Freeman's Classic Paper

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Centrality Measures

FINALLY! Yes, in today's SNA Wednesday, we will go over Freeman's 1977 classic paper on network centrality measures:

A Set of Measures of Centrality Based on Betweenness http://moreno.ss.uci.edu/23.pdf

Freeman sets the scene with a bit of history and motivation as to why centrality is an important property of (social) networks:

The earliest intuitive conception of point centrality in communication was based upon the structural property of betweenness. According to this view, a point in a communication network is central to the extent that it falls on the shortest path between pairs of other points.

. . .

... the importance of this conception of point centrality is in the potential of a point forcontrol of information flow in the network. Positions are viewed as structurally central to the degree that they stand between others and can therefore facilitate, impede or bias the transmission of messages.

In the 1^st installment discussing network centrality measures, we conceptualized centrality measures in two ways:

Holistically: From every- / no- one's perspective
Egotistically: From a particular node's perspective

The 2 measures introduced in that 1^st post and the 2 in the interlude were decidedly egotistic:

Instead, in his paper, Freeman is thinking holistically, as is shown by the focus on betweenness. While the difference is subtle, betweenness calculates an Ego's centrality measure based more on its position in the network, the paths it participates in, and thus its ability to control flow between nodes. This is what he means when he speaks about structure and position in the quote above.

Freeman comes up with a measure that begins to take into account

Ego's centrality by its participation in paths

between pairs of nodes in the network.

So, lets describe it:

Freeman

. . .

Betweenness requires Ego to lie on a geodesic (shortest path) between a node-pair, say (j, k). If there is only 1 geodesic between (j, k) and Ego does not lie on it, then Ego's betweenness is increased by 0. If it does, it is increased by 1. When Ego lies on a geodesic lets note the number of these as:

g_jk(ego)

It is not unlikely for there to be multiple geodesics between node-pairs in a (social) network. Lets note the number of these as:

g_jk

Ego's betweenness for this node-pair is thus the proportion or probability that it lies on any of its geodesics, ie. we must normalize the number of geodesics Ego lies on. Lets note this as:

g_jk(ego) / g_jk

We must then follow this process and sum for all node-pairs in the (social) network. Lets note this as:

∑ g_jk(ego) / g_jk

thus:

Betweenness Centrality

C^B_i(g) : ∑ g_jk(i) / g_jk

g: the network under analysis
i: the node under analysis, ie. Ego
g_jk: the number of geodesics between nodes j and k, where j and k are all node-pairs in g excluding i/ego
g_jk(i): the number of geodesics between nodes j and k that node i lies on

https://en.wikipedia.org/wiki/Betweenness_centrality
https://reference.wolfram.com/language/ref/BetweennessCentrality.html

Lets return to our 4 node network we have been analyzing so far and calculate Ego's betweenness centrality:

Betweenness centrality for each is:

Well, that was anticlimactic! The "problem" is that this is a pretty basic network, without multiple paths between nodes.

Lets try it with this "classic" network:

Hmm, still not terribly exciting.

Think about betweenness .. what is it about? The fact that nodes of type C have 0 betweenness should help.

It is a measure of how a nodes participates in paths in the network.

Lets modify the "classic" network just a bit:

Ah!! Ok .. it should be getting clearer. We added 2 nodes -- D -- in the same position within the network, and we removed an edge between the two C type nodes on the right (thus no longer a clique) changing the structure and hence impacting the betweenness of the B type node on the right, now indicated with B'.

What was the outcome? What is there to learn?

Notice that nodes of type D are part of more paths between node-pairs than type B, even though they look to be in basically the same place in the network, the measure highlights a subtle structural different yielding a very useful distinction that the other centrality measures do not.

B' is part of 1 more path than B due to the missing edge/link/relationship, which we will see below is only hinted at indirectly by differences in other centrality measures for the C and C' type nodes.

And node type A still sits "atop" the network bisecting it, participating in all paths between both "sides" of the network.

So, lets sum up all centrality measures so far for this network:

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