39,549 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
The one-dimensional contact process: duality and renormalisation
We study the one-dimensional contact process in its quantum version using a
recently proposed real space renormalisation technique for stochastic
many-particle systems. Exploiting the duality and other properties of the
model, we can apply the method for cells with up to 37 sites. After suitable
extrapolation, we obtain exponent estimates which are comparable in accuracy
with the best known in the literature.Comment: 15 page
Spectral Clustering of Graphs with the Bethe Hessian
Spectral clustering is a standard approach to label nodes on a graph by
studying the (largest or lowest) eigenvalues of a symmetric real matrix such as
e.g. the adjacency or the Laplacian. Recently, it has been argued that using
instead a more complicated, non-symmetric and higher dimensional operator,
related to the non-backtracking walk on the graph, leads to improved
performance in detecting clusters, and even to optimal performance for the
stochastic block model. Here, we propose to use instead a simpler object, a
symmetric real matrix known as the Bethe Hessian operator, or deformed
Laplacian. We show that this approach combines the performances of the
non-backtracking operator, thus detecting clusters all the way down to the
theoretical limit in the stochastic block model, with the computational,
theoretical and memory advantages of real symmetric matrices.Comment: 8 pages, 2 figure
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
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