Non-Local Centrality

Evaluating An Actor’S Network Position Features With Respect To General Structure Of Network

Each actor within a social network has different structural features in terms of their social environment. Some individuals/actors have more friends or other types of connections, immediate circles of some actors form a closely-tied group and others have relationships with individuals from other “circles”.

We have analyzed Krackhardt’s network of advice-giving relations among company managers before (Krackhardt, 1987). Let us look at the distribution of local centralities in this network. We can generate the degree distribution histogram in Figure 1 by using the following commands:

require(NetData)
data(kracknets)
require(igraph)
g<-graph_from_edgelist(
as.matrix(
advice_data_frame[advice_data_frame[3]==1,1:2])) 
hist(degree(g,mode=”out”),xlab=”Out-degree centrality”,ylab=”Frequency”,main=”Histogram”)

Figure 1.

Distribution of local centrality within Krackhardt’s network of company managers.

The degree values can also be plotted, which in this example forms some groups nicely, as seen in Figure 2, with the following code:

require(NetData)
data(kracknets)
require(igraph)
g<-graph_from_edgelist(
as.matrix(
advice_data_frame[advice_data_frame[3]==1,1:2])) 
plot(degree(g,mode=”out”),type=”l”)

Figure 2.

Plot of local centrality within Krackhardt’s network of company managers

The out-degree centrality values used in this directed example indicate the popularity of the actor in terms of “advice-giving”. This distribution roughly shows us that the individuals within the social group form four different clusters as far as degree centrality goes. Each cluster corresponds to certain roles in our social phenomenon (community leader, popular person, marginal person etc.). For instance, a few individuals with a high centrality value between 15-20 are community leaders whose advice is sought by many. Those with a centrality value between 10-15 have a relatively “normal” popularity. As for those with a value between 5-10, they are in the periphery in terms of “advice-giving”. There is a group with even lower centrality values, meaning that nobody asks their opinion. These individuals are socially isolated.

It is usually beneficial to analyze measures regarding individuals as a whole. Let us place the ego network density and local centrality together. The ego network density is calculated as the density of an ego’s (an actor or a node) network without the ego. The ego network is a first step to evaluating the structure that surrounds an actor. It is defined as the ties between the actors to whom the selected ego actor has connections. In other words it is the friendships between your friends, if you think of yourself as the ‘ego’ in this definition. The igraph library provides some routines to produce ego networks. The make_ego_graph() functions produces ego graphs for all nodes in the graph. Let us visualize one such ego network, for node 6, the plot from the code below is shown in Figure 3:

require(NetData)
data(kracknets)
require(igraph)
g<-graph_from_edgelist(
as.matrix(
advice_data_frame[advice_data_frame[3]==1,1:2])) 
g1 <- make_ego_graph(g,1)[[6]]
plot(g1)

Figure 3.

Extraction of ego network for node 6

Now we can proceed to calculate density of ego networks. What we will do here is to examine ego network densities together with degree centralities. These data are visualized in Figure 4. We use the sapply() function in R to compute ego network densities for all nodes, as follows: