Network Brownian Motion: A New Method to Measure Vertex-Vertex Proximity and to Identify Communities and Subcommunities

The networks considered here consist of sets of intercon- nected vertices, examples of which include social networks, technological networks, and biological networks. Two important issues are to measure the extent of proximity between vertices and to identify the community structure of a network. In this paper, the proximity index between two nearest-neighboring vertices of a network is measured by a biased Brown- ian particle which moves on the network. This proximity index integrates both the local and the global structural information of a given network, and it is used by an agglomerative hierarchical algorithm to identify the community structure of the network. This method is applied to several artificial or real-world networks and satisfying results are attained. Find- ing the proximity indices for all nearest-neighboring vertex pairs needs a computational time that scales as O(N 3 ), with N being the total number of vertices in the network.

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