e-05 0.162399415
## 289       193         5        3  1.296219e-04 0.164261931
## 290       194         6        1  0.000000e+00 0.162102957
## 291       195         6        1  0.000000e+00 0.162102957
## 292       196         3        2  0.000000e+00 0.187976291
## 293       197         5        3  7.205470e-06 0.176190476
## 294       198         1        1  0.000000e+00 0.158911954
## 295         8         1        5  4.479254e-03 0.188135593
## 296         8         1        1  0.000000e+00 0.158571429
## 297         8         1        2  0.000000e+00 0.187896741
## 298       199         5        3  0.000000e+00 0.191792657
## 299       200         4        2  0.000000e+00 0.187976291
## 300       201         8       15  4.780285e-03 0.194140796
## 301        22         2       14  2.176201e-03 0.192959583
## 302       202         1        5  2.205982e-06 0.188615123
## 303       203         4        7  6.020324e-05 0.192457737
## 304       204         4        2  0.000000e+00 0.187817259
## 305       205         7        2  0.000000e+00 0.188295165
## 306        25         2        4  1.164955e-04 0.165857303
## 307       206         2        6  7.882054e-04 0.175980975
## 308       207         3        8  6.997015e-05 0.189824711
## 309       208         2        1  0.000000e+00 0.164566345
## 310        26         5       30  2.296343e-02 0.192541197
## 311       209         5        1  0.000000e+00 0.161689731
## 312       210         5        1  0.000000e+00 0.161689731
## 313       211         5        1  0.000000e+00 0.161689731
## 314       212         5        2  0.000000e+00 0.188615123
## 315       213         5        2  0.000000e+00 0.162102957
## 316       214         5        1  0.000000e+00 0.161689731
## 317       215         5        6  8.873663e-05 0.192291035
## 318       216         1        4  0.000000e+00 0.191958495
## 319        27         7        4  5.642092e-05 0.175771971
## 320       217         3        3  0.000000e+00 0.191627104
## 321        34         7        2  3.223866e-06 0.161866569
## 322        34         7        5  8.752388e-04 0.189500640
## 323       218         2        3  0.000000e+00 0.176050753
## 324       219         2        4  1.069884e-05 0.188375053
## 325       220         8        2  0.000000e+00 0.188135593
## 326       221         8        3  0.000000e+00 0.191461837
## 327       222         8        1  0.000000e+00 0.162816282
## 328       223         7        1  0.000000e+00 0.161689731
## 329       224         1        9  5.630786e-05 0.192041522
## 330         2         6        3  7.858756e-04 0.188215345
## 331       225         1        6  1.271023e-06 0.188055909
## 332        19         6        1  0.000000e+00 0.158911954
## 333       226         8        2  1.819547e-05 0.187976291
## 334       227         3        2  0.000000e+00 0.169400992
## 335       228         2        6  3.508836e-05 0.189258312
## 336       229         2       12  2.630565e-04 0.194225722
## 337         7         6        4  1.347255e-03 0.188375053
## 338       230         1        4  0.000000e+00 0.188295165
## 339       231         2        2  0.000000e+00 0.188455008
## 340        35         1        3  2.514451e-04 0.187976291
## 341       232         5       11  2.492494e-04 0.179684338
## 342        33         1        4  7.466899e-04 0.188135593
## 343        35         1        2  0.000000e+00 0.187896741
## 344         7         6        2  1.122599e-03 0.187896741
## 345       233         7        3  0.000000e+00 0.188375053
## 346        31         1        3  9.418734e-04 0.187976291
## 347        36         2        5  7.612775e-04 0.192875760
## 348       234         2        1  0.000000e+00 0.164688427
## 349       235         2        1  0.000000e+00 0.164688427
## 350       236         8        6  5.820118e-05 0.189662537
## 351       237         3        6  4.412943e-05 0.176962933
## 352       238         5        1  0.000000e+00 0.164444444
## 353        36         4        2  1.992894e-06 0.166043381
## 354       239         4        3  0.000000e+00 0.191544435
## 355       240         1        4  0.000000e+00 0.187817259
## 356       241         1        5  4.248942e-06 0.188455008
## 357       242         2        2  5.191504e-05 0.187976291
## 358       243         3        2  0.000000e+00 0.188695283
## 359       244         4        1  0.000000e+00 0.160346696
## 360       245         2        4  6.108170e-05 0.188615123
## 361        20         8       21  1.206649e-03 0.175841584
## 362       246         1        1  0.000000e+00 0.187737844
## 363       247         1        1  0.000000e+00 0.187737844
## 364       248         1        1  0.000000e+00 0.187737844
## 365       249         1        1  0.000000e+00 0.187737844
## 366       250         1        2  0.000000e+00 0.187737844
## 367       251         1        1  0.000000e+00 0.187737844
## 368       252         1        1  0.000000e+00 0.187737844
## 369       253         1        1  0.000000e+00 0.187737844
## 370       254         4        1  0.000000e+00 0.161866569
## 371       255         1        1  0.000000e+00 0.187737844
## 372       256         1        1  0.000000e+00 0.187737844
## 373       257         1        1  0.000000e+00 0.187737844
## 374       258         1        1  0.000000e+00 0.187737844
## 375       259         1        1  0.000000e+00 0.187737844
## 376       260         1        3  0.000000e+00 0.191296855
## 377       261         2        2  0.000000e+00 0.191296855
## 378       262         2        2  0.000000e+00 0.191296855
## 379       263         2        2  0.000000e+00 0.191296855
## 380       264         1        1  0.000000e+00 0.187737844
## 381       265         1        1  0.000000e+00 0.187737844
## 382       266         2        2  0.000000e+00 0.191296855
## 383       267         1        1  0.000000e+00 0.187737844
## 384       268         1        1  0.000000e+00 0.187737844
## 385       269         1        1  0.000000e+00 0.187737844
## 386       270         1        1  0.000000e+00 0.187737844
## 387       271         1        1  0.000000e+00 0.187737844
## 388       272         2        2  0.000000e+00 0.191296855
## 389       273         1        1  0.000000e+00 0.187737844
## 390       274         1        1  0.000000e+00 0.187737844
## 391       275         1        1  0.000000e+00 0.187737844
## 392       276         1        1  0.000000e+00 0.187737844
## 393       277         1        1  0.000000e+00 0.187737844
## 394       278         1        1  0.000000e+00 0.187737844
## 395       279         1        1  0.000000e+00 0.187737844
## 396       280         1        1  0.000000e+00 0.187737844
## 397       281         1        3  0.000000e+00 0.191296855
## 398       282         1        1  0.000000e+00 0.187737844
## 399       283         2        2  0.000000e+00 0.191296855
## 400       284         1        1  0.000000e+00 0.187737844
## 401       285         1        1  0.000000e+00 0.187737844
## 402       286         1        1  0.000000e+00 0.187737844
## 403       287         2        2  0.000000e+00 0.191296855
## 404       288         1        1  0.000000e+00 0.187737844
## 405       289         1        1  0.000000e+00 0.187737844
## 406       290         1        3  0.000000e+00 0.191296855
## 407       291         1        1  0.000000e+00 0.187737844
## 408       292         1        1  0.000000e+00 0.187737844
## 409       293         2        2  0.000000e+00 0.191296855
## 410       294         2        3  0.000000e+00 0.191296855
## 411       295         1        1  0.000000e+00 0.187737844
## 412       296         1        1  0.000000e+00 0.187737844
## 413       297         1        2  0.000000e+00 0.187737844
## 414       298         1        1  0.000000e+00 0.187737844
## 415       299         1        2  0.000000e+00 0.187737844
## 416       300         2        1  0.000000e+00 0.175217048
## 417       301         2        1  0.000000e+00 0.175217048
## 418       302         2        1  0.000000e+00 0.175217048
## 419       303         2        1  0.000000e+00 0.175217048
## 420       304         2        1  0.000000e+00 0.175217048
## 421       305         2        1  0.000000e+00 0.175217048
## 422       306         3        4  6.389261e-06 0.166416792
## 423       307         2        1  0.000000e+00 0.175217048
## 424       308         1        1  0.000000e+00 0.187737844
## 425       309         1        1  0.000000e+00 0.187737844
## 426       310         1        1  0.000000e+00 0.187737844
## 427       311         1        1  0.000000e+00 0.187737844
## 428       312         1        1  0.000000e+00 0.187737844
## 429       313         1        1  0.000000e+00 0.187737844
## 430       314         1        1  0.000000e+00 0.187737844
## 431       315         1        1  0.000000e+00 0.187737844
## 432         1         3       43  4.114279e-03 0.183698800
## 433       316         2        2  0.000000e+00 0.175217048
## 434       317         3        3  1.626206e-06 0.176610979
## 435       318         2        1  0.000000e+00 0.175217048
## 436       319         2        1  0.000000e+00 0.175217048
## 437       320         7        2  0.000000e+00 0.162340037
## 438       321         7        2  9.470854e-06 0.175702414
## 439       322         1        2  0.000000e+00 0.162399415
## 440       323         1        1  0.000000e+00 0.187737844
## 441       324         9        0  0.000000e+00 0.002247191
## 442       325        10        0  0.000000e+00 0.002247191
## 443       326        11        0  0.000000e+00 0.002247191
## 444         1         1      581  7.142161e-01 0.230649351
## 445         1         2      332  1.850191e-01 0.212034384

keyego <- data.frame(
  dltl_stats_ego %>%
  arrange(-degree_c)%>%
  select(uid),
 dltl_stats_ego %>%
  arrange(-betweenness_c)%>%
  select(uid),
  dltl_stats_ego %>%
  arrange(-closeness_c)%>%
  select(uid))%>%
  setNames(c("degree","betweenness","closeness"))
keyego[]

##     degree betweenness closeness
## 1      444         444       444
## 2      445         445       445
## 3       11          19        44
## 4       44          44        11
## 5       30         115        30
## 6       19         310         7
## 7       60         234        19
## 8       68          61        24
## 9        7          30        36
## 10      24         116        60
## 11       1           7         1
## 12      61          60       116
## 13      36          11        61
## 14      64           1        29
## 15     432          29        54
## 16      49         243        58
## 17      54         223        15
## 18      34         150        49
## 19      58         192        67
## 20     223          54         6
## 21     234         109        34
## 22      63          49        68
## 23     198          88       223
## 24      13          36       100
## 25      29          27        92
## 26      41          24        62
## 27      53          67       137
## 28     116          59        64
## 29      26          41        26
## 30      35          35       198
## 31       5         103        17
## 32       6           5        59
## 33      62         176        50
## 34      92         300        14
## 35       8          10         5
## 36      15          64         8
## 37      27          69        10
## 38      50         248       195
## 39      67         247        88
## 40      88          95        63
## 41     137          84        83
## 42     219         110       202
## 43     310         295       226
## 44     100         245       336
## 45      98         120        13
## 46     115         261        22
## 47      14         284        39
## 48     136         171       300
## 49      12          68        18
## 50      17         432        87
## 51      56           6       155
## 52     128         249       201
## 53      22          92       136
## 54      33         137        27
## 55     109         219        35
## 56     193          13       121
## 57     201         167       142
## 58      59          83       193
## 59     104         278         9
## 60     144         155        51
## 61     361          87       185
## 62      10         217       199
## 63      18         154        91
## 64      42         100       101
## 65      51         301       147
## 66     107          72       184
## 67      39          15       115
## 68      91         173       163
## 69     155          26        66
## 70     161          94        75
## 71     192          53       301
## 72      52         184        96
## 73     205         147       176
## 74       4          22       347
## 75      74         141        69
## 76     207         279        85
## 77      32         128        98
## 78     158          63        99
## 79     177         101       103
## 80     226         337       157
## 81     300         129       211
## 82       9         361       219
## 83      83         158       154
## 84     103          71       234
## 85     129         136       247
## 86     142         344       249
## 87     167         157       310
## 88     176          34        77
## 89     185           8       212
## 90     199          78       278
## 91     301          58       303
## 92      71          91        82
## 93     147          50       102
## 94     202          85       128
## 95     217         104       138
## 96      25         205       317
## 97      87         346        71
## 98      96         239       109
## 99     101         322       276
## 100    163         185       281
## 101    183         178       161
## 102    184         307       192
## 103    195         330       329
## 104    203          74       166
## 105    212         347       318
## 106    249         342        33
## 107    279         160       162
## 108    336         198       179
## 109     37         123       235
## 110     46          33        94
## 111     57         183       298
## 112     77         201         4
## 113    114          56       190
## 114    121         197       232
## 115    157          17       250
## 116    341         221       258
## 117     69         133       320
## 118     75         152       354
## 119    113         119       146
## 120    154          18       221
## 121    182          52       326
## 122    211         107       279
## 123    221         177       376
## 124     38          32       377
## 125     81         265       378
## 126    133          46       379
## 127    138          96       382
## 128    170          42       388
## 129    173          75       397
## 130    248          39       399
## 131    268          62       403
## 132    329         193       406
## 133     43          98       409
## 134     66          12       410
## 135    148         207       107
## 136    165         211       144
## 137    172         226       165
## 138    178         161        38
## 139    243         199       203
## 140    247         203       248
## 141    281          14       167
## 142    308         336       217
## 143      2         218       158
## 144     80         340       205
## 145     82         165       129
## 146     85         341       133
## 147     94         142        81
## 148     95         170       308
## 149    119          25       113
## 150    162         271       173
## 151    171         258        32
## 152    197           4       105
## 153    208         251       243
## 154    216         179       350
## 155    231          81       172
## 156    303         144       322
## 157      3         200       194
## 158     23          51       196
## 159     48         113       218
## 160     72          37        25
## 161     78          77        45
## 162     99          82        57
## 163    102         268       335
## 164    106         166        72
## 165    117         121        78
## 166    118           3       170
## 167    132         289        48
## 168    143         194       200
## 169    166         306       143
## 170    179         134       152
## 171    194         250       178
## 172    246         138       131
## 173    277          55       246
## 174    285         195       257
## 175    307          99       112
## 176    317         172        70
## 177    331          66       110
## 178    335          38       267
## 179    350         105       256
## 180    351         114       259
## 181     47          48       358
## 182     76         212        76
## 183     84           2       120
## 184     97         208       171
## 185    105         231       302
## 186    112         317       314
## 187    123         182       360
## 188    131          57        65
## 189    141         112       141
## 190    152         202       188
## 191    159         281       251
## 192    169         262       265
## 193    175         163       124
## 194    190         259       126
## 195    196         216       127
## 196    200         257       139
## 197    215         308       169
## 198    218           9       245
## 199    251          80       339
## 200    256         190       356
## 201    258         360        90
## 202    266         303        95
## 203    275         350       239
## 204    276         319       242
## 205    278         329       324
## 206    295         118       337
## 207    302         357       345
## 208    322         266       236
## 209    347          97       240
## 210    356         351       241
## 211     55         209       252
## 212     70         235       253
## 213     79         256       280
## 214     90          70       305
## 215    108         146       338
## 216    110          43       214
## 217    139         148       330
## 218    150         287       150
## 219    188         132       160
## 220    191          45       206
## 221    235         162       271
## 222    242         335       295
## 223    250         191       325
## 224    257         117       342
## 225    262         122       331
## 226    265         159       189
## 227    271         230       204
## 228    273         151       292
## 229    287          93       299
## 230    306         135       333
## 231    318         145       340
## 232    319         139       346
## 233    324         131       357
## 234    337         333       134
## 235    338         196       181
## 236    342         232       187
## 237    355          47       255
## 238    360         106       261
## 239    422          16       284
## 240     16         264       297
## 241     20         215       343
## 242     45          79       344
## 243     65         175       186
## 244     93         285       304
## 245    120         288       355
## 246    122         324       362
## 247    126         273       363
## 248    140         246       364
## 249    146         438       365
## 250    153          23       366
## 251    156          76       367
## 252    160         293       368
## 253    181         153       369
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1. Which two actors have the highest betweenness and what is unique about their role in the course?

   • The top two actors that have the highest betweeness are 444 and 445. They are also the facilitators of the group

2. Do these same two actors also have the highest closeness? How might you interpret that?

   • Yes, they also have the highest number of closeness as well. They have the shortest distance path to all other nodes.

Again, we’ve only scratched the surface of the vast number of centrality meaures available in the {igraph} and {tidygraph} packages. Feel free to explore other functions to calculate node and edge centrality in these two packages.

4. MODEL

As highlighted in Chapter 3 of Data Science in Education Using R, the Model step of the data science process entails “using statistical models, from simple to complex, to understand trends and patterns in the data.” The authors note that while descriptive statistics and data visualization during the Explore step can help us to identify patterns and relationships in our data, statistical models can be used to help us determine if relationships, patterns and trends are actually meaningful.

We will not explore the use of models for SNA until Unit 3, but recall from A Social Network Perspective in MOOC-Eds was guided by the following questions:

1. What are the patterns of peer interaction and the structure of peer networks that emerge over the course of a MOOC-Ed?

2. To what extent do participant and network attributes (e.g., homophily, reciprocity, transitivity) account for the structure of these networks?

3. To what extent do these networks result in the co-construction of new knowledge?

To address Question 1, actors in the network were categorized into distinct mutually exclusive groups using the core-periphery and regular equivalence functions of UCINET. The former used the CORR algorithm to divide the network into actors that are part of a densely connected subgroup, or “core,” from those that are part of the sparsely connected periphery. Regular equivalence employs the REGE blockmodeling algorithm to partition, or group, actors in the network based on the similarity of their ties to others with similar ties. In essence, blockmodeling provides a systematic way for categorizing educators based on the ways in which they interacted with peers.

As we saw upon just a basic visual inspection of our network during the Explore section, there was a small core of highly connected participants surrounded by those on the “periphery,” or edge, of the network with very few connections. In the DLT 2 course, those on the periphery made up roughly 90% of network! The study also found relatively high levels of reciprocation, but also found that roughly a quarter of participants were characterized as “brodcasters” – educators who initiated a discussion thread, but neither reciprocated with those who replied, nor posted to threads initiated by others.

To address Question 2, this study use the exponential family of random graph models (ERGM; also known as p* models), which provide a statistical approach to network modeling that addresses the complex dependencies within networks. ERGMs predict network ties and determine the statistical likelihood of a given network structure, based on an assumed dependency structure, the attributes of the individuals (e.g., gender, popularity, location, previous ties) and prior states of the network.

👉 Your Turn ⤵

Recall that network analyses to identify groups, positions, and central actors are a means to an end rather than an end in itself (Carolan 2014). One primary goal of gathering data about a network’s groups, positions, and central actors – a goal which we’ll explore more deeply in later units – is test whether there is a relationship between these measures and educational outcomes of interest. That was ultimately the aim of the research article guiding this study, but sadly got neglected.

Thinking about the node and group level analyses we’ve conducted so far, and what you know about this educational context so far, write a research question or hypothesis you would like to test that would examine whether there is a relationship between these group or node-level measures and educational outcomes appropriate for this MOOC-Ed.

• How did reciprocity change over time within the MOOC-ED. Did this impact the density of the network. If so did the key actors change?

---

5. COMMUNICATE

For your final Your Turn, your goal is to distill our analysis from above into a simple “data product” designed to illustrate key findings about changes in the collaboration network over time. For the purposes of this task, imagine that your audience consists of the developers and facilitators of the DLT MOOC-Eds who have limited background in SNA and adapt the following steps accordingly:

1. Select. Select a group and/or node-level analysis from above, or a new analysis if so motivated, that you think would be interesting or relevant for the target audience and that helps answer our research questions.

2. Polish. Create and polish a data visualization and/or data table to communicate your selected findings.

3. Narrate. Write a brief narrative (2-3 paragraphs) to accompany your visualization and/or table that includes the following:

   • The question or questions guiding the analysis;

   • The conclusions you’ve reached based on our findings;

   • How your audience might use this information;

   • How you might revisit or improve upon this analysis in the future.

👉 Your Turn ⤵

Use the code chunk below create a polished table and/or visualization(s) and write a brief narrative in the space that follows.

Data Visualization or Table

dlt1_network |>
  activate(nodes) |>
  filter(region == 'South')|>
  mutate(degree=centrality_degree()) |> 
  filter(degree>0)|>
  activate(edges) |>
  mutate(betweenness = centrality_edge_betweenness()) |>
ggraph(layout = "stress") +
  geom_edge_link0(aes(edge_alpha = betweenness), edge_color = "grey66") +
  geom_node_point(aes(size = degree, shape = experience2, color = group)) +
  scale_edge_width_continuous(range = c(0.2, 10)) +
  scale_size_continuous(range = c(1, 8)) +
  ggtitle( "MOOC-Ed Southern Region Participants" ) +
  theme( legend.key = element_rect( fill = "white", colour = "black" ), 
  legend.title = element_text(face = "bold" )) +
  guides(color =FALSE)+
  theme_graph() +
  theme(legend.position = "right", legend.text = element_text(size = 6, colour = "black"))

Narrative

To be connected to others matters. To be able to share and learn from each other matters. Carolan( 2014) explains, “Teachers who share best practices with others expand their own teaching repertoires.” We explored the Southern region participants of a MOOC-Ed discussion through the visualization above.

Asking whether years of experience make a difference in a discussion forum? What ways are the participants connected?

• What we found was that the blue group seemed to have higher centrality degree and betweenness. The localized degree measurement captures many actors in the blue group with more years of experience are higher connected to other actors regardless of how those other actors are related to each other.

• We also looked at the gatekeeper function, and those same actors with the highest degree also had a higher betweenness score.

Using this type of analysis, the facilitator could reach out to those with high betweenness and closeness degrees to spread information and ideas to the rest of the group.

To improve this analysis, I would first look at the closeness degree and include it as an edge color. Next, I would like to explore each group separately. I would be able to analyze each network and the actors’ brokerage to find out if there are any consultants.

🧶 Knit & Check ✅

Congratulations - you’ve completed the Unit 3 case study! To share your work, click the drop down arrow next to the ball of yarn that says “Knit” at the top of this markdown file, then select “Knit top HTML.” Assuming your code contains no errors, this will create a web page in your Files pane that serves as a record of your work.

Once your file has been knitted, you can publish this file online using RPubs, or share the HTML file through another means.

References

Carolan, Brian. 2014. “Social Network Analysis and Education: Theory, Methods & Applications.” https://doi.org/10.4135/9781452270104.

Kellogg, Shaun, and Achim Edelmann. 2015. “Massively Open Online Course for Educators (MOOC-e d) Network Dataset.” British Journal of Educational Technology 46 (5): 977–83.

Krumm, Andrew, Barbara Means, and Marie Bienkowski. 2018. Learning Analytics Goes to School. Routledge. https://doi.org/10.4324/9781315650722.

Wickham, Hadley, and Garrett Grolemund. 2016. R for Data Science: Import, Tidy, Transform, Visualize, and Model Data. " O’Reilly Media, Inc.". https://r4ds.had.co.nz.