2 research outputs found
Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models
This paper is concerned with voting processes on graphs where each vertex
holds one of two different opinions. In particular, we study the
\emph{Best-of-two} and the \emph{Best-of-three}. Here at each synchronous and
discrete time step, each vertex updates its opinion to match the majority among
the opinions of two random neighbors and itself (the Best-of-two) or the
opinions of three random neighbors (the Best-of-three). Previous studies have
explored these processes on complete graphs and expander graphs, but we
understand significantly less about their properties on graphs with more
complicated structures.
In this paper, we study the Best-of-two and the Best-of-three on the
stochastic block model , which is a random graph consisting of two
distinct Erd\H{o}s-R\'enyi graphs joined by random edges with density
. We obtain two main results. First, if and
is a constant, we show that there is a phase transition in with
threshold (specifically, for the Best-of-two, and
for the Best-of-three). If , the process reaches consensus
within steps for any initial opinion
configuration with a bias of . By contrast, if , then there
exists an initial opinion configuration with a bias of from which
the process requires at least steps to reach consensus. Second,
if is a constant and , we show that, for any initial opinion
configuration, the process reaches consensus within steps. To the
best of our knowledge, this is the first result concerning multiple-choice
voting for arbitrary initial opinion configurations on non-complete graphs
Phase Transition of a Non-Linear Opinion Dynamics with Noisy Interactions
International audienceIn several real \emph{Multi-Agent Systems} (MAS), it has been observed that only weaker forms of\emph{metastable consensus} are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents. In this work, we take a step towards the investigation of metastable consensus for complex (non-linear) \emph{opinion dynamics} by considering the famous \undecided dynamics in the binary setting, which is known to reach consensus exponentially faster than the \voter dynamics. We propose a simple form of uniform noise in which each message can change to another one with probability and we prove that the persistence of a \emph{metastable consensus} undergoes a \emph{phase transition} for . In detail, below this threshold, we prove the system reaches with high probability a metastable regime where a large majority of agents keeps supporting the same opinion for polynomial time. Moreover, this opinion turns out to be the initial majority opinion, whenever the initial bias is slightly larger than its standard deviation.On the contrary, above the threshold, we show that the information about the initial majority opinion is ``lost'' within logarithmic time even when the initial bias is maximum.Interestingly, using a simple coupling argument, we show the equivalence between our noisy model above and the model where a subset of agents behave in a \emph{stubborn} way