No abstract

Abstract. 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).

This book treats the very special and fundamental mathematical properties that hold for a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, other parts of mathematics, statistics and theoretical physics. The emphasis throughout this book is on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.

A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level.Using the Athreya-Karlin embedding, these results yield asymptotic results for generalized Pólya urns. We investigate such results in detail and obtain explicit formulas for the asymptotic variances and covariances. The general formulas involve integrals of matrix functions; we show how they can be evaluated and simplified in important special cases. We also consider the numbers of drawn balls of different types and functional limit theorems for the urns.We illustrate our results by some examples, including several applications to random trees where our theorems and variance formulas give simple proofs of some known results; we also give some new results.Remark 3.5. By (3.1), U I is a stationary Gaussian process. It can be regarded as a multi-dimensional Ornstein-Uhlenbeck process.Remark 3.6. If λ ∈ Λ II with Im λ = 0, it follows from (3.2) and (2.16) thatConsequently, U λ,k is either real (when λ is real) or symmetric complex.Remark 3.7. We have no general description of the distributions of W λ,k for λ ∈ Λ III , and there seems to be no reason to expect any. They are (typically, at least) not normal, and not independent of each other. Moreover, their distributions (typically) depend on the initial state X (0), unlike U I and U λ,k .Taking x = 0 in (i) and x = 1 in (ii) and (iii), we obtain as a corollary some standard results, cf. [9].Corollary 3.8. Assume (A1)-(A6). Then, as t → ∞, with joint convergence, (i) e −λ 1 t/2 P I X (t) d → W 1/2 U I ; (ii) for every λ ∈ Λ II and k = 0, . . . , d λ ,Here, U I , U λ,k and W λ,k are vector-valued random variables with U I and U λ,k jointly Gaussian. The vector U I is real, while U λ,k and W λ,k are real for real λ but complex otherwise, with Uλ ,k = U λ,k and Wλ ,k = W λ,k . Furthermore, a.s., U I ∈ E I := λ∈Λ I E λ , U λ,k ∈ E λ,k , and W λ,k ∈ E λ,k .The random vector U I , the families {U λ,k } 0≤k≤d λ for different λ ∈ Λ II with Im λ ≥ 0, and the family {W λ,k } λ∈Λ III ,k≤d λ ∪ {W } are independent of each other.

Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately f n edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching cosh i= 0.9325 as n + m. The limiting probability that it consists o f trees, unicyclic components, and at most one another component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes 4n, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the evolution, approaches 5 wl18 = 0.8727. A "uniform" model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substantially simpler than the analogous formulas for the classical random graphs of ErdBs and RCnyi. The notions of "excess" and "deficiency," which are significant characteristics of the generating function as well as of the graphs themselves, lead to a 233 the multigraph process, because it can generate graphs with self-loops x-x, and it can also generate multiple edges. Notice that a self-loop x-x is generated with probability 1 ln', while an edge x-y with x # y is generated with probability 2 ln' because it can occur either as ( x , y ) or ( y, x ) .The second evolution procedure, introduced by Erdos and Rknyi [12], is called the permutation model or the graph process. In this case we consider all N = ( ) possible edges x-y with x < y and introduce them in random order, with all N! permutations considered equally likely. In this model there are no self-loops or multiple edges.A multigraph M on n labeled vertices can be defined by a symmetric n X n matrix of nonnegative integers m x y , where mxy = myx is the number of undirected edges x-y in G. For purposes of analysis, we shall assign a compensation factor to M ; if m = Ez=, E:=, mxy is the total number of edges, the number of sequences ( x , , y , ) ( x 2 , y z ) . . . ( x , , y, ) that lead to M is then exactly (The factor 2" accounts for choosing either ( x , y ) or ( y , x ) ; the 2mxx in the denominator of K ( M ) compensates for the case x = y . The other factor m ! accounts for permutations of the pairs, with mxy! in K ( M ) to compensate for permutations between multiple edges.) Equation (1.2) tells us that K ( M ) is a natural weighting factor for a multigraph M , because it corresponds to the relative frequency with which M tends to occur in applications. For example, consider multigraphs on three vertices (1, 2, 3) having exactly three edges. The edges will form the cycle M , = {1-2, 2-3, 3-1) much more often than they will form three identical self-loops M2 = { 1-1, 1-1, 1-1}, when the multigraphs are generated in a uniform way. For if we consider the 36 possible sequences ( x , , y...

Simply generated trees and Galton-Watson treesWe suppose that we are given a fixed weight sequence w = (w k ) k 0 of non-negative real numbers. We then define the weight of a finite rooted and ordered (a.k.a. plane) tree T bytaking the product over all nodes v in T , where d + (v) is the outdegree of v. Trees with such weights are called simply generated trees and were introduced by Meir and Moon [24]. We let T n be the random simply generated tree obtained by picking a tree with n nodes at random with probability proportional to its weight. (To avoid trivialities, we assume that w 0 > 0 and that there exists some k 2 with w k > 0. We consider only n such that there exists some tree with n vertices and positive weight.)One particularly important case is when ∞ k=0 w k = 1, so the weight sequence (w k ) is a probability distribution on Z 0 . (We then say that (w k ) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = w k . It is easily seen that the simply generated random tree T n equals the conditioned Galton-Watson tree with offspring distribution ξ, i.e., the random Galton-Watson tree defined by ξ conditioned on having exactly n vertices.One of the reasons for the interest in these trees is that many kinds of random trees occuring in various applications (random ordered trees, unordered trees, binary trees, . . . ) can be seen as simply generated random trees and conditioned Galton-Watson trees, see e.g. Aldous [3,4], Devroye [9] and Drmota [10].It is easily seen that if a, b > 0 and we change w k to

We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With X n denoting position at time n, we show that X n /n converges weakly as n → ∞ to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.

Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^*$; we also prove a central limit theorem for $A^*$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.Comment: Published in at http://dx.doi.org/10.1214/11-AAP822 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org