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July 15

A Few Notes on Centrality and Flow
Stephen P. Borgatti
University of Kentucky, USA
Centrality is often described as measuring the “importance” of a node in a
network. Dozens of measures have been proposed in the literature, most of
which can be characterized quite directly in terms of a node’s involvement
in the walk structure of a network (e.g., the average distance from the node
to all other nodes; the number of walks of all lengths emanating from a
node, weighted inversely by their length). Under a general rubric of social
capital, measures of centrality are often used to predict node outcomes,
such as career success. Unfortunately, almost none of these measures are
explicitly derived from any kind of model of what is happening in the
network and how these processes lead to the outcome state of each node. It
is true that it has been shown that some centrality measures implicitly
contain (or are consistent with) underlying models of network flows. For
example, the formula for betweenness centrality yields precisely the
expected values of the number of times that a token flowing through the
network will reach a given node, given that the token takes only shortest
paths, can only be in one place at one time, and chooses randomly between
equally short paths. The problem is, this implicit model virtually never
makes sense in the empirical settings where these measures are employed. For
example, in the management field, we commonly study flows of information
among members of an organization. Yet information does not travel only along
shortest paths. Indeed, information does not try to get anywhere in
particular, and no mind is guiding it, through intermediaries, to a specific
target. It also does not so much “move” from node to node as “copy” – i.e.,
it can be in more than place at a time. There are several other models that
are implicit in the set of extant centrality measures, but it is safe to say
that they too are inappropriate for the vast majority of empirical settings
social scientists wish to model. The objective of this paper is to explore
new measures that are based on more plausible flow processes.

