Subtleties of Centrality

“Take me to your leader.”

Centrality is one of the most basic social network concepts across the galaxy. And yet even for us professional social network analysts, this basic concept seems surprisingly slippery.

For those of you who have never thought about centrality formally, let me introduce you to the slipperiness with a little game: Who is the most central person in the network below?

Did you guess (1)? This person has more direct connections than anyone else. Or perhaps you guessed (6)? This person has fewer direct connections than (1) but is more central in the sense of influencing network-wide communication as it transfers through him.

The above puzzle is just the beginning of the surprising subtleties of centrality. To help us explore more deeply, Steve Borgatti recently published a basic checklist of questions we should ask before measuring centrality in our networks. These questions break into two families, approximated below:

One: What is the fundamental commodity that flows through the network? Is the commodity like a physical object that can only be in one place at a time? Or does it generate copies of itself (like an idea) and so occupy many places at once?

Two: What kinds of trajectories does the commodity follow through the network? Can it sensibly visit the same person or relationship more than once (like a dollar bill) or does it proceed directly from source to destination?

After presenting his checklist, Borgatti then surveys the literature on centrality. Here we find a bewildering array of formulae with names like degree, closeness, betweenness, information, and eigenvalue. Borgatti explains how each centrality measure assumes a certain set of answers to his two-dimensional list of questions. The resulting table exposes how centrality measures developed with one set of assumptions in mind are often applied to networks where these assumptions completely break down. For example, betweenness centrality assumes a network where flow moves along the shortest possible path from source to destination (like a package), and yet it is commonly used to measure influence in scenarios where shortest paths are hardly the only ones that matter. (Rhetorical question: Is the shortest path between HIV and you the only one you care about? Or do you care about all possible paths?) Borgatti’s table also exposes gaping holes in our understanding of centrality—for example, there are no established centrality measures for gossip or support networks.

For more on centrality, see Borgatti’s “Centrality and network flow” and other excellent articles in Social Networks 27 (2005) 55-71.