Neighborhood and Position: Local Centrality

Centrality - Local

One of the first definitions to be used for centrality measurement defines centrality as the number of ties of an actor with the reest of the actors. In the statement above, we actually define the social notion of “popularity” in a measurable manner. In graph terms then number of relations of a vertex is called its degree. Thus this particular centrality measure is called degree centrality

Let us take the network shown in Figure 1 into consideration (you can find information on how to generate the visual with R commands in the box below, using igraph library by Csardi and Nepusz, 2006). Upon looking at the graph visualization, we can see that d and b are the actors with the highest degree centrality values (3 for each).

# The small network in the example is generated with R commands
# In a real data analysis we would upload the network from the data file
library(igraph) #load the library
# the function below forms the network
g1 <- graph.formula(a--b, b--c, c--g, b--d, d--e, d--f)
V(g1)$size <- 25 # arrange the size of “Vertex” or nodes
V(g1)$color <- “white”
plot(g1) # Network visualization function

Figure 1.

A sample personal friendship network

It is easy to count the centralities in this network manually, but let us see how do we do it with a computer. The degree() function in the igraph library provides the degree centrality metric. This function computes the metric for each of the nodes in the graph and reports the node names and centrality measurements as a numeric vector whose rows are named with vertex/node names:

degree(g1)
## a b c g d e f 
## 1 3 2 1 3 1 1

I would like to emphasize that our definition here is quite local. If we think of the social network as a wide geographical area encompassing many individuals, this definition takes into consideration only the relations in the individual’s closest neighborhood. Therefore, it does not provide information on the position of the individual within the broader structure of the network. For instance, the social network visualization above, in a way, suggests that b is “slightly more central” than d. But the centrality measurement does not indicate such a difference: The centrality values are the same for both individuals. It would not be possible to notice this difference only by looking at the centrality values without visualization if we were dealing with a large social network. Despite this fact, local centrality was widely used for social network research and is still a good criterion for certain features of a social phenomenon. We will see other centrality measurements taking the general structure of the network into consideration in the next chapter.

Local centrality may be used for comparing different actors within the same social network. However, it is not suitable for making comparisons between social networks of different qualities or sizes. For example, the density of business partnership relations among a group of people might be different from that of friendship network between them. The sizes of these two networks are the same as they concern the same group, but it is not possible to make a centrality comparison. This is the main problem in this respect even though more comparable criteria were also tested (e.g. Freeman relative centrality measurement, Scott (2000)).