Is it true that centrality measures in SNA are indicative for most important vertices only?

Question

I read about the limitations of centrality measures on Wikipedia. It says that centrality measures are good only for identifying top most important nodes in a social network. Their relative values can not be used to measure how much one node is more important than the other. The values of less important nodes are not indicative of anything at all. This made me thinking if the thing I am trying to do makes sense.

What I have is a graph (1K nodes), which has edges that dynamically change in time. New edges appear, old ones disappear. Changes are more or less subtle, gradual.

With centrality measures, I wanted to observe the following two things:

1. Follow top 10 important nodes at the moment and how their centrality is changing over time. For example, is closeness centrality growing in the social network or not.

2. Follow selected 20 nodes which might not be most important ones. I wanted to observe how their centralities are changing over time.

As you see, I would not compare nodes between each other but rather with themselves. Is this approach valid? Also, is it valid to use average values of centralities of all nodes or top N nodes?

Answer 1

There are a lot of centrality measures that capture one or more properties of interest in the system described by a graph. Typically these measures are heuristically inspired, and the meaning of importance is system-dependent.

For example, in transport and communication networks, the concept of the shortest path is typically more relevant that amount of connections.

In social systems, is the same. Depends on the kind of interaction that you are modeling (there are several kinds of social interactions). Therefore, there is a lot of information that you need to extract from the system under study to complement the information that gives you the centrality measures.

In dynamical graphs, there are a lot of possibilities to implement centrality measures. A node can be relevant in a temporal window and lose its relevance over time and vice-versa.

About your comment:

Changes are more or less subtle, gradual.

It depends on several factors and the used centrality measure (properties of your interest). Imagine two dense graph/clusters of 500k nodes each one connected by one link. If you remove this link, e.g., the distances between nodes in the different clusters will be infinite. Therefore, some deletions and additions can change global properties (you can think in the small-world transition).

About your comment:

As you see, I would not compare nodes between each other but rather with themselves. Is this approach valid? Also, is it valid to use average values of centralities of all nodes or top N nodes?

The reason is that the centrality measures presented in Wikipedia are mainly focused on static graphs and not time-dependent graphs. But there are a lot of papers in centrality for dynamical networks/graphs.

Read: https://en.wikipedia.org/wiki/Temporal_network