The augmented multiplicative coalescent,bounded size rules and critical dynamics of random graphs

Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of \(n\) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed \(K\ge 1\) , all components of size greater than \(K\) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erd?s–Rényi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from Bhamidi et al. (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime.     

Aggregation models with limited choice and the multiplicative coalescent

The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many different communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing configurations, and choice, wherein one selects from a “limited” set of edges at random to use in the configuration. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman‐Frieze model, and provide first results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain infinite‐type branching processes with types taking values in the space of cadlag paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth‐first search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite different set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 55–116, 2015     

Random k‐SAT and the power of two choices

We study an Achlioptas‐process version of the random k‐SAT process: a bounded number of k‐clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well‐studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2‐SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k‐SAT for . We then propose a gap decision problem based upon this semi‐random model. The aim of the problem is to investigate the hardness of the random k‐SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 163–173, 2015     

On the normality of giant components

We consider a general family of random graph processes, which begin with an empty graph, and where at every step an edge is added at random according to some rule. We show that when certain general conditions are satisfied, the order of the giant component tends to a normal distribution. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 43, 452–485, 2013     

Delaying satisfiability for random 2SAT

Let (C₁,C′_(*)),(C₂,C′_(*)),…,(C _m,C′_(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \begin{align*}\binom{n}{2}\end{align*} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on‐line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on‐line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well‐known fact that a random m ‐clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on‐line setting above, we also consider an off‐line version in which all m clause‐pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off‐line setting, we show that the two‐choice satisfiability threshold for k ‐SAT for any fixed k coincides with the standard satisfiability threshold for random 2k ‐SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Avoiding a giant component

Let e₁, e′₁; e₂, e′₂;…;e_i, e′_i;??? be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph K_n (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from e_i,e′_i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75–85, 2001     

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2‐balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs.     

Sparse random graphs with clustering

In 2007, we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this article is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with (conditionally) independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non‐Poisson) multi‐type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power‐law degree sequences with a wide range of degree exponents and clustering coefficients. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 269–323, 2011     

The Bohman‐Frieze process near criticality

The Erd?s‐Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd?s and Rényi states that the Erd?s‐Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd?s and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman‐Frieze process, a simple modification of the Erd?s‐Rényi process. The Bohman‐Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman‐Frieze process. We show that it has a qualitatively similar phase transition to the Erd?s‐Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c ? ? (that is, when the number of edges are (t_c ? ?)n/2) are trees or unicyclic components and that the largest component is of size Ω(?^‐2log n). Further, at t_c + ?, all components apart from the giant component are trees or unicyclic and the size of the second‐largest component is Θ(?^‐2log n). Each of these results corresponds to an analogous well‐known result for the Erd?s‐Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi‐linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Hamiltonicity thresholds in Achlioptas processes

In this article, we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on nlabeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K = o(log n), the threshold for Hamiltonicity is n log n, i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K = ω(log n) we can essentially waste almost no edges, and create a Hamilton cycle in n + o(n) rounds with high probability. Finally, in the intermediate regime where K = Θ(log n), the threshold has order nand we obtain upper and lower bounds that differ by a multiplicative factor of 3. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Emergence of the giant component in special Marcus–Lushnikov processes

Component sizes in the usual random graph process are a special case of the Marcus–Lushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the Marcus–Lushnikov process. Essentially no rigorous results are known; we make a start by proving a weak result, but our main purpose is to draw this topic to the attention of random graph theorists. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 179–196, 1998     

A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near‐critical coalescent processes, including the classical Erd?s‐Rényi process. We show tightness of the fixation time in the “Brownian” regime, explicitly determining the median value of the fixation time to within an optimal O(1) window. This generalizes ?uczak's result for the Erd?s‐Rényi random graph using completely different techniques. In the heavy‐tailed case, in which the limit of the component sizes can be encoded using a thinned pure‐jump Lévy process, we prove that only one‐sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.     

The construction of extensible polynomial lattice rules with small weighted star discrepancy

In this paper we introduce a construction algorithm for extensible polynomial lattice rules and we prove that the construction algorithm yields generating vectors of polynomials which are optimal for a range of moduli chosen in advance. The construction algorithm uses a sieve where the generating vectors are extended by one coefficient in each component at each step and where one keeps a certain number of good ones and discards the rest. We also show that the construction can be done component by component.

     

Time evolution of dense multigraph limits under edge‐conservative preferential attachment dynamics

We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the notion of convergence of dense graph sequences, defined by Lovász and Szegedy (J. Combin. Theory Ser B 96 (2006) 933–957). We investigate how the limit object evolves under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limit object from its initial state up to the stationary state, which is described in the companion paper (Ráth and Szakács, in press). In our proofs we use the theory of exchangeable arrays, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits subaging. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Markov random fields,Markov cocycles and the 3-colored chessboard

The well-known Hammersley–Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley–Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce “Markov cocycles”, reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley–Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview where the underlying graph is Z_d. This family includes all Markov random fields whose support is the d-dimensional “3-colored chessboard”. On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.     

Addendum to ‘avoiding a giant component’

We take this opportunity, which is kindly provided by the editors, to add a theorem and to correct the proof of one of the lemmas in the article ‘Avoiding a Giant Component’ [Bohman and Frieze, Avoiding a giant component, Random Struct Alg 19 (2001), 75–85]. The subject of the said article is the following guided random process. Let e₁, $e^{\prime}_{1}$; e₂, $e^{\prime}_{2}$;…;e_i, $e^{\prime}_{i}$;… be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph K_n. This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from e_i, $e_{i}^{\prime}$ to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. In Bohman and Frieze [Avoiding a giant component, Random Struct Alg 19 (2001), 75–85], we proved that these choices can be made so that whp (A sequence of events ?_n is said to occur with high probability (whp) if lim_n→∞ Pr (?_n)=1.), the size of the largest component of the graph formed at stage. 535n, is polylogarithmic in n. Here, we make a correction to that proof and prove that the graph formed at stage cn for any constant c>1 will contain a component of size Ω(n) (regardless of how the edges are chosen at each stage). © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 126–130, 2002     

Birth control for giants

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v ₁, v ₂}, {v ₃, v ₄} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t _c so that S(t) → ∞ as t approaches t _c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t _c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t _c we show the existence of a giant component, so that t = t _c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component. Research supported by the Australian Research Council, the Canada Research Chairs Program and NSERC. Research partly carried out while the author was at the Department of Mathematics and Statistics, University of Melbourne.     

Changes of leadership in a random graph process

Let be a random graph process in which in each step we add to a graph a new edge, chosen at random from all available pairs. Define the leader of G(n, M) as either the unique largest component or, if G(n, M) contains many components of the maximum size, the one from the largest components which emerged first during the process. We show that the longest period between two changes of leaders in the random graph process is, with probability tending to 1 as n →∞, of the order of n/log log n/log n. © 1994 John Wiley & Sons, Inc.     

Large deviation analysis for layered percolation problems on the complete graph

We analyze the large deviation properties for the (multitype) version of percolation on the complete graph – the simplest substitutive generalization of the Erd&0151;s‐Rènyi random graph that was treated in article by Bollobás et al. (Random Structures Algorithms 31 (2007), 3–122). Here the vertices of the graph are divided into a fixed finite number of sets (called layers) the probability of {u,v} being in our edge set depends on the respective layers of u and v. We determine the exponential rate function for the probability that a giant component occupies a fixed fraction of the graph, while all other components are small. We also determine the exponential rate function for the probability that a particular exploration process on the random graph will discover a certain fraction of vertices in each layer, without encountering a giant component.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 460–492, 2012