Randomized rumor spreading in poorly connected small‐world networks

The Push‐Pull protocol is a well‐studied round‐robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random ‐trees, a class of power law graphs, which are small‐world and have large clustering coefficients, built as follows: initially we have a ‐clique. In every step a new node is born, a random ‐clique of the current graph is chosen, and the new node is joined to all nodes of the ‐clique. When is fixed, we show that if initially a random node is aware of the rumor, then with probability after rounds the rumor propagates to nodes, where is the number of nodes and is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion and constant treewidth, these results demonstrate that Push‐Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability the protocol needs at least rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random ‐Apollonian networks, for which we prove an upper bound of rounds for informing nodes with probability when is fixed. Here, © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 185–208, 2016     

The random connection model: Connectivity,edge lengths,and degree distributions

Consider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter M_n for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .     

Multipass greedy coloring of simple uniform hypergraphs

Let be the minimum number of edges in an n‐uniform simple hypergraph that is not two colorable. We prove that . Our result generalizes to r‐coloring of b‐simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b‐simple n‐uniform hypergraph that is not r‐colorable must be . By trimming arguments it implies that every such graph has edges. For any fixed our techniques yield also a lower bound for van der Waerden numbers W(n, r). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 125–146, 2016     

High degrees in random recursive trees

For , let T_n be a random recursive tree (RRT) on the vertex set . Let be the degree of vertex v in T_n, that is, the number of children of v in T_n. Devroye and Lu showed that the maximum degree Δ_n of T_n satisfies almost surely; Goh and Schmutz showed distributional convergence of along suitable subsequences. In this work we show how a version of Kingman's coalescent can be used to access much finer properties of the degree distribution in T_n. For any , let . Also, let be a Poisson point process on with rate function . We show that, up to lattice effects, the vectors converge weakly in distribution to . We also prove asymptotic normality of when slowly, and obtain precise asymptotics for when and is not too large. Our results recover and extends the previous distributional convergence results on maximal and near‐maximal degrees in RRT.     

Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter and interpolating between the deterministic rotor walk () and the simple random walk (). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each the sequence of successive exits from is a Markov chain. The main result of this paper identifies the scaling limit of the p‐rotor walk with two‐sided i.i.d. initial rotors. The limiting process takes the form , where is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation (1) for all . Here is a standard Brownian motion and are constants depending on the marginals of the initial rotors on and respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution , and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, and . This last result, together with the main result of this paper, implies that the p‐rotor walk is recurrent for any two‐sided i.i.d. initial rotors and any .     

Inside the critical window for cohomology of random k‐complexes

We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for ‐cohomology of a random k‐dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k‐dimensional simplicial complex with each k‐simplex appearing i.i.d. with probability with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last ‐simplex gets covered by a k‐simplex, a higher‐dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016     

Clairvoyant embedding in one dimension

Let v, w be infinite 0‐1 sequences, and a positive integer. We say that is ‐embeddable in , if there exists an increasing sequence of integers with , such that , for all . Let and be coin‐tossing sequences. We will show that there is an with the property that is ‐embeddable into with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 520–560, 2015     

On replica symmetry of large deviations in random graphs

The following question is due to Chatterjee and Varadhan (2011). Fix and take , the Erd?s‐Rényi random graph with edge density p, conditioned to have at least as many triangles as the typical . Is G close in cut‐distance to a typical ? Via a beautiful new framework for large deviation principles in , Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d‐regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all such that lies on the convex minorant of where is the rate function of a binomial with parameter p. In particular, the answer for triangles involves rather than the natural guess of where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of is conditioned to exceed the typical value of the largest eigenvalue of . Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109–146, 2015     

On induced Ramsey numbers for k‐uniform hypergraphs

Let denote the complete k‐uniform k‐partite hypergraph with classes of size t and the complete k‐uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = s^o(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k‐partite k‐uniform hypergraph with classes of size t and an arbitrary k‐graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

σ‐algebras for quasirandom hypergraphs

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of . We associate each notion of quasirandomness with a collection of hypergraphs, the ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ‐adapted hypergraph. We then identify, for each , a particular ‐adapted hypergraph with the property that if G contains roughly the correct number of copies of then G is quasirandom in the sense of . This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017     

On the variational problem for upper tails in sparse random graphs

What is the probability that the number of triangles in , the Erd?s‐Rényi random graph with edge density p , is at least twice its mean? Writing it as , already the order of the rate function r (n, p ) was a longstanding open problem when p = o (1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that for ; the exact asymptotics of r (n, p ) remained unknown. The following variational problem can be related to this large deviation question at : for δ > 0 fixed, what is the minimum asymptotic p‐relative entropy of a weighted graph on n vertices with triangle density at least (1 + δ )p ³? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for fixed p . A very recent breakthrough of Chatterjee and Dembo extended its validity to for an explicit α > 0, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the variational problem is when vs. when (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that for has twice as many triangles as its expectation is where . Our results further extend to k‐cliques for any fixed k , as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 420–436, 2017     

Efficient algorithms for discrepancy minimization in convex sets

A result of Spencer states that every collection of n sets over a universe of size n has a coloring of the ground set with of discrepancy . A geometric generalization of this result was given by Gluskin (see also Giannopoulos) who showed that every symmetric convex body with Gaussian measure at least , for a small , contains a point where a constant fraction of coordinates of y are in . This is often called a partial coloring result. While the proofs of both these results were inherently non‐algorithmic, recently Bansal (see also Lovett‐Meka) gave a polynomial time algorithm for Spencer's setting and Rothvoß gave a randomized polynomial time algorithm obtaining the same guarantee as the result of Gluskin and Giannopoulos. This paper contains several related results which combine techniques from convex geometry to analyze simple and efficient algorithms for discrepancy minimization. First, we prove another constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function. This implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer. Our second result suggests a new approach to obtain partial colorings, which is also valid for the non‐symmetric case. It shows that every (possibly non‐symmetric) convex body , with Gaussian measure at least , for a small , contains a point where a constant fraction of coordinates of y are in . Finally, we give a simple proof that shows that for any there exists a constant c > 0 such that given a body K with , a uniformly random x from is in cK with constant probability. This gives an algorithmic version of a special case of the result of Banaszczyk.     

A Bernoulli mean estimate with known relative error distribution

Suppose that are independent identically distributed Bernoulli random variables with mean p , so and . Any estimate of p has relative error . This paper builds a new estimate of p with the remarkable property that the relative error of the estimate does not depend in any way on the value of p . This allows the easy construction of exact confidence intervals for p of any desired level without needing any sort of limit or approximation. In addition, is unbiased. For ? and δ in (0, 1), to obtain an estimate where , the new algorithm takes on average at most samples. It is also shown that any such algorithm that applies whenever requires at least samples on average. The same algorithm can also be applied to estimate the mean of any random variable that falls in . The used here employs randomness external to the sample, and has a small (but nonzero) chance of being above 1. It is shown that any nontrivial where the relative error is independent of p must also have these properties. Applications of this methodology include finding exact p‐values and randomized approximation algorithms for # P complete problems. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 173–182, 2017     

Almost‐spanning universality in random graphs

A graph G is said to be ‐universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any and any natural number Δ there exists such that the random graph G(n, p) is asymptotically almost surely ‐universal for . Bypassing this natural boundary, we show that for the same conclusion holds when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 380–393, 2017     

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer , the binomial ‐uniform random hypergraph contains edge‐disjoint perfect matchings, provided , where is an integer depending only on . Our result for is asymptotically optimal and for is optimal up to the factor. This significantly improves a result of Frieze and Krivelevich.     

Chromatic thresholds in dense random graphs

The chromatic threshold of a graph H with respect to the random graph G (n, p ) is the infimum over d > 0 such that the following holds with high probability: the family of H‐free graphs with minimum degree has bounded chromatic number. The study of the parameter was initiated in 1973 by Erd?s and Simonovits, and was recently determined for all graphs H . In this paper we show that for all fixed , but that typically if . We also make significant progress towards determining for all graphs H in the range . In sparser random graphs the problem is somewhat more complicated, and is studied in a separate paper. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 185–214, 2017     

Arboricity and spanning‐tree packing in random graphs

We study the arboricity and the maximum number of edge‐disjoint spanning trees of the classical random graph . For all , we show that, with high probability, is precisely the minimum of and , where is the minimum degree of the graph and denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for such that the following holds. Above this threshold, equals and equals . Below this threshold, equals , and we give a two‐value concentration result for the arboricity in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are randomly added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most in the two‐choice load balancing problem, where .     

Counting dense connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on with m edges, whenever and . We give an asymptotic formula for the number of connected r‐uniform hypergraphs on with m edges, whenever is fixed and with , that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja‐Oghlan, and Kang (the case ) and the present authors (the case , ie, “nullity” or “excess” o(n)). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use “smoothing” techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.     

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least infected neighbours becomes infected and remains so forever. Assume that initially vertices are randomly infected, where is the total number of vertices of the graph. Suppose also that , where is the average degree. We determine a critical function such that when , complete infection occurs with high probability as , but when , then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to .