160 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
Fast plurality consensus in regular expanders
Pull voting is a classic method to reach consensus among vertices with
differing opinions in a distributed network: each vertex at each step takes on
the opinion of a random neighbour. This method, however, suffers from two
drawbacks. Even if there are only two opposing opinions, the time taken for a
single opinion to emerge can be slow and the final opinion is not necessarily
the initially held majority.
We refer to a protocol where 2 neighbours are contacted at each step as a
2-sample voting protocol. In the two-sample protocol a vertex updates its
opinion only if both sampled opinions are the same. Not much was known about
the performance of two-sample voting on general expanders in the case of three
or more opinions. In this paper we show that the following performance can be
achieved on a -regular expander using two-sample voting. We suppose there
are opinions, and that the initial size of the largest and second
largest opinions is respectively.
We prove that, if ,
where is the absolute second eigenvalue of matrix and
is a suitable constant, then the largest opinion wins in steps with high probability.
For almost all -regular graphs, we have for some
constant . This means that as increases we can separate an opinion
whose majority is , whereas majority is required for
constant.
This work generalizes the results of Becchetti et. al (SPAA 2014) for the
complete graph
The Power of Two Choices in Distributed Voting
Distributed voting is a fundamental topic in distributed computing. In pull
voting, in each step every vertex chooses a neighbour uniformly at random, and
adopts its opinion. The voting is completed when all vertices hold the same
opinion. On many graph classes including regular graphs, pull voting requires
expected steps to complete, even if initially there are only two
distinct opinions.
In this paper we consider a related process which we call two-sample voting:
every vertex chooses two random neighbours in each step. If the opinions of
these neighbours coincide, then the vertex revises its opinion according to the
chosen sample. Otherwise, it keeps its own opinion. We consider the performance
of this process in the case where two different opinions reside on vertices of
some (arbitrary) sets and , respectively. Here, is the
number of vertices of the graph.
We show that there is a constant such that if the initial imbalance
between the two opinions is ?, then with high probability two sample voting completes in a random
regular graph in steps and the initial majority opinion wins. We
also show the same performance for any regular graph, if where is the second largest eigenvalue of the transition
matrix. In the graphs we consider, standard pull voting requires
steps, and the minority can still win with probability .Comment: 22 page
Quasi-Majority Functional Voting on Expander Graphs
Consider a distributed graph where each vertex holds one of two distinct
opinions. In this paper, we are interested in synchronous voting processes
where each vertex updates its opinion according to a predefined common local
updating rule. For example, each vertex adopts the majority opinion among 1)
itself and two randomly picked neighbors in best-of-two or 2) three randomly
picked neighbors in best-of-three. Previous works intensively studied specific
rules including best-of-two and best-of-three individually.
In this paper, we generalize and extend previous works of best-of-two and
best-of-three on expander graphs by proposing a new model, quasi-majority
functional voting. This new model contains best-of-two and best-of-three as
special cases. We show that, on expander graphs with sufficiently large initial
bias, any quasi-majority functional voting reaches consensus within
steps with high probability. Moreover, we show that, for any initial opinion
configuration, any quasi-majority functional voting on expander graphs with
higher expansion (e.g., Erd\H{o}s-R\'enyi graph with
) reaches consensus within with high
probability. Furthermore, we show that the consensus time is
of best-of- for
Biased Consensus Dynamics on Regular Expander Graphs
Consensus protocols play an important role in the study of distributed
algorithms. In this paper, we study the effect of bias on two popular consensus
protocols, namely, the {\em voter rule} and the {\em 2-choices rule} with
binary opinions. We assume that agents with opinion update their opinion
with a probability strictly less than the probability with which
update occurs for agents with opinion . We call opinion as the superior
opinion and our interest is to study the conditions under which the network
reaches consensus on this opinion. We assume that the agents are located on the
vertices of a regular expander graph with vertices. We show that for the
voter rule, consensus is achieved on the superior opinion in time
with high probability even if system starts with only agents
having the superior opinion. This is in sharp contrast to the classical voter
rule where consensus is achieved in time and the probability of
achieving consensus on any particular opinion is directly proportional to the
initial number of agents with that opinion. For the 2-choices rule, we show
that consensus is achieved on the superior opinion in time with
high probability when the initial proportion of agents with the superior
opinion is above a certain threshold. We explicitly characterise this threshold
as a function of the strength of the bias and the spectral properties of the
graph. We show that for the biased version of the 2-choice rule this threshold
can be significantly less than that for the unbiased version of the same rule.
Our techniques involve using sharp probabilistic bounds on the drift to
characterise the Markovian dynamics of the system
Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks
Consider the following process on a network: Each agent initially holds
either opinion blue or red; then, in each round, each agent looks at two random
neighbors and, if the two have the same opinion, the agent adopts it. This
process is known as the 2-Choices dynamics and is arguably the most basic
non-trivial opinion dynamics modeling voting behavior on social networks.
Despite its apparent simplicity, 2-Choices has been analytically characterized
only on networks with a strong expansion property -- under assumptions on the
initial configuration that establish it as a fast majority consensus protocol.
In this work, we aim at contributing to the understanding of the 2-Choices
dynamics by considering its behavior on a class of networks with core-periphery
structure, a well-known topological assumption in social networks. In a
nutshell, assume that a densely-connected subset of agents, the core, holds a
different opinion from the rest of the network, the periphery. Then, depending
on the strength of the cut between the core and the periphery, a
phase-transition phenomenon occurs: Either the core's opinion rapidly spreads
among the rest of the network, or a metastability phase takes place, in which
both opinions coexist in the network for superpolynomial time. The interest of
our result is twofold. On the one hand, by looking at the 2-Choices dynamics as
a simplistic model of competition among opinions in social networks, our
theorem sheds light on the influence of the core on the rest of the network, as
a function of the core's connectivity towards the latter. On the other hand, to
the best of our knowledge, we provide the first analytical result which shows a
heterogeneous behavior of a simple dynamics as a function of structural
parameters of the network. Finally, we validate our theoretical predictions
with extensive experiments on real networks
Discrete Incremental Voting
We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 (âdisagree stronglyâ), 2 (âdisagreeâ), . . ., 5 (âagree stronglyâ). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbourâs opinion, if different. For opinions drawn from a set {1, 2, . . ., k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by â1, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i †c †i + 1 for some integer i, then the asymptotic probability that opinion i wins is i + 1 â c, and the probability that opinion i + 1 wins is c â i. With high probability, the winning opinion cannot be other than i or i + 1. To contrast this, we show that for a path and opinions 0, 1, 2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.</p
Bounds on the Voter Model in Dynamic Networks
In the voter model, each node of a graph has an opinion, and in every round
each node chooses independently a random neighbour and adopts its opinion. We
are interested in the consensus time, which is the first point in time where
all nodes have the same opinion. We consider dynamic graphs in which the edges
are rewired in every round (by an adversary) giving rise to the graph sequence
, where we assume that has conductance at least
. We assume that the degrees of nodes don't change over time as one can
show that the consensus time can become super-exponential otherwise. In the
case of a sequence of -regular graphs, we obtain asymptotically tight
results. Even for some static graphs, such as the cycle, our results improve
the state of the art. Here we show that the expected number of rounds until all
nodes have the same opinion is bounded by , for any
graph with edges, conductance , and degrees at least . In
addition, we consider a biased dynamic voter model, where each opinion is
associated with a probability , and when a node chooses a neighbour with
that opinion, it adopts opinion with probability (otherwise the node
keeps its current opinion). We show for any regular dynamic graph, that if
there is an difference between the highest and second highest
opinion probabilities, and at least nodes have initially the
opinion with the highest probability, then all nodes adopt w.h.p. that opinion.
We obtain a bound on the convergences time, which becomes for
static graphs
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