Computing the Probability Vectors for Random Walks on Graphs with Bounded Arboricity

Abstract

The problem of detecting dense subgraphs (\emph{communities}) in large sparse graphs is inherent to many real world domains like social networking. A popular approach of detecting these communities involves first computing the \emph{probability~vectors} for \emph{random~walks} on the graph for a fixed number of steps, and then using these probability vectors to detect the communities. Such an approach has been discussed by Latapy and Pons in \cite{latapypons}. They compute the probability vectors using simple matrix multiplication and define a measure of the structural similarity between vertices which they call \emph{distance}. Based on the probability vectors, they compute the distances between vertices and then based on these distances group the vertices into communities. Their algorithm takes $O(n^2\log n)$ time where $n$ is the number of vertices in the graph. We focus on the first part of the approach i.e. computation of the probability vectors for the random walks, and propose a more efficient algorithm (than matrix multiplication) for computing these vectors in time complexity that is linear in the size of the output