The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Molloy, Michael; Reed, Bruce

doi:10.1017/s0963548398003526

Cited by 720 publications

(728 citation statements)

References 11 publications

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“…We assume that all N i nodes in network i are randomly assigned a degree k from a probability distribution P i (k), and are randomly connected with the only constraint that the node with degree k has exactly k links 91 . We define the generating function of the degree distribution…”

Section: Generating Functions For a Single Networkmentioning

confidence: 99%

Networks formed from interdependent networks

et al. 2011

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Complex networks appear in almost every aspect of science and technology. Although most results in the field have been obtained by analysing isolated networks, many real-world networks do in fact interact with and depend on other networks. The set of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presence of other networks can be justified. Recently, an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review this framework and the results obtained so far for connectivity properties of 'networks of networks' formed by interdependent random networks.T he interdisciplinary field of network science has attracted a great deal of attention in recent years . This development is based on the enormous number of data that are now routinely being collected, modelled and analysed, concerning social 31-39 , economic 14,36,40,41 , technological 40,42-48 and biological 9,13,49,50 systems. The investigation and growing understanding of this extraordinary volume of data will enable us to make the infrastructures we use in everyday life more efficient and more robust.The original model of networks, random graph theory, was developed in the 1960s by ErdAEs and Rényi, and is based on the assumption that every pair of nodes is randomly connected with the same probability, leading to a Poisson degree distribution. In parallel, in physics, lattice networks, where each node has exactly the same number of links, have been studied to model physical systems. Although graph theory is a well-established tool in the mathematics and computer science literature, it cannot describe well modern, real-life networks. Indeed, the pioneering 1999 observation by Barabasi 2 , that many real networks do not follow the ErdAEs-Rényi model but that organizational principles naturally arise in most systems, led to an overwhelming accumulation of supporting data, new models and computational and analytical results, and to the emergence of a new science, that of complex networks.Complex networks are usually non-homogeneous structures that in many cases obey a power-law form in their degree (that is, number of links per node) distribution. These systems are called scale-free networks. Real networks that can be approximated as scale-free networks include the Internet 3 , the World Wide Web 4 , social networks 31-39 representing the relations between individuals, infrastructure networks such as those of airlines 51 , networks in biology 9,13,49,50 , in particular networks of proteinprotein interactions 10 , gene regulation and biochemical pathways, and networks in physics, such as polymer networks or the potentialenergy-landscape network. The discovery of scale-free networks led to a re-evaluation of the basic properties of networks, such as their robustness, which exhibit a drastically different character than those of ErdAEs-Rényi networks. For example, whereas homogeneous ErdAEs-Rényi networks are extremely vulnerable to random fa...

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Section: Generating Functions For a Single Networkmentioning

confidence: 99%

Networks formed from interdependent networks

et al. 2011

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“…However, by using techniques in Lemma 1 of [15], it can be shown that if a random graph with a given degree sequence a. s. has property P under one of these two models, then it a. s. has property P under the other model, provided some general conditions are satisfied.…”

Section: I{vldeg(v) = X} I= Y = ~-Z"mentioning

confidence: 99%

A Random Graph Model for Massive Graphs

Aiello¹,

Chung²,

Lü³

2011

The Structure and Dynamics of Networks

173

303

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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 log-log 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.

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“…. , n), the configuration model produces a random graph with specified degrees [32,36,37]: initially, each vertex i is assigned d i "stubs" or "half edges." Then, the edge set is constructed as a uniformly random matching on the stubs, i.e., each matching is selected with equal probability.…”

Section: K-nearest Neighbor Random Graphsmentioning

confidence: 99%

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

Kabkab

2015

Internet Mathematics

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We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is (log(n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log(n)/ log(log(n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β log(n), where β ≈ 2.4626, there exists a connected Erdös-Rényi random graph that is embedded in our random graph, with high probability.

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The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Cited by 720 publications

References 11 publications

Networks formed from interdependent networks

Networks formed from interdependent networks

A Random Graph Model for Massive Graphs

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

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