## Eigenvector Measures of Centrality

I’m working with the social networks in the Longitudinal Study of Adolescent Health.  Students are asked to name their five best male friends and five best female friends.  I’m interested in something like a measure of popularity.  In-degree, the number of times others nominate you as their friend, is a simple measure, but I think I can do a bit better if I can capture the intuition that people with popular friends are themselves more popular.  This is one potential use of eigenvector measures of centrality.  In working with such measures, I’m learning a thing or two.  For example, the weight parameter can matter a lot.

I used the igraph package (see p. 8), in R to calculate Bonacich’s alpha measure of centrality for directed networks.  The default weight (variable: acent) is 1.  I compared this measure to in-degree (idgx2) and out-degree (odgx2) with a matrix of scatterplots and was a bit surprised not to see a clear positive relationship with in-degree.

When I tried alpha weights of 0.2 and 0.4  I found fairly strong non-linear relationships due to a handful of outliers.  While I think different alpha weights are worth exploring empirically, I’m inclined to emphasize ones which a positive monotonic relationship with in-degree.  The reason is that, to me, in-degree itself seems like a fairly good measure of popularity or social prominence.  I feel that moving to a measure quite different from in-degree requires justification in the form of strong theory or empirics.  I lack both.

In other contexts though, higher or lower (including negative) alpha weights might be justified.  For more on applying these measures to social networks, I recommend the work of Phillip Bonacich.

### 4 Responses to Eigenvector Measures of Centrality

1. Eric says:

Having sort of read a Bonacich paper from 2001, I think you are not using the right values of alpha. I think you want alpha times the largest eigenvalue of the adjacency matrix to be strictly less than one. Given the setup you described, if you are using a 0-1 adjacency matrix with 10 neighbors, your largest eigenvalue will be exactly 10. So, alpha=0.1 is the only “good” choice you looked at, and even it is a little bit too high. The relationship between in-degree and centrality looks to be pretty linear when alpha=0.1.

Essentially, I think the issue is that when I give my status away to people, it should total less than I have. Otherwise there is an increasing amount of status at every time, most of the statuses go to infinity, and the eigenvector solution is probably garbage, although I’m not sure.

This stuff is basically the coolest thing ever!

2. Michael Bishop says:

You know what you are talking about sir. And I think I’m starting to as well 🙂 Thanks for the intuition.

3. Michael Bishop says:

My friend, Matt Rocklin, just wrote a wikipedia entry on alpha centrality

4. […] of alpha-centrality (which my co-authors and I used to measure Hollywood […]