We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erdős–Renyi model, where it settles a conjecture of Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398–1414]. Our proof consists in extending the notion of balanced loads from finite graphs to their local weak limits, using unimodularity. This is a new illustration of the objective method described by Aldous and Steele [In Probability on Discrete Structures (2004) 1–72 Springer].
This work is a new illustration of the general principles exposed in the by Aldous and Steele @cite_2 . The latter provides a powerful framework for the unified study of sparse random graphs and has already led to several remarkable results. Two prototypical examples are the celebrated @math limit in the random assignment problem due to Aldous @cite_14 , and the asymptotic enumeration of spanning trees in large graphs by Lyons @cite_21 . Since then, the method has been successfully applied to various other combinatorial enumeration optimization problems on graphs, including (but not limited to) @cite_17 @cite_11 @cite_15 @cite_25 @cite_18 @cite_5 @cite_26 @cite_0 .
@cite_18 @cite_14 @cite_11 @cite_26 @cite_21 @cite_0 @cite_2 @cite_5 @cite_15 @cite_25 @cite_17
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Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. Recently, in [Random Struct Algorithm 41 2012, 441-450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 2007, 133-142], in turn based on ideas of Martin-Lof [J Appl Probab 23 1986, 265-282], Karp [Random Struct Alg 1 1990, 73-93] and Aldous [Ann Probab 25 1997, 812-854], to prove asymptotic normality of the number L1 of vertices in the largest component L1 of the random r-uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L1, and joint asymptotic normality of L1 and the number M1 of edges of L1 in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 2016, 21-75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325-352, 2017 Abstract Building upon the theory of graph limits and the Aldous–Hoover representation and inspired by Panchenko’s work on asymptotic Gibbs measures [Annals of Probability 2013], we construct continuous embeddings of discrete probability distributions. We show that the theory of graph limits induces a meaningful notion of convergence and derive a corresponding version of the Szemeredi regularity lemma. Moreover, complementing recent work (2015), we apply these results to Gibbs measures induced by sparse random factor graphs and verify the “replica symmetric solution” predicted in the physics literature under the assumption of non-reconstruction. Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphs