2,865 research outputs found
Sampling in Potts Model on Sparse Random Graphs
We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014].
Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 03(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs
Factor models on locally tree-like graphs
We consider homogeneous factor models on uniformly sparse graph sequences
converging locally to a (unimodular) random tree , and study the existence
of the free energy density , the limit of the log-partition function
divided by the number of vertices as tends to infinity. We provide a
new interpolation scheme and use it to prove existence of, and to explicitly
compute, the quantity subject to uniqueness of a relevant Gibbs measure
for the factor model on . By way of example we compute for the
independent set (or hard-core) model at low fugacity, for the ferromagnetic
Ising model at all parameter values, and for the ferromagnetic Potts model with
both weak enough and strong enough interactions. Even beyond uniqueness regimes
our interpolation provides useful explicit bounds on . In the regimes in
which we establish existence of the limit, we show that it coincides with the
Bethe free energy functional evaluated at a suitable fixed point of the belief
propagation (Bethe) recursions on . In the special case that has a
Galton-Watson law, this formula coincides with the nonrigorous "Bethe
prediction" obtained by statistical physicists using the "replica" or "cavity"
methods. Thus our work is a rigorous generalization of these heuristic
calculations to the broader class of sparse graph sequences converging locally
to trees. We also provide a variational characterization for the Bethe
prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Statistical Mechanics of Community Detection
Starting from a general \textit{ansatz}, we show how community detection can
be interpreted as finding the ground state of an infinite range spin glass. Our
approach applies to weighted and directed networks alike. It contains the
\textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the
modularity as defined by Newman and Girvan \cite{Girvan03} as special
cases. The community structure of the network is interpreted as the spin
configuration that minimizes the energy of the spin glass with the spin states
being the community indices. We elucidate the properties of the ground state
configuration to give a concise definition of communities as cohesive subgroups
in networks that is adaptive to the specific class of network under study.
Further we show, how hierarchies and overlap in the community structure can be
detected. Computationally effective local update rules for optimization
procedures to find the ground state are given. We show how the \textit{ansatz}
may be used to discover the community around a given node without detecting all
communities in the full network and we give benchmarks for the performance of
this extension. Finally, we give expectation values for the modularity of
random graphs, which can be used in the assessment of statistical significance
of community structure
Minimizing Unsatisfaction in Colourful Neighbourhoods
Colouring sparse graphs under various restrictions is a theoretical problem
of significant practical relevance. Here we consider the problem of maximizing
the number of different colours available at the nodes and their
neighbourhoods, given a predetermined number of colours. In the analytical
framework of a tree approximation, carried out at both zero and finite
temperatures, solutions obtained by population dynamics give rise to estimates
of the threshold connectivity for the incomplete to complete transition, which
are consistent with those of existing algorithms. The nature of the transition
as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional
explanatio
Sidorenko's conjecture, colorings and independent sets
Let denote the number of homomorphisms from a graph to a
graph . Sidorenko's conjecture asserts that for any bipartite graph , and
a graph we have where
and denote the number of vertices and edges of the graph and
, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs : for the complete graph on vertices, for a
with a loop added at one of the end vertices, and for a path on vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph . For instance,
for a bipartite graph the number of -colorings
satisfies
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
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