Coalescing random walks and voting on graphs

Details Coalescing random walks and voting on graphs Coalescing random walks and voting on graphs

2012-04-17

arxiv.org/abs/1204.4106v1

Computer Science

Data Structures and Algorithms

18 pages

Scientific paper

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 general result is that C(n)= O(n/(A (1-lambda_2))), 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), 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. The result holds provided the maximum node degree is O(m^{1-epsilon}).

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