We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Table Of Integrals Series And Products

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Subgraphs reveal information about the geometry and functionalities of complex networks For scale-free networks with unbounded degree fluctuations, we count the number of times a small connected graph occurs as a subgraph (motif counting) or as an induced subgraph (graphlet counting) We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems For any given subgraph, the unique optimizer describes the degrees of the nodes that together span the subgraph We find that every subgraph occurs typically between vertices with specific degree ranges In this way, we can count and characterize {\it all} subgraphs We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model

/pdf/optimal-subgraph-structures-in-scale-free-configuration-5dlnmf1o53.pdf

Optimal subgraph structures in scale-free configuration models

The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals. 
The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in $d\geq 11$, for lattice trees in $d\geq 16$ and for lattice animals in $d\geq 21$.

Generalized approach to the non-backtracking lace expansion

We consider a repulsion{attraction model for a random polymer of flnite length in Z d . Its law is that of a flnite simple random walk path in Z d receiving a penalty e i2fl for every self-intersection, and a reward e ∞=d for every pair of neighbouring monomers. The non-negative parameters fl and ∞ measure the strength of repellence and attraction, respectively. We show that for ∞ > fl the attraction dominates the repulsion, i.e., with high probability the polymer is contained in a flnite box whose size is independent of the length of the polymer. For ∞ < fl the behaviour is difierent. We give a lower bound for the rate at which the polymer extends in space. Indeed, we show that the probability for the polymer consisting of n monomers to be contained in a cube of side length †n 1=d tends to zero as n tends to inflnity.

/pdf/self-attractive-random-polymers-2krw9exvub.pdf

Self{attractive random polymers

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Optimal subgraph structures in scale-free configuration models

Generalized approach to the non-backtracking lace expansion

Self{attractive random polymers