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24
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1407 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
A Random Graph Model for Massive Graphs
, 2000
"... We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from the ..."
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Cited by 335 (26 self)
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We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.
A Critical Point For Random Graphs With A Given Degree Sequence
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 the ..."
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Cited by 289 (7 self)
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Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 then almost surely all components in such graphs are small. We can apply these results to G n;p ; G n;M , and other wellknown models of random graphs. There are also applications related to the chromatic number of sparse random graphs.
Models of Random Regular Graphs
 In Surveys in combinatorics
, 1999
"... In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular g ..."
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Cited by 157 (33 self)
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In a previous paper we showed that a random 4regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random dregular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1
The phase transition in inhomogeneous random graphs, preprint available from http://www.arxiv.org/abs/math.PR/0504589
"... Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. ..."
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Cited by 101 (30 self)
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Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with
Random Evolution in Massive Graphs
, 2001
"... Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for generating p ..."
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Cited by 89 (7 self)
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Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by socalled the "power law". In this paper, we will first briefly survey the history and previous work on power law graphs. Then we will give four evolution models for generating power law graphs by adding one node/edge at a time. We will show that for any given edge density and desired distributions for indegrees and outdegrees (not necessarily the same, but adhered to certain general conditions), the resulting graph will almost surely satisfy the power law and the in/outdegree conditions. We will show that our most general directed and undirected models include nearly all known models as special cases. In addition, we consider another crucial aspects of massive graphs that is called "scalefree" in the sense that the f requency of sampling (w.r.t. the growth rate) is independent of the parameter of the resulting power law graphs. We will show that our evolution models generate scalefree power law graphs. 1
A Random Graph Model for Power Law Graphs
 Experimental Math
, 2000
"... We propose a random graph m del which is a special case of sparse random graphs with given degree sequences which satisfy a power law. Thism odel involves only asm all num ber of param eters, called logsize and loglog growth rate. These param eters capturesom e universal characteristics ofm assive ..."
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Cited by 73 (4 self)
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We propose a random graph m del which is a special case of sparse random graphs with given degree sequences which satisfy a power law. Thism odel involves only asm all num ber of param eters, called logsize and loglog growth rate. These param eters capturesom e universal characteristics ofm assive graphs. Furtherm re, from these paramfi ters, various properties of the graph can be derived. Forexam)(( for certain ranges of the paramJ?0CM we willcom?C7 the expected distribution of the sizes of the connectedcom onents which almJC surely occur with high probability. We will illustrate the consistency of our m del with the behavior of so m m ssive graphs derived from data in telecom unications. We will also discuss the threshold function, the giant com ponent, and the evolution of random graphs in thism del. 1
Relevance of Massively Distributed Explorations of the Internet Topology: Simulation Results
, 2005
"... Internet maps are generally constructed using the traceroute tool from a few sources to many destinations. It appeared recently that this exploration process gives a partial and biased view of the real topology, which leads to the idea of increasing the number of sources to improve the quality of ..."
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Cited by 33 (10 self)
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Internet maps are generally constructed using the traceroute tool from a few sources to many destinations. It appeared recently that this exploration process gives a partial and biased view of the real topology, which leads to the idea of increasing the number of sources to improve the quality of the maps. In this paper, we present a set of experiments we have conduced to evaluate the relevance of this approach. It appears that the statistical properties of the underlying network have a strong influence on the quality of the obtained maps, which can be improved using massively distributed explorations. Conversely, we show that the exploration process induces some properties on the maps. We validate our analysis using realworld data and experiments and we discuss its implications.
Bipartite Graphs as Models of Complex Networks
 Aspects of Networking
, 2004
"... It appeared recently that the classical random graph model used to represent realworld complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here the first model which achieves the following challenges: it produces ..."
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Cited by 29 (4 self)
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It appeared recently that the classical random graph model used to represent realworld complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here the first model which achieves the following challenges: it produces graphs which have the three main wanted properties (clustering, degree distribution, average distance), it is based on some realworld observations, and it is sufficiently simple to make it possible to prove its main properties. This model consists in sampling a random bipartite graph with prescribed degree distribution. Indeed, we show that any complex network can be viewed as a bipartite graph with some specific characteristics, and that its main properties can be viewed as consequences of this underlying structure. We also propose a growing model based on this observation. Introduction.
Random Regular Graphs of NonConstant Degree: Connectivity And Hamiltonicity
, 2002
"... Let Gr denote a graph chosen uniformly at random from the set of rregular graphs with vertex set {1, 2,... , n} where 3 < r < con for some small constant co. We prove that with probability tending to 1 as n  o, Gr is rconnected and Hamiltonian. ..."
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Cited by 28 (6 self)
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Let Gr denote a graph chosen uniformly at random from the set of rregular graphs with vertex set {1, 2,... , n} where 3 < r < con for some small constant co. We prove that with probability tending to 1 as n  o, Gr is rconnected and Hamiltonian.