From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

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Introduction to random graphs

Let G be a locally compact (second countable) group and π:G → U(ℋ) a unitary representation of G on the (separable) Hilbert space ℋ.Of course,ℋ may or may not contain non-trivial vectors invariant under π(G). We wish to define a notion of π almost containing invariant vectors.

Kazhdan’s Property (T)

Are biological networks different from other large complex networks? Both large biological and non-biological networks exhibit power-law graphs (number of nodes with degree k, N(k) ~ k-b) yet the exponents, b, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks), and is fundamentally different from the mechanisms thought to dominate the growth of most non-biological networks (such as the internet [1-4]). The preferential choice models non-biological networks like web graphs can only produce power-law graphs with exponents greater than 2 [1-4,8]. We use combinatorial probabilistic methods to examine the evolution of graphs by duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.

Duplication Models for Biological Networks

Limit Theorems of Probability Theory.

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein’s method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.

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Degree asymptotics with rates for preferential attachment random graphs

Let G be a d-regular graph of sufficiently large-girth depending on parameters k and r and µ be a random process on the vertices of G produced by a randomized local algorithm of radius r. We prove the upper bound k+1-2k/d1d-1k for the absolute value of the correlation of values on pairs of vertices of distance k and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the d-regular tree. In that case we give an explicit description for the closure of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infinite d-regular tree has spectral radius 2d-1. Graphings with this spectral gap are infinite analogues of finite Ramanujan graphs and they are interesting on their own right. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 424-435, 2015

Ramanujan graphings and correlation decay in local algorithms

Let $d\geq3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree. In particular, we obtain that if an invariant eigenvector process on the infinite $d$-regular tree is in the weak closure of factor of i.i.d. processes then it has Gaussian distribution.

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On the almost eigenvectors of random regular graphs

We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random $d$-reguar graphs and ergodic theory on $T_d$. Using this correspondence principle together with entropy inequalities for typical processes we prove a family of combinatorial statements about random $d$-regular graphs.

On large girth regular graphs and random processes on trees

We present a new approach to graph limit theory which unifies and generalizes the two most well developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini--Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called $P$-operators) of the form $L^\infty(\Omega)\to L^1(\Omega)$ for probability spaces $\Omega$. We introduce a metric to compare $P$-operators (for example finite matrices) even if they act on different spaces. We prove a compactness result which implies that in appropriate norms, limits of uniformly bounded $P$-operators can again be represented by $P$-operators. We show that limits of operators representing graphs are self-adjoint, positivity-preserving $P$-operators called graphops. Graphons, $L^p$ graphons and graphings (known from graph limit theory) are special examples for graphops. We describe a new point of view on random matrix theory using our operator limit framework.

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Action convergence of operators and graphs

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c(d) > 0 almost surely as the number of steps goes to infinity, and c(d) similar to (e pi)(1/2)d(1/4)e(-2)root d holds as d -> infinity.

Ágnes Backhausz

Papers

Ramanujan graphings and correlation decay in local algorithms

On the almost eigenvectors of random regular graphs

On large girth regular graphs and random processes on trees

Action convergence of operators and graphs

Asymptotic properties of a random graph with duplications