Handbook of large-scale random networks
type: Item
instance of: scholarly article
title: Handbook of large-scale random networks
zbMATH Open document ID: 1170.05053
DOI: 10.1007/978-3-540-69395-6
published in: Bolyai Society Mathematical Studies
publication date: 2008-08-26T00:00:00Z
review text: This book is intended to give an overview of various recent methods for the study of large-scale complex networks. It comes in two main parts, the first emphasising relevant mathematical theory, the second a discussion of applications, mainly in biology/brain modelling, but also in (e.g.) physics, computer networks and social science. Chapter 1 by Bollobás and Riordan, discusses various random graph models, including a very general model of random graphs with independence studied by these two authors and S. Janson. Much attention is paid to the the idea that the set of vertices in a sparse Erdős-Rényi random graph \(G(n,c/n)\) at distance \(i\) from a fixed vertex can be approximated by the \(i\)th generation of a branching process with offspring size having Poisson distribution with parameter \(c\), which underlies the idea of a giant (linear in the number \(n\) of vertices) component forming when \(c>1\), only small components forming if \(c<1\) and medium sized components for \(c=1\) (the `phase transition'), and some discussion of the generalisations of this idea to other models, e.g. that of Bollobás-Janson-Riordan. Chapter 2 by Balister, Bollobás and Sarkar discussses random geometric graphs where one has a Poisson process of rate 1 in \({\mathbb R}^{2}\) and two points are adjacent if and only if the distance between them is less than \(r\) (closely related to the model discussed in M. D. Penrose's book Random Geometric Graphs). It also discusses the \(k\)-nearest neighbour model where instead each point of the process is joined to its \(k\) nearest neighbours, and various random tessellation models. Again thresholds, where the qualitative behaviour of the system changes suddenly are discussed. Chapter 3 by Cohen and Havlin discusses (not always fully rigorously) links between systems at a threshold -- corresponding to \(c=1\) in the second paragraph -- and the distance between vertices. In particular, the so-called scale free graphs, where (roughly) the prob…
reviewed by: David B. Penman
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 05-02
zbMATH DE Number: 5316760
zbMATH Keywords: random graph
zbMATH Keywords: network
zbMATH Keywords: random geometric graph
zbMATH Keywords: random tree
zbMATH Keywords: critical point
zbMATH Keywords: cortical network
zbMATH Keywords: k-clique percolation
Wikidata QID: Q64117667
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-69395-6
OpenAlex ID: W577299679
description: scientific article; zbMATH DE number 5316760