Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $\Delta$ of the input graph $G$ and hold when $k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx 1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on the constant $\alpha$ hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of $n$ and the goal is to obtain results in terms of the expected degree $d$. The following rapid mixing results for $G(n,d/n)$ hold with high probability over the choice of the random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where $\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain $O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in $d$. As in previous results for random graphs our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks

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

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs

Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p

Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average

In this work we show that for every $d < \infty$ and the Ising model defined on $G(n,d/n)$, there exists a $\beta_d > 0$, such that for all $\beta < \beta_d$ with probability going to 1 as $n \to \infty$, the mixing time of the dynamics on $G(n,d/n)$ is polynomial in $n$. Our results are the first polynomial time mixing results proven for a natural model on $G(n,d/n)$ for $d > 1$ where the parameters of the model do not depend on $n$. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than n \polylog(n). Our proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex $v$ of the graph has a neighborhood $N(v)$ of radius $O(\log n)$ in which the induced sub-graph is a tree union at most $O(\log n)$ edges and where for each simple path in $N(v)$ the sum of the vertex degrees along the path is $O(\log n)$. Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for $G(n,d/n)$ it applies for all external fields and $\beta < \beta_d$, where $d \tanh(\beta_d) = 1$ is the critical point for decay of correlation for the Ising model on $G(n,d/n)$.Comment: Corrected proof of Lemma 2.

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org