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 thatHere λ 2 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 ν is given by νis the degree of vertex v, and d = 2m/n is the average degree. The parameter ν is an indicator of the variability of vertex degrees: 1 ≤ ν = O(n), with ν = 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 2 )) for d-regular graphs with expansion parameterized by the eigenvalue gap 1 − λ 2 . 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(Cv ) 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(Cv ) = C(n). Thus our result stated above for C(n) also gives general bounds for E(Cv ).
Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts the opinion of that neighbour. The voting is said to be completed when all vertices hold the same opinion. On many graph classes including regular graphs, irrespective of the expansion properties, pull voting requires Ω(n) expected time steps to complete, even if initially there are only two distinct opinions with the minority opinion being sufficiently large.In this paper we consider a related process which we call two-sample voting. In this process every vertex chooses two random neighbors in each step. If the opinions of these neighbors coincide, then the vertex revises its opinion according to the chosen sample. Otherwise, it keeps its own opinion. We consider the performance of this process in the case where two different opinions reside on vertices of some (arbitrary) sets A and B, respectively. Here, |A| + |B| = n is the number of vertices of the graph.We show that there is a constant K such that if the initial imbalance between the two opinions is ν 0 = (|A| − |B|)/n ≥ K (1/d) + (d/n), then with high probability two sample voting completes in a random d regular graph in O(log n) steps and the initial majority opinion wins. We also show the same performance for any regular graph, if ν 0 ≥ Kλ 2 , where λ 2 is the second largest eigenvalue of the transition matrix. In the graphs we consider, standard pull voting requires Ω(n) steps, and the minority can still win with probability |B|/n. Our results hold even if an adversary is able to rearrange the opinions in each step, and has complete knowledge of the graph structure.
a b s t r a c tWe study the cover time of multiple random walks on undirected graphs G = (V , E). We consider k parallel, independent random walks that start from the same vertex. The speedup is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of Ω(k) for several graphs; however, for many of them, k has to be bounded by O(log n). They also conjectured that, for any 1 ⩽ k ⩽ n, the speed-up is at most O(k) on any graph. We prove the following main results:• We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of Ω(k) on many graphs, even if k is as large as n. • We prove that the speed-up is O(k log n) on any graph. For a large class of graphs we can also improve this bound to O(k), matching the conjecture of Alon et al.• We determine the order of the speed-up for any value of 1 ⩽ k ⩽ n on hypercubes, random graphs and degree restricted expanders. For d-dimensional tori with d > 2, our bounds are tight up to logarithmic factors.• Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d-dimensional torus with d > 2 and hypercubes: there is a value T such that the speed-up is approximately min{T , k} for any 1 ⩽ k ⩽ n.
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