# `igraph` is a package that provides tools for the analysis and visualization of networks
library(igraph)
##
## Attaching package: 'igraph'
## The following objects are masked from 'package:stats':
##
## decompose, spectrum
## The following object is masked from 'package:base':
##
## union
# SOME DEFINITIONS
# Graph: A collection of vertices (or nodes) and undirected edges (or ties), denoted g(V, E), where V is the vertex set and E is the edge set.
# Digraph (Directed graph): A collection of vertices (or nodes) and directed graphs
# Bipartite graph: Graph where all the nodes of a graph can be partitioned into two sets V1 and V2 such that for all edges in the graph connects and unordered pair where one vertex comes from V1 and the other from V2. Often called as an "affiliation graph" as bipartite graphs are used to represent people's affiliations to organizations or events.
# FROM GRAPHS TO PEOPLE AND RELATIONSHIPS
# The vertices of the graph represent the actors in the social system. These are usually individual people, but they could be households, geographical localities, institutions, or other social entities.
# The edges of the graph represents the relations between these entities (e.g., "is friends with" or "has sexual intercourse with" or "sends money to"). These edges can be undirected (e.g., "within 2 meters of"), or directed (e.g., "sends money to"), in the case of a digraph.
# Graphs (and digraphs) can be binary (i.e., presence/absence of a relationship) or valued (e.g, "groomed five times in the observation period", "sent $100").
# A graph (with no self-loops) with n vertices has n(n-1)/2 possible unordered pairs. This number is important for defining the density of a graph, i.e., the fraction of all possible relations that actually exists in a network.
# ENCODING A GRAPH BY HAND
# Create a small, undirected graph of five vertices from a vector of vertex pairs.
require(igraph)
g <- make_graph(c(1,2, 1,3, 2,3, 2,4, 3,5, 4,5), n=5, dir=FALSE)
plot(g, vertex.color="lightblue")
# Create a small graph using `graph_from_literal()`.
# Undirected edges are indicated with one or more dashes, `-`, `--`, etc. It doesn't matter how many dashes you use - you can use as many as you want to make your code more readable.
# The colon operator `:` links "vertex sets" - i.e., creates ties between all members of two groups of vertices.
g <- graph_from_literal(Fred-Daphne:Velma-Shaggy, Fred-Shaggy-Scooby)
plot(g, vertex.shape="none", vertex.label.color="black")
# Make directed edges using `-+` where the plus indicates the direction of the arrow, i.e., `A--+B` creates a directed edge from `A` to `B`.
# A mutual edge can be created using `+-+`.
# empty graph
g0 <- make_empty_graph(20)
plot(g0, vertex.size=10, vertex.label=NA)
# full graph
g1 <- make_full_graph(20)
plot(g1, vertex.size=10, vertex.label=NA)
# ring
g2 <- make_ring(20)
plot(g2, vertex.size=10, vertex.label=NA)
# lattice
g3 <- make_lattice(dimvector=c(10,10))
plot(g3, vertex.size=10, vertex.label=NA)
# tree
g4 <- make_tree(20, children=3, mode="undirected")
plot(g4, vertex.size=10, vertex.label=NA)
# star
g5 <- make_star(20, mode="undirected")
plot(g5, vertex.size=10, vertex.label=NA)
# Erods-Renyi Random Graph
g6 <- sample_gnm(n=100, m=50)
plot(g6, vertex.size=5, vertex.label=NA)
# Power Law
g7 <- sample_pa(n=100, power=1.5, m=1, directed=FALSE)
plot(g7, vertex.size=5, vertex.label=NA)
# Sometimes you want to plot two or more graphs together
# The disjoint union operator allows you to merge two graphs with different vertex sets
plot(g4 %du% g7, vertex.size=5, vertex.label=NA)
# Rewiring means rearranging the ties in a graph. It randomizes the connections between nodes without changing the degree distribution.
gg <- g4 %du% g7
gg <- rewire(gg, each_edge(prob=0.3))
plot(gg, vertex.size=5, vertex.label=NA)
# retain only the connected component
gg <- induced.subgraph(gg, subcomponent(gg,1))
plot(gg, vertex.size=5, vertex.label=NA)
# You can add arbitrary attributes to both vertices and edges. Generally, you do this to store information for plotting: colors, edge weights, names, etc.
# Some attributes are automatically created when you construct a graph object (e.g., 'name' or 'weight' if you load a weighted adjacency matrix)
# `V(g)` accesses vertex attributes.
# `E(g)` accesses edge attributes.
g4 # look at the structure
## IGRAPH 8383f90 U--- 20 19 -- Tree
## + attr: name (g/c), children (g/n), mode (g/c)
## + edges from 8383f90:
## [1] 1-- 2 1-- 3 1-- 4 2-- 5 2-- 6 2-- 7 3-- 8 3-- 9 3--10 4--11 4--12 4--13
## [13] 5--14 5--15 5--16 6--17 6--18 6--19 7--20
V(g4)$name <- LETTERS[1:20]
## see how it's changed
g4
## IGRAPH 8383f90 UN-- 20 19 -- Tree
## + attr: name (g/c), children (g/n), mode (g/c), name (v/c)
## + edges from 8383f90 (vertex names):
## [1] A--B A--C A--D B--E B--F B--G C--H C--I C--J D--K D--L D--M E--N E--O E--P
## [16] F--Q F--R F--S G--T
# Do some other stuff
V(g4)$vertex.color <- "Pink"
E(g4)$edge.color <- "SkyBlue2"
plot(g4, vertex.size=10, vertex.label=NA, vertex.color=V(g4)$vertex.color, edge.color=E(g4)$edge.color, edge.width=3)
# Most primatologists/behavioral ecologists probably have experience thinking in terms of adjacency matrices.
# An example of an adjacency matrix is the pairwise interaction matrices (e.g., agonistic or affiliative interactions) that we construct from behavioral observations.
# A very important potential gotcha: when you read data into R, it will be in the form of a data frame. Converting an adjacency matrix to an `igraph` graph object requires the data to be in the `matrix` class.
# Therefore, you need to coerce the data you read in by wrapping your `read.table()` in an `as.matrix()` command.
kids <- as.matrix(read.table("http://web.stanford.edu/class/ess360/data/strayer_strayer1976-fig2.txt", header=FALSE))
kid.names <- c("Ro","Ss","Br","If","Td","Sd","Pe","Ir","Cs","Ka",
"Ch","Ty","Gl","Sa", "Me","Ju","Sh")
colnames(kids) <- kid.names
rownames(kids) <- kid.names
g <- graph_from_adjacency_matrix(kids, mode="directed", weighted=TRUE)
lay <- layout_with_fr(g)
plot(g, edge.width=log2(E(g)$weight)+1, layout=lay, vertex.color="lightblue")
# Adjacency matrices are actually very inefficient.
# Most sociomatrices are quire sparce.
# Cost of an adjacency matrix increases as k^2.
# Edge lists are much more efficient.
# Various algorithms for detecting clusters of similar vertices - i.e., "communities".
# Use `fastgreedy.community()` to identify clusters in Kapferer's tailor shop and cokor the vertices based on their membership.
# `fastgreedy.community()` identifies four clusters.
# These clusters are listed as numbers in `fg$membership`.
# Use this vector to index vertex colors.
A <- as.matrix(
read.table(file="http://web.stanford.edu/class/ess360/data/kapferer-tailorshop1.txt",
header=TRUE, row.names=1)
)
G <- graph.adjacency(A, mode="undirected", diag=FALSE)
fg <- fastgreedy.community(G)
cols <- c("blue", "red", "black", "magenta")
plot(G, vertex.shape="none", vertex.label.cex=0.75, edge.color=grey(0.85), edge.width=1, vertex.label.color=cols[fg$membership])
# another approach to visualizing
plot(fg, G, vertex.label=NA)
davismat <- as.matrix(
read.table(file="http://web.stanford.edu/class/ess360/data/davismat.txt", row.names=1, header=TRUE))
(davismat)
## E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
## EVELYN 1 1 1 1 1 1 0 1 1 0 0 0 0 0
## LAURA 1 1 1 0 1 1 1 1 0 0 0 0 0 0
## THERESA 0 1 1 1 1 1 1 1 1 0 0 0 0 0
## BRENDA 1 0 1 1 1 1 1 1 0 0 0 0 0 0
## CHARLOTTE 0 0 1 1 1 0 1 0 0 0 0 0 0 0
## FRANCES 0 0 1 0 1 1 0 1 0 0 0 0 0 0
## ELEANOR 0 0 0 0 1 1 1 1 0 0 0 0 0 0
## PEARL 0 0 0 0 0 1 0 1 1 0 0 0 0 0
## RUTH 0 0 0 0 1 0 1 1 1 0 0 0 0 0
## VERNE 0 0 0 0 0 0 1 1 1 0 0 1 0 0
## MYRNA 0 0 0 0 0 0 0 1 1 1 0 1 0 0
## KATHERINE 0 0 0 0 0 0 0 1 1 1 0 1 1 1
## SYLVIA 0 0 0 0 0 0 1 1 1 1 0 1 1 1
## NORA 0 0 0 0 0 1 1 0 1 1 1 1 1 1
## HELEN 0 0 0 0 0 0 1 1 0 1 1 1 0 0
## DOROTHY 0 0 0 0 0 0 0 1 1 0 0 0 0 0
## OLIVIA 0 0 0 0 0 0 0 0 1 0 1 0 0 0
## FLORA 0 0 0 0 0 0 0 0 1 0 1 0 0 0
southern <- graph_from_incidence_matrix(davismat)
V(southern)$shape <- c(rep("circle", 18), rep("square", 14))
V(southern)$color <- c(rep("blue", 18), rep("red", 14))
plot(southern, layout=layout.bipartite)
# The incidence matrix is n*k, where n is the number of actors and k is the number of events
# Project the incidence matrix X into social space, creating a sociomatrix A, A=XX^T
# This transforms the n*k into an n*n sociomatrix.
# Sociomatrix
(f2f <- davismat %*% t(davismat))
## EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
## EVELYN 8 6 7 6 3 4 3 3 3
## LAURA 6 7 6 6 3 4 4 2 3
## THERESA 7 6 8 6 4 4 4 3 4
## BRENDA 6 6 6 7 4 4 4 2 3
## CHARLOTTE 3 3 4 4 4 2 2 0 2
## FRANCES 4 4 4 4 2 4 3 2 2
## ELEANOR 3 4 4 4 2 3 4 2 3
## PEARL 3 2 3 2 0 2 2 3 2
## RUTH 3 3 4 3 2 2 3 2 4
## VERNE 2 2 3 2 1 1 2 2 3
## MYRNA 2 1 2 1 0 1 1 2 2
## KATHERINE 2 1 2 1 0 1 1 2 2
## SYLVIA 2 2 3 2 1 1 2 2 3
## NORA 2 2 3 2 1 1 2 2 2
## HELEN 1 2 2 2 1 1 2 1 2
## DOROTHY 2 1 2 1 0 1 1 2 2
## OLIVIA 1 0 1 0 0 0 0 1 1
## FLORA 1 0 1 0 0 0 0 1 1
## VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
## EVELYN 2 2 2 2 2 1 2 1 1
## LAURA 2 1 1 2 2 2 1 0 0
## THERESA 3 2 2 3 3 2 2 1 1
## BRENDA 2 1 1 2 2 2 1 0 0
## CHARLOTTE 1 0 0 1 1 1 0 0 0
## FRANCES 1 1 1 1 1 1 1 0 0
## ELEANOR 2 1 1 2 2 2 1 0 0
## PEARL 2 2 2 2 2 1 2 1 1
## RUTH 3 2 2 3 2 2 2 1 1
## VERNE 4 3 3 4 3 3 2 1 1
## MYRNA 3 4 4 4 3 3 2 1 1
## KATHERINE 3 4 6 6 5 3 2 1 1
## SYLVIA 4 4 6 7 6 4 2 1 1
## NORA 3 3 5 6 8 4 1 2 2
## HELEN 3 3 3 4 4 5 1 1 1
## DOROTHY 2 2 2 2 1 1 2 1 1
## OLIVIA 1 1 1 1 2 1 1 2 2
## FLORA 1 1 1 1 2 1 1 2 2
gf2f <- graph_from_adjacency_matrix(f2f, mode="undirected", diag=FALSE, add.rownames=TRUE)
gf2f <- simplify(gf2f)
plot(gf2f)
# who is the most central?
cb <- betweenness(gf2f)
# plot (gf2f, vertex.size=cb*10)
plot(gf2f, vertex.label.cex=1+cb/2, vertex.shape="none")
# Project the matrix into event space
# This gives you the number of women at each event (diagonal) or mutually at 2 events.
(e2e <- t(davismat) %*% davismat)
## E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
## E1 3 2 3 2 3 3 2 3 1 0 0 0 0 0
## E2 2 3 3 2 3 3 2 3 2 0 0 0 0 0
## E3 3 3 6 4 6 5 4 5 2 0 0 0 0 0
## E4 2 2 4 4 4 3 3 3 2 0 0 0 0 0
## E5 3 3 6 4 8 6 6 7 3 0 0 0 0 0
## E6 3 3 5 3 6 8 5 7 4 1 1 1 1 1
## E7 2 2 4 3 6 5 10 8 5 3 2 4 2 2
## E8 3 3 5 3 7 7 8 14 9 4 1 5 2 2
## E9 1 2 2 2 3 4 5 9 12 4 3 5 3 3
## E10 0 0 0 0 0 1 3 4 4 5 2 5 3 3
## E11 0 0 0 0 0 1 2 1 3 2 4 2 1 1
## E12 0 0 0 0 0 1 4 5 5 5 2 6 3 3
## E13 0 0 0 0 0 1 2 2 3 3 1 3 3 3
## E14 0 0 0 0 0 1 2 2 3 3 1 3 3 3
ge2e <- graph_from_adjacency_matrix(e2e, mode="undirected", diag=FALSE, add.rownames=TRUE)
ge2e <- simplify(ge2e)
plot(ge2e)
