An index for the “local” influence in social networks

In this paper we define a novel index of node centrality in social networks that extends the recently proposed Harmonic Influence Centrality (HIC) and that we call Local-Harmonic Influence centrality (L-HIC). Indeed, when compared with the HIC, our index shows a local nature that rules out one pathological behavior of the HIC. Similarly to the HIC, the L-HIC can be approximated by a distributed message passing algorithm that is inspired by an analogy between electrical and social networks on tree graphs. We prove a result that guarantees convergence on graphs containing at most one cycle.