Fast Low-Cost Estimation of Network Properties Using Random Walks

Abstract We study the use of random walks as an efficient method to estimate global properties of large connected undirected graphs. Typical examples of the properties of interest include the number of edges, vertices, and triangles, and more generally, the number of small fixed subgraphs. We consider two methods based on first returns of random walks: (1) the cycle formula of regenerative processes and (2) weighted random walks with edge weights defined by the property under investigation. We review the theoretical foundations for these methods and indicate how they can be adapted for the general nonintrusive investigation of large online networks. The expected value and variance of the time of the first return of a random walk decrease with increasing vertex weight, so for a given time budget, returns to high-weight vertices should give the best property estimates. We present theoretical and experimental results on the rate of convergence of the estimates as a function of the number of returns of a random walk to a given start vertex. We made experiments to estimate the number of vertices, edges, and triangles for two test graphs.