Why ERGMs?

The fundamental problem of network analysis:

• Is the network we observe an Erdos-Renyi network?

  • In an Erdos-Renyi network, the probability of each edge is independent of that of other edges.

• If not, what are the endogeneous network features that define our network?

Example: Friendship Netwok

library(igraph)
library(sna)
data(coleman) #Use friendship data
coleman<-coleman[1,,]
#convert to an -igraph- object, we'll treat it as a directed graph for now:
coleman<-graph_from_adjacency_matrix(coleman, mode="directed", diag=FALSE) 
edge_density(coleman)

## [1] 0.04623288

reciprocity(coleman) #Note that -igraph- default is an undirected graph

## [1] 0.5102881

Friendship Data

• What are some endogenous network features of this network?

Erdos-Renyi Networks

• In order to answer whether an observed network is different from a random network, it would help to know what a random network would look like.

• In a random network, all edges have the same probability of realization, $p$. Moreover, the probability of realization of edge $i$, $p_i$, does not depend on $p_j$.

Simulate A Random Graph to Compare to the Coleman Data

Need:

• number of nodes $N$ in the friendship network
• the probability that each two nodes are connected $p$

vcount(coleman) #returns the number of vertices

## [1] 73

gsize(coleman) #returns the number of edges

## [1] 243

#or
summary(coleman)

## IGRAPH fe3973e DN-- 73 243 -- 
## + attr: name (v/c)

Simulate A Random Graph to Compare to the Coleman Data

• Let $p$ denote the probability that any two vertices are connected by an edge. Then, we can calculate the value of $p$ in the friendship network as the number of observated edges over the number of all possible edges.

  • With 73 vertices, we have $73\ast 72/2=2628$ possible undirected edges or $73\ast 72=5256$ directed edges, as each vertex can connect to each other vertex, but there are no self-loops. (Since the friendship network is directed, let's focus on simulating a directed network.)

  • Then $p=243/5256=0.046$. Does this value seem high or low? Note that $p$ is the clustering coefficient of a random network with a given number of nodes and edges.

Simulate A Random Graph (Continued)

set.seed(45765) #since a simulation involves randomness, set the seed for 
#reproducibility.
#Step i--start with a matrix of 73 unconnected nodes.
N=73 #Set the number of nodes:
rnet<-matrix(0, nrow=N,ncol=N) 
#Step ii:
p<-243/5256
for (i in 1:N) {
  for (j in 1:N){
    if (i!=j) {
    rnet[i,j]=as.numeric(runif(1)<p)
    }}}

Check Our Work

summary(g<-graph_from_adjacency_matrix(rnet, mode="directed", weighted=NULL))

## IGRAPH fe59946 D--- 73 232 --

edge_density(coleman)

## [1] 0.04623288

reciprocity(coleman) #Note that -igraph- default is an undirected graph

## [1] 0.5102881

edge_density(g)

## [1] 0.04414003

reciprocity(g) #Note that -igraph- default is an undirected graph

## [1] 0.03448276

Visualize

Summarize the Observed and Simulated Networks

library(statnet)
data(coleman)
coleman<- as.network.matrix(coleman[1,,],    matrix.type='adjacency', directed=TRUE)
rnet<- as.network.matrix(rnet, matrix.type='adjacency',    directed=TRUE)
summary(coleman~ edges+idegree(6)+ triangles+ mutual+ostar(2)+istar(2))

##    edges idegree6 triangle   mutual   ostar2   istar2 
##      243        5      460       62      383      542

summary(rnet~edges+idegree(6)+ triangles+ mutual+ostar(2)+istar(2))

##    edges idegree6 triangle   mutual   ostar2   istar2 
##      232        5       37        4      343      345

Your Turn

1. Simulate a random network that we could compare to the Sampson data.

2. Plot the two side-by-side

3. Use summary to further explore the differences between the two.

4. Based on this analysis, what model specification would you propose?