Metropolis-Hastings Algorithms for Estimating Betweenness Centrality in Large Networks
  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.
