1 research outputs found
Global Majority Consensus by Local Majority Polling on Graphs of a Given Degree Sequence
Suppose in a graph vertices can be either red or blue. Let be odd. At
each time step, each vertex in polls random neighbours and takes
the majority colour. If it doesn't have neighbours, it simply polls all of
them, or all less one if the degree of is even. We study this protocol on
graphs of a given degree sequence, in the following setting: initially each
vertex of is red independently with probability , and
is otherwise blue. We show that if is sufficiently biased, then with
high probability consensus is reached on the initial global majority within
steps if , and steps
if . Here, is the effective minimum degree, the smallest
integer which occurs times in the degree sequence. We further show
that on such graphs, any local protocol in which a vertex does not change
colour if all its neighbours have that same colour, takes time at least
, with high probability. Additionally, we demonstrate
how the technique for the above sparse graphs can be applied in a
straightforward manner to get bounds for the Erd\H{o}s-R\'enyi random graphs in
the connected regime.Comment: Discrete Applied Mathematics 180:1-10 201