Explaining diffusion in finite time: “diffusion centrality”

Many applied mathematicians like to put together their own indicators. In this paper, Banerjee and co-authors made a simple and interesting move that I want to make a note of.

They are interested in the diffusion of participation to microfinance in 43 Indian villages. In each of this case, a microfinance institution would move into the village and identify a person likely to drive participation in that village (they call this person “leader”): the local schoolteacher, or some other respected person. Before the bank moved in, they mapped out social networks in the village. The question they were interested in was: does the centrality of the person the bank talks to first influence the outcome in terms of participation? (Notice how this question brushes aside textbook economics, in that it assumes that participation in the microfinance program spreads across the social network rather than resulting from utility-maximizing decisions made in isolation in response to price signals. We are already in the future of economics.)

To test this hypothesis, you have to pick one measure of centrality. The authors do one better, and make up their own. They imagine a process in which one person is initially informed about the microfinance program. This person than tells her contacts about it with a certain probability p. This is iterated over time periods: in the first period only a fraction p of the initial person’s neighbors learn about the microfinance program, in the second period the information reaches a fraction of those guys neighbors etc. This results in a measure of centrality they call diffusion centrality:

 DC (p, T)=\sum_{t=1}^{T}{p\mathbf{g}^t}

Where g is the adjacency matrix. My gut reaction to this was: “this is supposed to capture the notion of being connected to central nodes – you are central if you are connected to people who are central, and so on, recursively – so why don’t they just use eigenvector centrality?” Indeed, for T large and p large, this measure converges to eigenvector centrality. But the way diffusion centrality is construed encodes the idea that this all happens in finite time, so that the initial information might not really reach every corner of the network. This makes plenty of sense, because we live in finite time and so do real-world diffusion processes. Statistical tests on the data, indeed, show that diffusion centrality is a better predictor of microfinance participation in villages than eigenvector centrality. The other centrality measures are not even contenders.

What I like about this is that it injects some dynamics in eigenvector centrality, which is essentially a topological (hence static) measure. Moreover, it does so with minimum computational fuss. Well done!

I learned this in yet another MOOC on networks I am taking, this one.

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