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 (log⁴ n+n/v)). Here λ 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 required. The value of v is given by v = ΣvandV d₂(v)/(d₂n), where d(v) is the degree of vertex v, and d = 2m/n is the average degree. The parameter ? is an indicator of the variability of vertex degrees: 1 = ≤ = O(n), with v = 1 for regular graphs. Our general bound on C(n) holds for all connected graphs. This implies, for example, that C(n) = O(n/(1 - λ)) for d-regular graphs with expansion parameterized by the eigenvalue gap 1 - λ. The bound on C(n) given above is sublinear for some classes of graphs with skewed degree distributions. In the voter model, initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. Let E(C_v) be the expected time for voting to complete, that is, for a unique opinion to emerge. A system of coalescing particles, where initially one particle is located at each vertex, corresponds to the voter model in that E(C_v) = C(n). Thus our result stated above for C(n) also gives general bounds for E(C_v).