Network landscape from a Brownian particle’s perspective

Given a complex biological or social network, how many clusters should it be decomposed into? We define the distance d(i,j) from node i to node j as the average number of steps a Brownian particle takes to reach j from i. Node j is a global attractor of i if d(i,j)< or =d(i,k) for any k of the graph; it is a local attractor of i if j in E(i) (the set of nearest neighbors of i) and d(i,j)< or =d(i,l) for any l in E(i). Based on the intuition that each node should have a high probability to be in the same community as its global (local) attractor on the global (local) scale, we present a simple method to uncover a network's community structure. This method is applied to several real networks and some discussion on its possible extensions is made.