Working from the example R syntax, calculate network density, degree centrality and centralization measures for your network data. Write up your results briefly, focusing on interpretation of the results. Don’t list statistics or provide code, the goal is to choose a few interesting results and provide an interpretation.
Briefly describe the dataset you are using: identify initial network format, describe and identify the nodes (including how many nodes are in the dataset), what constitutes a tie or edge (including how many ties, whether ties are directed/undirected and weighted/binary, and how to interpret the value of the tie if any), whether or not there are edge attributes that might be used to subset data or stack multiple networks (e.g., tie type, year, etc). Not every feature of the network needs to be described, but description should orient reader to the network data and provide any necessary context for the results provided.
Provide at least two or three noteworthy results, including the relevant statistics and interpretation. For example, explaining which node(s) are most central and which are least central. Discuss (with any related evidence) whether or not the node(s) behavior is in line with or violates expectations based on the degree centrality measure. What do you make of network density and centralization measures?
In this homework, I will provide a preliminary exploration of Padgett (1994)’s dataset on the network of weddings among leading Florentine families from 1282- 1500. This dataset restricts its focus to 16 families, the most famous of which is the Medicis. In the dataset, the nodes represent each family throughout the time period and the edges signal the presence of at least one marriage between children in respective families. It is important to note that this dataset is but a subset of the larger marriage network compiled by Padgett which includes 38 Renaissance Florentine family-trees and “about 10,500 dated marriages among Florentine surnamed families” (See: http://home.uchicago.edu/jpadgett/papers/unpublished/maelite.pdf).That said, there is enough information in the abridged dataset to begin an exploration.
In total, there are 20 mutual ties between the Florentine families in the network Given the nature of marriages as “co-equal” relationships, these edges are not bipartite, weighted, or directed, and there are no edge attributes (i.e. each edge indicates that there was a inter-family marriage at some point; there is no indication of how many marriages between the families there arer or when the marriages took place). This last point is critically important, as we cannot extrapolate beyond a 0,1 relationship between the families. We do not know how often across the three hundred year time period the families inter-married. This would be a worthwhile addition to the dataset.
There are a few noticeable facts that can gleaned from the network via descriptive analysis. First, the network of Florentine families is not complete. There are two components. The first has fifteen families in it. The second has a single isolate. This suggests that, putting aside the isolate, the network is quite connected. That said, the network density is only .17 (20 edges out of 120), which suggests that while the network is rather connected, it is not per se dense. Information does flow through the network efficiently despite the lack of density; the average path length is a mere 2.49, which means the average distance between any two nodes is under 2.5.
Connectedness can tell us much about the shape of the network, but to really understand what is happening among the families, it is important to assess how centralized the network and individual nodes are. The centralization of the network is .27, while the indegree and outdegree centralization are both .25 (they are equivalent because the edges are not weighted and are not directed). What about individual families? The Medici family has the highest degree centrality (12), with six in degrees and out degrees respectively (i.e. 6 edges connecting the family with other families). Only two families, the Guadagni and Strozzi approach the centrality of the Medici family, with a total of 8 degrees of connection (4 edges). This suggests that the Medici family has the most connections in the network, beating out their rivals. The periphery nodes are clear without having to contruct a plot, as a number of families (Acciaiuoli, Ginori, Lamberteschi, and Pazzi) only have a degree of 2, meaning that they only have one connection to the major component of the network. Pucci is the only family with no degrees, which signals that they are the isolate in the network and have no marriage ties to any other family.
## [1] 16## [1] 40## [1] FALSE## [1] TRUE## [1] FALSE## [1] "name"## character(0)## $mut
## [1] 20
##
## $asym
## [1] 0
##
## $null
## [1] 100## [1] 324 0 195 0 0 0 0 0 0 0 38 0 0 0 0 3## [1] 0.1914894## [1] 0.1914894## [1] 0.2181818## [1] 2.485714## [1] "membership" "csize" "no"## [1] 2## [1] 15 1## [1] 0.1666667## [1] 0.15625## [1] 0.1666667flo.nodes<-data.frame(name=V(flograph)$name, degree=igraph::degree(flograph))
flodata.nodes <- data.frame( name = V(flograph)$name, degree = igraph::degree(flograph), indegrees = igraph::degree(flograph, mode = “in”, loops = FALSE), outdegrees = igraph::degree(flograph, mode = “out”, loops = FALSE) )
flo.total_centralization <-centr_degree(flograph, loops = FALSE, mode = “total”)$centralization
flo.indegree_centralization <- centr_degree(flograph, loops = FALSE, mode = “in”)$centralization
flo.outdegree_centralization <- centr_degree(flograph, loops = FALSE, mode = “out”)$centralization