Consensus on the Initial Global Majority by Local Majority Polling for a Class of Sparse Graphs

نویسندگان

  • Mohammed Amin Abdullah
  • Moez Draief
چکیده

We study the local majority protocol on simple graphs of a given degree sequence, for a certain class of degree sequences. We show that for almost all such graphs, subject to a sufficiently large bias, within time A logd logd n the local majority protocol achieves consensus on the initial global majority with probability 1− n−Ω((logn)ε), where ε > 0 is a constant. A is bounded by a universal constant and d is a parameter of the graph; the smallest integer which is the degree of Θ(n) vertices in the graph. We further show that under the assumption that a vertex v does not change its colour if it and all of its neighbours are the same colour, any local protocol P takes time at least (1 − o(1)) logd logd n, with probability 1 − e ) on such graphs. We further show that for almost all d-regular simple graphs with d constant, we can get a stronger probability to convergence of initial majority at the expense of time. Specifically, with probability 1−O ( c ε ) , the local majority protocol achieves consensus on the initial majority by time O(log n). Finally, we show how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erdős–Renyi random graphs in the connected regime.

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تاریخ انتشار 2012