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 $d_{i,j}<~d_{i,k}$ 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 $d_{i,j}<~d_{i,l}$ 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.