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This link describes betweenness: "Betweenness centrality measures the extent to which a vertex lies on paths between other vertices."

This link describes brokerage: "Brokerage is a state or situation in which an actor connects otherwise unconnected actors or fills gaps or network holes in the social structure."

They are evidently described very similarly and many publications or descriptions of one use the other term synonymously. This link even acknowledges that many people define brokerage by betweenness.

However, the popular network graph package igraph has a function for estimating the betweenness centrality here. Meanwhile, another popular (but less so) package called sna has a function for estimating the brokerage of a node here. Brokerage roles are described in different categories such as coordinator, itinerant, gatekeeper, representative, and liason. I have estimated this brokerage function and betweenness function on an undirected graph and I find they provide slightly different values. They're very correlated, but still somewhat different.

So what gives? My best guess has been that brokerage and betweenness are more or less the same but brokerage can refer to some betweenness-type roles in a directed network graph. But then why would I be getting different values between brokerage and betweenness in an undirected network graph?

Any feedback here would be greatly appreciated.

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The brokerage measure introduced by Gould and Fernandez (referenced in the sna documentation) is a constrained version of betweenness centrality. Betweenness centrality counts shortest paths between all nodes. Gould-Fernandez brokerage only counts paths between nodes assigned to different groups/classes, for different combinations of group membership among triads (giving rise to the gatekeeper, coordinator, etc roles) and only (if memory serves) two-step paths. Hence they are correlated, but not perfectly so. My experience is that the correlation can be fairly high, despite the apparent conceptual distinction.

See: www.jstor.org/stable/270949

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