Blow-up Lemma for Cycles in Sparse Random Graphs

An efficient container lemma

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 0 3(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.

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.

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.

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.

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.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

Information dissemination is a fundamental problem in parallel and distributed computing. In its simplest variant, known as the broadcasting problem, a single message has to be spread among all nodes of a graph. A prominent communication protocol for this problem is based on the socalled random phone call model (Karp et al., FOCS 2000). In each step, every node opens a communication channel to a randomly chosen neighbor, which can then be used for bidirectional communication. Motivated by replicated databases and peer-to-peer networks, Berenbrink et al., ICALP 2010, considered the so-called gossiping problem in the random phone call model. There, each node starts with its own message and all messages have to be disseminated to all nodes in the network. They showed that any O(log n)-time algorithm in complete graphs requires Ω(log n) message transmissions per node to complete gossiping, with high probability, while it is known that in the case of broadcasting the average number of message transmissions per node is O(log log n). Furthermore, they explored different possibilities on how to reduce the communication overhead of randomized gossiping in complete graphs. It is known that the O(nloglogn) bound on the number of message transmissions produced by randomized broadcasting in complete graphs cannot be achieved in sparse graphs even if they have best expansion and connectivity properties. In this paper, we analyze whether a similar influence of the graph density also holds w.r.t. the performance of gossiping. We study analytically and empirically the communication overhead generated by gossiping algorithms w.r.t. the random phone call model in random graphs and also consider simple modifications of the random phone call model in these graphs. Our results indicate that, unlike in broadcasting, there seems to be no significant difference between the performance of randomized gossiping in complete graphs and sparse random graphs. Furthermore, our simulations illustrate that by tuning the parameters of our algorithms, we can significantly reduce the communication overhead compared to the traditional push-pull approach in the graphs we consider.

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.

Finding hamilton cycles in sparse random graphs

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.

The genus of the binomial random graph $G(n,p)$ is well understood for a wide range of $p=p(n)$. Recently, the study of the genus of the random bipartite graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

LetG(n,c/n) andGr(n) be ann‐node sparse random graph and a sparse randomr‐regular graph, respectively, and letI(n,r) andI(n,c) be the sizes of the largest independent set inG(n,c/n) andGr(n). The asymptotic value ofI(n,c)/nasn→ ∞, can be computed using the Karp‐Sipser algorithm whenc≤e. For random cubic graphs,r= 3, it is only known that .432 ≤ lim infnI(n,3)/n≤ lim supnI(n,3)/n≤ .4591 with high probability (w.h.p.) asn→ ∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit limnI(n,c)/ncan be computed exactly even whenc>e, and limnI(n,r)/ncan be computed exactly for somer≥ 1. For example, when the weights are exponentially distributed with parameter 1, limnI(n,2e)/n≈ .5517, and limnI(n,3)/n≈ .6077. Our results are established using the recently developedlocal weak convergencemethod further reduced to a certainlocal optimalityproperty exhibited by the models we consider. We extend our results to maximum weight matchings inG(n,c/n) andGr(n). For the case of exponential distributions, we compute the corresponding limits for everyc> 0 and everyr≥ 2. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

We consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satises a uniform X 2 logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Centered by a multiple of logn, where the constant is the inverse of the Malthusian rate of growth of the associated branching process, the minimal weight converges in distribution. The limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices, and a Gumbel random variable, and thus shows a remarkably universal behavior. The proofs rely on a rened coupling between the shortest path problems on these graphs and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination. The results extend to a host of related random graph models, ranging from random rregular graphs, inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence.