Metropolis-Hastings Algorithms for Estimating Betweenness Centrality in Large Networks

Abstract: Betweenness centrality is an important index widely used in different domains such as social networks, traffic networks and the world wide web. However, even for mid-size networks that have only a few hundreds thousands vertices, it is computationally expensive to compute exact betweenness scores. Therefore in recent years, several approximate algorithms have been developed. In this paper, first given a network $G$ and a vertex $r \in V(G)$, we propose a Metropolis-Hastings MCMC algorithm that samples from the space $V(G)$ and estimates betweenness score of $r$. The stationary distribution of our MCMC sampler is the optimal sampling proposed for betweenness centrality estimation. We show that our MCMC sampler provides an $(\epsilon,\delta)$-approximation, where the number of required samples depends on the position of $r$ in $G$ and in many cases, it is a constant. Then, given a network $G$ and a set $R \subset V(G)$, we present a Metropolis-Hastings MCMC sampler that samples from the joint space $R$ and $V(G)$ and estimates relative betweenness scores of the vertices in $R$. We show that for any pair $r_i, r_j \in R$, the ratio of the expected values of the estimated relative betweenness scores of $r_i$ and $r_j$ respect to each other is equal to the ratio of their betweenness scores. We also show that our joint-space MCMC sampler provides an $(\epsilon,\delta)$-approximation of the relative betweenness score of $r_i$ respect to $r_j$, where the number of required samples depends on the position of $r_j$ in $G$ and in many cases, it is a constant.