Book ChapterDOI
Surveys in Combinatorics, 1999: Models of Random Regular Graphs
Nicholas Charles Wormald
- pp 239-298
TLDR
This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results.Abstract:
This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results. Related results on asymptotic enumeration are also presented, as well as various generalisations to random graphs with given degree sequence. A major feature in this area is the small subgraph conditioning method. When applicable, this establishes a relationship between random regular graphs with uniform distribution, and non-uniform models of random regular graphs in which the probability of a graph G is weighted according to the number of subgraphs G has of a certain type. Information can be obtained in this way on the probability of existence of various types of spanning subgraphs, such as Hamilton cycles and decompositions into perfect matchings. Uniformly distributed labelled random regular graphs receive most of the attention, but also included are several non-uniform models which come about in a natural way. Some of these appear as spin-offs from the small subgraph conditioning method, and some arise from algorithms which use simple approaches to generating random regular graphs. A quite separate role played by algorithms is in the derivation of random graph properties by analysing the performance of an appropriate greedy algorithm on a random regular graph. Many open problems and conjectures are given.read more
Citations
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Journal ArticleDOI
Evolutionary games on graphs
György Szabó,Gábor Fáth +1 more
TL;DR: The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
Journal ArticleDOI
Expander graphs and their applications
S Hoory,Nathan Linial +1 more
TL;DR: Expander graphs were first defined by Bassalygo and Pinsker in the early 1970s, and their existence was proved in the late 1970s as discussed by the authors and early 1980s.
Journal ArticleDOI
Weight-conserving characterization of complex functional brain networks.
Mikail Rubinov,Olaf Sporns +1 more
TL;DR: This study generalizes measures of modularity and centrality to fully connected and weighted complex networks, describes the detection of degenerate high-modularity partitions of these networks, and introduces a weighted-connectivity null model of these Networks.
MonographDOI
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Journal ArticleDOI
A random graph model for power law graphs
TL;DR: A random graph model is proposed which is a special case of sparserandom graphs with given degree sequences which satisfy a power law and involves only a small number of parameters, called logsize and log-log growth rate, which capture some universal characteristics of massive graphs.
References
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Book
Graph theory with applications
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
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The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
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Graph Theory
TL;DR: Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.
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Probability and random processes
TL;DR: In this article, the authors present a survey of the history and varieties of probability for the laws of chance and their application in the context of Markov chains convergence of random variables.