text: In this paper, we consider lightweight decentralised algorithms for achieving consensus in distributed systems. Each member of a distributed group has a private value from a fixed set consisting of, say, two elements, and the goal is for all members to reach consensus on the majority value. We explore variants of the voter model applied to this problem. In the voter model, each node polls a randomly chosen group member and adopts its value. The process is repeated until consensus is reached. We generalize this so that each member polls a (deterministic or random) number of other group members and changes opinion only if a suitably defined super-majority has a different opinion. We show that this modification greatly speeds up the convergence of the algorithm, as well as substantially reducing the probability of it reaching consensus on the incorrect value.
  @cite_2: In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon. Let G=(V,E), be an undirected, connected n vertex graph with m edges. Let C(n) be the expected time for all particles to coalesce, when initially one particle is located at each vertex of an n vertex graph. We study the problem of bounding the coalescence time C(n) for general classes of graphs. Our main result is that C(n)= O(1 (1-lambda_2))*((log n)^4 +n A)), where lambda_2 is the absolute value of the second largest eigenvalue of the transition matrix of the random walk, A= (sum d^2(v)) (d^2 n), d(v) is the degree of vertex v, and d is the average node degree. The parameter A is an indicator of the variability of node degrees. Thus 1 <= A =O(n), with A=1 for regular graphs.
summary: There is a well known duality between the voter model and coalescing random walks, which can be exploited to obtain bounds on the time to reach consensus in the voter model. On the complete graph, it shows that the expected time to reach consensus scales linearly in the number of nodes. Recent work in @cite_2 has used this approach to derive bounds on the consensus time in general graphs in terms of the number of nodes, the variance of node degrees, and the spectral gap of the transition probability matrix for the random walk associated with the voter model. Unfortunately, the duality with random walks does not extend to the generalised voter model considered in this paper. There also doesn't seem to be a natural martingale associated with it; in the voter model, this martingale is intimately related to the dual random walk. Hence, we need a quite different approach for our model, which doesn't extend to general graphs.