Consider the Erdős–Renyi random graph on (n ) vertices where each edge is present independently with probability ( n ), with ( >0 ) fixed. For large (n ), a typical random graph locally behaves like a Galton–Watson tree with Poisson offspring distribution with mean ( ). Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.
Large deviations in random graphs is a rapidly growing topic. For dense graphs, e.g. @math with fixed @math , a thorough treatment has been given recently by Chatterjee and Varadhan @cite_12 , in the framework of the cut topology introduced by Lov 'a sz and coauthors, see Borgs, Chayes, Lov 'a sz, S 'o s and Vesztergombi @cite_25 - @cite_14 . In the sparse regime, only a few partial results are known. O'Connell @cite_7 , Biskup, Chayes and Smith @cite_11 and Puhalskii @cite_4 have proven large deviation asymptotics for the connectivity and for the size of the connected components. Large deviations for degree sequences of Erd o s-R 'enyi graphs has been studied in Doku-Amponsah and M "o rters @cite_23 and Boucheron, Gamboa and L 'eonard [Theorem 7.1] BGL2002 . Closer to our approach, large deviations in the local weak topology were obtained for critical multi-type Galton-Watson trees by Dembo, M "o rters and Sheffield @cite_6 . Finally, large deviations for other models of statistical physics on Erd o s-R 'enyi graphs have been considered in Rivoire @cite_17 and Engel, Monasson, and Hartmann @cite_28 .
@cite_14 @cite_4 @cite_7 @cite_28 @cite_17 @cite_6 @cite_23 @cite_25 @cite_12 @cite_11
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We consider sequences of graphs (Gn) and dene various notions of convergence related to these sequences including -convergence," dened in terms of the densities of homomorphisms from small graphs into Gn, and -convergence," dened in terms of the densities of homomorphisms from Gn into small graphs. We show that right-convergence is equivalent to left-convergence, both for simple graphs Gn, and for graphs Gn with nontrivial nodeweights and edgeweights. Other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics, such as maximum cuts and Szemer edi partitions, and fundamental notions from statistical physics, like energies and free energies. We thereby relate local and global properties of graph sequences. Quantitative forms of these results express the relationships among dierent measures of similarity of large graphs. We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components, of the sizes of the giant components and of the numbers of the excess edges of the giant components. For the supercritical case, we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components, of the size of the largest component and of the number of the excess edges of the largest component. For the critical case, we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges. Some related asymptotics are also established. The proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory. As a byproduct of the results, we provide some additional insight into the nature of phase transitions in sparse random graphs. We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. We show that large deviation properties of Erdos-Renyi random graphs can be derived from the free energy of the q-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to ln q is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the q→ 1 limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model. The one-step replica symmetry breaking cavity method is proposed as a new tool to investigate large deviations in random graph ensembles. The procedure hinges on a general connection between negative complexities and probabilities of rare samples in spin glass like models. This relation between large deviations and replica theory is explicited on different models where it is confronted to direct combinatorial calculations. Given a finite typed rooted tree T with n vertices, the empirical subtree measure is the uniform measure on the n typed subtrees of T formed by taking all descendants of a single vertex. We prove a large deviation principle in n, with explicit rate function, for the empirical subtree measures of multitype Galton–Watson trees conditioned to have exactly n vertices. In the process, we extend the notions of shift-invariance and specific relative entropy—as typically understood for Markov fields on deterministic graphs such as Zd—to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with n vertices. For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erdos-Renyi graphs near criticality. We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences: “left convergence” defined in terms of the densities of homomorphisms from small graphs into Gn; “right convergence” defined in terms of the densities of homomorphisms from Gn into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs Gn, and for graphs Gn with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition. We present a large-deviations thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are small.'' One consequence is an immediate derivation of the cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only small'' components.