Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees

The broadcasting models on trees arise in many contexts such as discrete mathematics, biology statistical physics and cs. In this work, we consider the colouring model. A basic question here is whether the root's assignment affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called "reconstruction problem". For a d-ary tree it is well known that d/ln (d) is the reconstruction threshold. That is, for k=(1+eps)d/ln(d) we have non-reconstruction while for k=(1-eps)d/ln(d) we have. Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, our focus is on the well-known Galton-Watson trees. This model arises naturally in many contexts, e.g. the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The aforementioned study focuses on Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial with parameters n and d/n, where d is fixed. Here we consider a broader version of the problem, as we assume general offspring distribution, which includes B(n,d/n) as a special case. Our approach relates the corresponding bounds for (non)reconstruction to certain concentration properties of the offspring distribution. This allows to derive reconstruction thresholds for a very wide family of offspring distributions, which includes B(n,d/n). A very interesting corollary is that for distributions with expected offspring d, we get reconstruction threshold d/ln(d) under weaker concentration conditions than what we have in B(n,d/n). Furthermore, our reconstruction threshold for the random colorings of Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for the random colourings of G(n,d/n)

A simple algorithm for sampling colourings of G(N,D/N) up to Gibbs Uniqueness threshold

Approximate random $k$-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,d/n), where d is a sufficiently large constant. We propose a novel efficient algorithm for approximate random k-colouring G(n,d/n) for any k>(1+\epsilon)d. To be more specific, with probability at least 1-n^{-\Omega(1)} over the input instances G(n,d/n) and for k>(1+\epsilon)d, the algorithm returns a k-colouring which is distributed within total variation distance n^{-\Omega(1)} from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a ``simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution

Spectral Independence Beyond Uniqueness using the topological method

We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. In our results, the parameters of the Gibbs distribution are expressed in terms of the spectral radius of the adjacency matrix of $G$, or that of the Hashimoto non-backtracking matrix. The analysis relies on new techniques that we introduce to bound the maximum eigenvalue of the pairwise influence matrix $\mathcal{I}^{\Lambda,\tau}_{G}$ for the two spin Gibbs distribution $\mu$. There is a common framework that underlies these techniques which we call the topological method. The idea is to systematically exploit the well-known connections between $\mathcal{I}^{\Lambda,\tau}_{G}$ and the topological construction called tree of self-avoiding walks. Our approach is novel and gives new insights to the problem of establishing spectral independence for Gibbs distributions. More importantly, it allows us to derive new -- improved -- rapid mixing bounds for Glauber dynamics on distributions such as the Hard-core model and the Ising model for graphs that the spectral radius is smaller than the maximum degree

MCMC sampling colourings and independent sets of G(n,d/n) near the uniqueness threshold

Sampling from the Gibbs distribution is a well studied problem in computer science as well as in statistical physics. In this work we focus on the k-colouring model and the hard-core model with fugacity \lambda when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed. We use the Markov Chain Monte Carlo method for sampling from the aforementioned distributions. In particular, we consider Glauber (block) dynamics. We show a dramatic improvement on the bounds for rapid mixing in terms of the number of colours and the fugacity for the corresponding models. For both models the bounds we get are only within small constant factors from the conjectured ones by the statistical physicists. We use Path Coupling to show rapid mixing. For k and \lambda in the range of our interest the technical challenge is to cope with the high degree vertices, i.e. vertices of degree much larger than the expected degree d. The usual approach to this problem is to consider block updates rather than single vertex updates for the Markov chain. Taking appropriately defined blocks the effect of high degree vertices diminishes. However devising such a block construction is a non trivial task. We develop for a first time a weighting schema for the paths of the underlying graph. Only, vertices which belong to "light" paths can be placed at the boundaries of the blocks. The tree-like local structure of G(n,d/n) allows the construction of simple structured blocks

Deterministic counting of graph colourings using sequences of subgraphs

In this paper we propose a deterministic algorithm for approximately counting the $k$-colourings of sparse random graphs $G(n,d/n)$. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$approximation of the logarithm of the number of $k$-colourings of $G(n,d/n)$ for $k\geq (2+\epsilon) d$ with high probability over the graph instances. Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA '06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation decay} to compute {\em deterministically} marginals of {\em Gibbs distribution}. We develop a scheme whose accuracy depends on {\em non-reconstruction} of the colourings of $G(n,d/n)$, rather than {\em uniqueness} that are required in previous works. This leaves open the possibility for our schema to be sufficiently accurate even for $k<d$. The set up for establishing correlation decay is as follows: Given $G(n,d/n)$, we alter the graph structure in some specific region $\Lambda$ of the graph by deleting edges between vertices of $\Lambda$. Then we show that the effect of this change on the marginals of Gibbs distribution, diminishes as we move away from $\Lambda$. Our approach is novel and suggests a new context for the study of deterministic counting algorithms

Local convergence of random graph colorings

Let $G=G(n,m)$ be a random graph whose average degree $d=2m/n$ is below the $k$-colorability threshold. If we sample a $k$-coloring $\sigma$ of $G$ uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} $d_c(k)$, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007]. We prove this conjecture for $k$ exceeding a certain constant $k_0$. More generally, we investigate the joint distribution of the $k$-colorings that $\sigma$ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem

Broadcasting with Random Matrices

Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem for the related distributions on the tree, and on the sparse random graph $G(n,d/n)$. Both cases, reduce naturally to studying broadcasting models on the tree, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution. In this context, we study the effect of the configuration at the root to that of the vertices at distance $h$, as $h\to\infty$. We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound which arises in the classic version of the reconstruction problem. Our results imply, as a special case, the reconstruction threshold for the well-known Edward-Anderson model of spin-glasses on the tree. Also, we extend our analysis to the setting of the Galton-Watson tree, and the random graph $G(n,d/n)$, where we establish the corresponding thresholds.Interestingly, for the Edward-Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established in [Guerra and Toninelli:2004], coincides with the reconstruction threshold. Compared to the classical Gibbs distributions, the spin-glasses have a lot of unique features. In that respect, their study calls for new ideas, e.g., we introduce novel estimators for the reconstruction problem. Furthermore, note that the main technical challenge in the analysis is the presence of (too) many levels of randomness. We manage to circumvent this problem by utilising recently proposed tools coming from the analysis of Markov chains