Degree centrality

What nodes have more ties?

degree( net )

## [1] 4 4 6 4 2

This is vector with the degree centrality for each node in the network. Degree centrality is also called Freeman degree centrality (Freeman 1978).

Betweeness centrality

Betweeness is related to connectivity and flow in a network (Freeman 1978; Borgatti and Everett 2006). This measure requires that you know the network structure of the entire network.

betweenness( net , gmode ='graph' )

## [1] 0 0 4 3 0

Node 3 is on the shortest path between 4 pairs of nodes:

These nodes will only be able to ‘communicate’ via 3

Brokers

A cutpoint is a node that when removed results in the network having more components. In simpler terms, a cutpoint is a node that connects otherwise dissconnected nodes

cutpoints( net , mode = "graph")

## [1] 3 4

The extent to which nodes broker between other nodes has been the fucus of a large part of Ron Burt’s career. He proposes some elaborations for measuring bokerness in Burt (2009) called constraint and effective size. These are local measures that do not, as opposed to cutpoints and betweeness, require that you know the network structure of the entire network.

Triad census

The triad census tabulates different types of triangles or tripples (???; Holland and Leinhardt 1972)

triad.census( net , mode = "graph")

##      0 1 2 3
## [1,] 0 6 3 1

There are 6 tripples, with one tie. For example the subgraph consisting of nodes 1,4 , and 5.

plot(as.network( ADJ[c(1,4,5),c(1,4,5)] , directed = FALSE),
     vertex.cex = degree(net)[c(1,4,5)],
     vertex.col = c('red','blue','grey','green','yellow')[c(1,4,5)])

Open triad: There are 3 tripples, with two ties. For example the subgraph consisting of nodes 3, 4 , and 5.

plot(as.network( ADJ[c(3,4,5),c(3,4,5)] , directed = FALSE),
     vertex.cex = degree(net)[c(3,4,5)],
     vertex.col = c('red','blue','grey','green','yellow')[c(3,4,5)])

Closed triad: There is 1 tripple, with three ties. For example the subgraph consisting of nodes 1, 2 , and 3.

plot(as.network( ADJ[c(1,2,3),c(1,2,3)] , directed = FALSE ),
     vertex.cex = degree(net)[c(1,2,3)],
     vertex.col = c('red','blue','grey','green','yellow')[c(1,2,3)] )

Cliques

A \(k\)-clique is a maximally connected subgraph. This means that a \(k\)-clique is a network with \(k\) nodes that are all connected to each other. In net nodes 5 and 4 consitute a 2-clique and nodes 1, 2, and 3 constitute a 3-clique. Like the triad census, we can calulate a clique census for a graph

cc <- clique.census( net , mode = "graph")

The object cc is of class

class(cc)

## [1] "list"

The list has the following objects

names(cc)

## [1] "clique.count" "cliques"

The object clique.count lists the membership of nodes in cliques

cc$clique.count

##   Agg 1 2 3 4 5
## 1   0 0 0 0 0 0
## 2   2 0 0 1 2 1
## 3   1 1 1 1 0 0

A list of lists of cliques and their members is provided in

cc$cliques

## [[1]]
## NULL
## 
## [[2]]
## [[2]][[1]]
## [1] 4 5
## 
## [[2]][[2]]
## [1] 3 4
## 
## 
## [[3]]
## [[3]][[1]]
## [1] 1 2 3

Note that subgraphs of cliques are not listed. For example, the 2-cliques with nodes 1 and 2 is not listed as both are part of the larger 3-clique.