Results 1  10
of
10
Girth of Sparse Graphs
 2002), 194  200. ILWOO CHO
"... Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the d ..."
Abstract

Cited by 76 (6 self)
 Add to MetaCart
(Show Context)
Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a
Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
, 2010
"... ..."
Sparse random graphs with clustering
, 2008
"... In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (nonPoisson) multitype branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with powerlaw degree sequences with a wide range of degree exponents and clustering coefficients.
The cut metric, random graphs, and branching processes
, 2009
"... In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an app ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in [4], as well as related results of Bollobás, Borgs, Chayes and Riordan [3], all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering [5]. 1 Introduction and
CONNECTEDNESS IN GRAPH LIMITS
"... Abstract. We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is wellknown, graph limits may be represented by symmetric functions on a probability space; there are natural defin ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is wellknown, graph limits may be represented by symmetric functions on a probability space; there are natural definitions of direct sums and connectedness for such functions, and there is a perfect correspondence with the corresponding properties of the graph limit. Similarly, every graph limit determines an infinite random graph, which is a.s. connected if and only if the graph limit is connected. There are also characterizations in terms of the asymptotic size of the largest component in the corresponding finite random graphs, and of minimal cuts in sequences of graphs converging to a given limit. 1. Introduction and
Duality in inhomogeneous random graphs, and the cut metric
, 2009
"... The classical random graph model G(n, λ/n) satisfies a ‘duality principle’, in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The classical random graph model G(n, λ/n) satisfies a ‘duality principle’, in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric.
CAN NETWORKS SUBSTITUTE FOR CONTRACTS? EVIDENCE FROM A LAB EXPERIMENT IN THE FIELD
"... Abstract. This paper investigates the effects of social network position on the ability to overcome contract incompleteness using an experiment in Karnataka, India. We randomly pair individuals (who live in the same village and are known to one another) to play informal risksharing games across thr ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. This paper investigates the effects of social network position on the ability to overcome contract incompleteness using an experiment in Karnataka, India. We randomly pair individuals (who live in the same village and are known to one another) to play informal risksharing games across three contracting environments which vary the level of formal commitment and ability to save. We consider two measures of network position – path length, i.e. social distance between partners; and relative centrality, i.e. relative importance. We find that social proximity substitutes for commitment: closer pairs achieve significantly more consumption smoothing under limited commitment than do more distant pairs. Conditional on distance, limited commitment binds significantly more when the members of a pair differ in centrality. The introduction of savings crowds out insurance more when players of different levels of centrality are paired. These facts are consistent with a model in which the continuation value of a relationship is reduced by social
Complex networks
 Handbook of Graph Theory, chapter 12.1
, 2013
"... 1.1.1 Examples of complex networks.......................... 2 1.1.2 Properties of complex networks......................... 2 1.1.3 Random graphs with general degree distributions...... 5 1.1.4 Online models of complex networks.................... 7 1.1.5 Geometric models for complex networks..... ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
1.1.1 Examples of complex networks.......................... 2 1.1.2 Properties of complex networks......................... 2 1.1.3 Random graphs with general degree distributions...... 5 1.1.4 Online models of complex networks.................... 7 1.1.5 Geometric models for complex networks............... 9 1.1.6 Percolation in a general host graph..................... 11 1.1.7 PageRank for ranking nodes............................ 12 1.1.8 Network games.......................................... 14