Coalescing Random Walks and Voting on Connected Graphs

In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let G = (V,E), be an undirected and connected graph, with n vertices and m edges. The coalescence time, C(n), is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs, and prove that C(n) = O ( 1 1− λ2 ( log n+ n ν )) . Here λ2 is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from e.g. lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if Partially supported by the Royal Society International Joint Project grant JP090592 ”Random Walks, Interacting Particles and Faster Network Exploration,” and the EPSRC grant EP/J006300/1 ”Random walks on computer networks.” A Preliminary version of the results in this paper was presented in the Proceedings of PODC 2012 [4]. Department of Informatics, King’s College London, UK Department of Computer Science, University of Salzburg, Austria Department of Economic Engineering, University of Kyushu, Fukuoka, Japan Department of Informatics, King’s College London, UK