Sampling in Potts Model on Sparse Random Graphs

Mean-field monomer-dimer models, on sparse random graphs or on the complete graph, can be considered as an approximation of finite-dimensional physical models involving particles of different sizes. On the other hand they have a particular interest for the emerging applications to Computer Science and Social Sciences, since the real-world networks are often modelled by particular families of random graphs. We give a rigorous proof of Zdeborova-Mezard's picture of the monomer-dimer model with pure hard-core interaction on sparse random graphs. As shown by Heilmann and Lieb, the hard-core interaction is not sufficient to cause a phase transition in monomer-dimer models. We study monomer-dimer models on the complete graph and in particular we add an attractive interaction to the hard-core one. We provide the solution of this model, showing that a phase transition occurs. The critical exponents are the standard mean-field ones and the central limit theorem breakdowns. Finite-dimensional monomer-dimer models (and more general hard-rods models) are still interesting also for applications to Physics, in the theory of liquid crystals. Heilmann and Lieb proposed some monomer-dimer models on Z^2 with attractive interactions that favour the presence of clusters of neighbouring parallel dimers. They showed the presence of orientational order at low temperatures, while they conjectured the absence of translational order. We prove the absence of translational order in a different framework, when the dimer potential favours one of the two orientations.

An efficient container lemma

This thesis is concerned with the study of random graphs and random algorithms. There are three overarching themes. One theme is sparse random graphs, i.e. random graphs in which the average degree is bounded with high probability. A second theme is that of finding spanning subsets such as spanning trees, perfect matchings and Hamilton cycles. A third theme is solving optimization problems on graphs with random edge costs.

It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size of the network? We study routing in families of sparse random graphs whose degrees follow heavy tailed distributions. Instantiations of such random graphs have been proposed as models for the topology of the Internet at the level of Autonomous Systems as well as at the level of routers. Let n be the number of nodes. Suppose that for each pair of nodes with degrees du and dv we have O(du dv) units of demand. Thus the total demand is O(n2). We argue analytically and experimentally that in the considered random graph model such demand patterns can be routed so that the flow through each link is at most O(n log2 n). This is to be compared with a bound O(n2) that holds for arbitrary Similar results were previously known for sparse random regular graphs, a.k.a. expander graphs. The significance is that Internet-like topologies, which grow in a dynamic, decentralized fashion and appear highly inhomogeneous, can support routing with performance characteristics comparable to those of their regular counterparts, at least under the assumption of uniform demand and capacities. Our proof uses approximation algorithms for multicommodity flow and establishes strong bounds of a generalization of expansion, namely conductance. Besides routing, our bounds on conductance have further implications, most notably on the gap between first and second eigenvalues of the stochastic normalization of the adjacency matrix of the graph.

It has been observed that the degrees of the topologies of several communication networks follow heavy tailed statistics. What is the impact of such heavy tailed statistics on the performance of basic communication tasks that a network is presumed to support? How does performance scale with the size of the network? We study routing in families of sparse random graphs whose degrees follow heavy tailed distributions. Instantiations of such random graphs have been proposed as models for the topology of the Internet at the level of Autonomous Systems as well as at the level of routers. Let n be the number of nodes. Suppose that for each pair of nodes with degrees d u and d v we have O(d u d v ) units of demand. Thus the total demand is O(n 2 ) . We argue analytically and experimentally that in the considered random graph model such demand patterns can be routed so that the flow through each link is at most O(n log 2 n) . This is to be compared with a bound O(n 2 ) that holds for arbitrary graphs. Similar results were previously known for sparse random regular graphs, a.k.a. "expander graphs." The significance is that Internet-like topologies, which grow in a dynamic, decentralized fashion and appear highly inhomogeneous, can support routing with performance characteristics comparable to those of their regular counterparts, at least under the assumption of uniform demand and capacities. Our proof uses approximation algorithms for multicommodity flow and establishes strong bounds of a generalization of "expansion," namely "conductance." Besides routing, our bounds on conductance have further implications, most notably on the gap between first and second eigenvalues of the stochastic normalization of the adjacency matrix of the graph.

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

Consider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs (extending the known results for ), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, \cup_{i=1}^{\alpha} E_{i}), where the edge set of G is partitioned into \alpha color classes, is called an \alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, \ldots, F_\alpha)-Deletion problem. In the latter problem, we are given an \alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 \leq i \leq \alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = \ldots = F_\alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and \alpha, and can be solved in O(2^{O(\alpha k)}n^{O(1)}) time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, \ldots, F_\alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i's is the class of bipartite graphs and the rest (if any) are forests. We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = \ldots = F_\alpha is the family of all forests then the problem is fixed-parameter tractable parameterized by k and \alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard.

Let G(n,c/n) and G r (n) be an n-node sparse random and a sparse random r-regular graph, respectively, and let \({\cal I}(n,c)\) and \({\cal I}(n,r)\) be the sizes of the largest independent set in G(n,c/n) and G r (n). The asymptotic value of \({\cal I}(n,c)/n\) as n→∞, can be computed using the Karp-Sipser algorithm when c≤ e. For random cubic graphs, r=3, it is only known that \(.432\leq\liminf_n {\cal I}(n,3)/n \leq \limsup_n {\cal I}(n,3)/n\leq .4591\) with high probability (w.h.p.) as n→∞, as shown in [FS94] and [Bol81], respectively.

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs. Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erd\H{o}s-R\'enyi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance. In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the $n$ vertices is labeled $+$ or $-$ independently and uniformly at random; each pair of vertices is connected independently with probability $a/n$ if both of them have the same label or $b/n$ otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree $a/b$ is $\Theta(1)$ and the average degree $ (a+b) / 2 = n^{o(1)}$, we characterize three different regimes under which a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly. Thus, in contrast to the case of symmetric block models without side information, we show that local algorithms can achieve optimal performance for the block model with side information.

We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erdos-Renyi (ER) graph G(N,c/N) and random r-regular graph G(N,r).For a variety of models, including independent sets, MAX-CUT, Coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem.For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p., thus resolving an open problem, (see Conjecture 2.20 in Wormald's Random Graph Models, or Aldous's list of open problems.Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli. Among other applications, this method was used to prove the existence of free energy limits for Viana-Bray and K-SAT models on ER graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of ER graph G(N,c/N) and random regular graph G(N,r).In addition we establish the large deviations principle for the satisfiability property for constraint satisfaction problems such as Coloring, K-SAT and NAE-K-SAT.

We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let 1 n be the eigenvalues of an n-vertex graph, and let = max[2,|n|]. Let c be a large enough constant. For graphs...

In graph sparsification, the goal has almost always been of global nature: compress a graph into a smaller subgraph (sparsifier) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus locally sparse, aiming to improve local measures of runtime and memory. To improve those local measures, it is important to be able to compute such sparsifiers locally. We initiate the study of local algorithms for bounded degree sparsifiers in unweighted sparse graphs, focusing on the problems of vertex cover, matching, and independent set. Let \eps > 0 be a slack parameter and \alpha \ge 1 be a density parameter. We devise local algorithms for computing: 1. A (1+\eps)-vertex cover sparsifier of degree O(\alpha / \eps), for any graph of arboricity \alpha.\footnote{In a graph of arboricity \alpha the average degree of any induced subgraph is at most 2\alpha.} 2. A (1+\eps)-maximum matching sparsifier and also a (1+\eps)-maximal matching sparsifier of degree O(\alpha / \eps, for any graph of arboricity \alpha. 3. A (1+\eps)-independent set sparsifier of degree O(\alpha^2 / \eps), for any graph of average degree \alpha. Our algorithms require only a single communication round in the standard message passing model of distributed computing, and moreover, they can be simulated locally in a trivial way. As an immediate application we can extend results from distributed computing and local computation algorithms that apply to graphs of degree bounded by d to graphs of arboricity O(d / \eps) or average degree O(d^2 / \eps), at the expense of increasing the approximation guarantee by a factor of (1+\eps). In particular, we can extend the plethora of recent local computation algorithms for approximate maximum and maximal matching from bounded degree graphs to bounded arboricity graphs with a negligible loss in the approximation guarantee. The inherently local behavior of our algorithms can be used to amplify the approximation guarantee of any sparsifier in time roughly linear in its size, which has immediate applications in the area of dynamic graph algorithms. In particular, the state-of-the-art algorithm for maintaining (2-\eps)-vertex cover (VC) is at least linear in the graph size, even in dynamic forests. We provide a reduction from the dynamic to the static case, showing that if a t-VC can be computed from scratch in time T(n) in any (sub)family of graphs with arboricity bounded by \alpha, for an arbitrary t \ge 1, then a (t+\eps)-VC can be maintained with update time \frac{T(n)}{O((n / \alpha) \cdot \eps^2)}, for any \eps > 0. For planar graphs this yields an algorithm for maintaining a (1+\eps)-VC with constant update time for any constant \eps > 0.

In this thesis we prove several results in extremal combinatorics from areas including Ramsey theory, random graphs and graph saturation. We give a random graph analogue of the classical Andr´asfai, Erd˝os and S´os theorem showing that in some ways subgraphs of sparse random graphs typically behave in a somewhat similar way to dense graphs. In graph saturation we explore a ‘partite’ version of the standard graph saturation question, determining the minimum number of edges in H-saturated graphs that in some way resemble H themselves. We determine these values for K4, paths, and stars and determine the order of magnitude for all graphs. In Ramsey theory we give a construction from a modified random graph to solve a question of Conlon, determining the order of magnitude of the size-Ramsey numbers of powers of paths. We show that these numbers are linear. Using models from statistical physics we study the expected size of random matchings and independent sets in d-regular graphs. From this we give a new proof of a result of Kahn determining which d-regular graphs have the most independent sets. We also give the equivalent result for matchings which was previously unknown and use this to prove the Asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow and Markstrom. Using these methods we give an alternative proof of Shearer’s upper bound on off-diagonal Ramsey numbers.

Finding hamilton cycles in sparse random graphs