Network Centrality: Rob Cross Braintrust Keynote and Density

As an example of network-cluster-driven-behavior, last time I suggested a simple way to stereotype the work of Rob Cross. The first row of the table below, from his "Braintrust Keynote" presentation, was my Exhibit A:
The other rows of the above table deserve comment as well. Let's focus today on the third row, Centrality, with apologies to those who thought that my recent series on network centrality was finished.

In all my posts on centrality, I never actually described a mathematical formula for calculating it. There are quite a few reasonable ways to define centrality. See this post for links to a few of them. We see above that Cross's Braintrust Keynote describes centrality as the "average # of relationships per person." Unfortunately, this notion of centrality has nothing at all to do with what other people mean when they say "centrality."

First, a preliminary clarification: "Centrality" is most commonly used to describe a single node in a network, but it is also used to describe a global property of an entire network (much like "centralization" in the bottom row of the Braintrust Keynote table above). So we should be clear that "average # of relationships per person" is a global property of an entire network.

With that in mind, observe the following two networks that have exactly the same number of nodes, exactly the same number of edges, and hence exactly the same value of "centrality" or "average # of relationships per person":
I don't think too many people would describe the above two networks as having equal centrality, despite the Braintrust Keynote assertion.

It's a shame to equate "centrality" and "average # of relationships per person." They are two of my most favorite network metrics. I have devoted enough recent bandwidth to centrality to make clear my affinity for that metric. Soon, I will explain why I like "average # of relationships per person" as an alternative to density (top row of the Braintrust Keynote table) that is much less susceptible to the network size bias noted by Kathleen Carley.

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