The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when . We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability , where is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014     

The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n,p) for np = 1 + ε, where ε = ε(n) = o(1). We show that if ε³n →—∞, then a.a.s. D(n,p) consists of components which are either isolated vertices or directed cycles, each of size O_p(|ε|^?1). On the other hand, if ε³n →∞, then a.a.s. the structure of D(n,p) is dominated by the unique complex component of size (4 + o(1))ε²n, whereas all other components are of size O_p(ε^?1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Branching random walks II

A general method is developed with which various theorems on the mean square convergence of functionals of branching random walks are proven. The results cover extensions and generalizations of classical central limit analogues as well as a result of a different type.     

Branching random walks I

A general method is developed with which various theorems on the mean square convergence of functionals of branching random walks are proven. The results cover extensions and generalizations of classical central limit analogues as well as a result of a different type.     

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs in the so‐called rank‐1 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W. In this case, the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power‐law case, i.e., the case where \begin{align*}{\mathbb{P}}(W\geq k)\end{align*} is proportional to k^‐(τ‐1) for some power‐law exponent τ > 3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when \begin{align*}{\mathbb{P}}(W > k) \leq ck^{-(\tau-1)}\end{align*} for all k ≥ 1 and some τ > 4 and c > 0, the largest critical connected component in a graph of size n is of order n^2/3, as it is for the critical Erd?s‐Rényi random graph. When, instead, \begin{align*}{\mathbb{P}}(W > k)=ck^{-(\tau-1)}(1+o(1))\end{align*} for k large and some τ∈(3,4) and c > 0, the largest critical connected component is of the much smaller order n^{(τ‐2)/(τ‐1)}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480–508, 2013     

The degree sequences and spectra of scale‐free random graphs

We investigate the degree sequences of scale‐free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non‐rigorous arguments of Dorogovtsev, Mendes, and Samukhin ( 14 ). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.     

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     

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

Semi‐random graph process

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties , we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies . Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.     

The diameter of inhomogeneous random graphs

In this paper, we study the diameter of inhomogeneous random graphs that are induced by irreducible kernels . The kernels we consider act on separable metric spaces and are almost everywhere continuous. We generalize results known for the Erdős–Rényi model G(n, p) for several ranges of p. We find upper and lower bounds for the diameter of in terms of the expansion factor and two explicit constants that depend on the behavior of the kernel over partitions of the metric space.     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

On the size of the algebraic difference of two random Cantor sets

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

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.     

Levels of a scale‐free tree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices the graph will be a tree. These graphs have been shown to have power law degree distributions, the same as observed in some large real‐world networks. We show that the degree distribution is the same on every sufficiently high level of the tree. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

Two models of a random digraph on n vertices, and are studied. In 1990, Karp for D(n, p) and independently T. ?uczak for D(n,m = cn) proved that for c > 1, with probability tending to 1, there is an unique strong component of size of order n. Karp showed, in fact, that the giant component has likely size asymptotic to nθ², where θ = θ(c) is the unique positive root of . In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector , and two distinct 2 × 2 covariance matrices, and . To this end, we introduce and analyze a randomized deletion process which terminates at the directed (1, 1)‐core, the maximal digraph with minimum in‐degree and out‐degree at least 1. This (1, 1)‐core contains all non‐trivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the (1, 1)‐core, those outside the largest strong component, are of polylog order, thus dwarfed by anticipated fluctuations, on the scale of n^1/2, of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier‐based techniques. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 3–64, 2016     

Harmonic moments of branching processes in random environments

We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.