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Percolation and Random Graphs

Remco van der Hofstad

in New Perspectives in Stochastic Geometry

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199232574
eISBN:
9780191716393
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232574.003.0006
Subject:
Mathematics, Geometry / Topology

In this chapter, we define percolation and random graph models, and survey the features of these models.


Random graphs: An introduction to the most basic of network models, the random graph

M. E. J. Newman

in Networks: An Introduction

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.003.0012
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the basic mathematics of the random graph G(n, p), focusing particularly on the degree distribution and component sizes, which are two of the model's most illuminating ... More


Factor graphs and graph ensembles

Marc Mézard and Andrea Montanari

in Information, Physics, and Computation

Published in print:
2009
Published Online:
September 2009
ISBN:
9780198570837
eISBN:
9780191718755
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570837.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Probabilistic systems involving a large number of simple variables with mutual dependencies appear recurrently in several fields of science. It is often the case that such dependencies can be ... More


Other network models: A brief introduction to two specialized network models, the small-world model and the exponential random graph

M. E. J. Newman

in Networks: An Introduction

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.003.0015
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter describes two of the best-known additional types of network models: the small-world model and exponential random graphs. Exercises are provided at the end of the chapter.


Random Directed and on‐Line Networks

Mathew D. Penrose and Andrew R. Wade

in New Perspectives in Stochastic Geometry

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199232574
eISBN:
9780191716393
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232574.003.0007
Subject:
Mathematics, Geometry / Topology

Various random spatial graphs defined on partially ordered point sets have been treated recently in the literature: these are discussed in the framework of the minimal directed spanning forest. ... More


EQUILIBRIUM NETWORKS

S. N. Dorogovtsev and J. F. F. Mendes

in Evolution of Networks: From Biological Nets to the Internet and WWW

Published in print:
2003
Published Online:
January 2010
ISBN:
9780198515906
eISBN:
9780191705670
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198515906.003.0005
Subject:
Physics, Soft Matter / Biological Physics

This chapter discusses the organization of equilibrium networks. These include the classical random graphs, uncorrelated and correlated networks, and the Watts–Strogatz model (small-world networks). ... More


Random graphs with general degree distributions: This chapter describes more sophisticated random graph models that mimic networks with arbitrary degree distributions

M. E. J. Newman

in Networks: An Introduction

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The previous chapter looked at the classic random graph model, in which pairs of vertices are connected at random with uniform probabilities. Although this model has proved tremendously useful as a ... More


Networks: An Introduction

Mark Newman

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet ... More


When Formulas Freeze: Phase Transitions in Computation

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0014
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Certain formulas, such as the 3-SAT formula, undergo a phase transition from almost certain satisfiability to almost certain unsatisfiability when the number of constraints per variable reaches a ... More


Ising models on random graph

Marc Mézard and Andrea Montanari

in Information, Physics, and Computation

Published in print:
2009
Published Online:
September 2009
ISBN:
9780198570837
eISBN:
9780191718755
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570837.003.0017
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter studies two problems of statistical physics: the ferromagnet and the spin glass, on large random graphs with fixed degree profile. It describes the use of the replica symmetric cavity ... More


Outlook

Sergey N. Dorogovtsev

in Lectures on Complex Networks

Published in print:
2010
Published Online:
May 2010
ISBN:
9780199548927
eISBN:
9780191720574
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199548927.003.0015
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of ... More


Analysis of Crossword Puzzle Difficulty Using a Random Graph Process

John K. McSweeney

in The Mathematics of Various Entertaining Subjects: Research in Recreational Math

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691164038
eISBN:
9781400881338
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691164038.003.0008
Subject:
Mathematics, History of Mathematics

This chapter quantifies the dynamics of a crossword puzzle by using a network structure to model it. Specifically, the chapter determines how the interaction between the structure of cells in the ... More


Exponential Random Graph Models, P* and Actor Oriented Models

Thomas W. Valente

in Social Networks and Health: Models, Methods, and Applications

Published in print:
2010
Published Online:
May 2010
ISBN:
9780195301014
eISBN:
9780199777051
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195301014.003.0009
Subject:
Public Health and Epidemiology, Epidemiology

This chapter describes in non-technical terms how researchers can determine whether empirical networks exhibit certain structural properties (centrality, triadic transitivity, etc.) using a ... More


Network economy

František Slanina

in Essentials of Econophysics Modelling

Published in print:
2013
Published Online:
January 2014
ISBN:
9780199299683
eISBN:
9780191747038
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199299683.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter is devoted to structural modelling of economic phenomena, based on the theory of complex networks. Its mathematical basis is the graph theory, which is briefly introduced. Then, we ... More


Random graphs

Mark Newman

in Networks

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805090
eISBN:
9780191843235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805090.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties ... More


Classifying Networks with dk-Series

Marija Mitrović Dankulov, Guido Caldarelli, Santo Fortunato, and Dmitri Krioukov

in Multiplex and Multilevel Networks

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198809456
eISBN:
9780191847073
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198809456.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

“Classifying Networks with dk-Series” discusses the dk-series and how it can be extended to describe the structure of multiplex networks. One way to address the problem of interdependence among ... More


Random graph ensembles

A.C.C. Coolen, A. Annibale, and E.S. Roberts

in Generating Random Networks and Graphs

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198709893
eISBN:
9780191780172
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198709893.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents some theoretical tools for defining random graph ensembles systematically via soft or hard topological constraints including working through some properties of the Erdös-Rényi ... More


Soft constraints: exponential random graph models

A.C.C. Coolen, A. Annibale, and E.S. Roberts

in Generating Random Networks and Graphs

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198709893
eISBN:
9780191780172
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198709893.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Exponential random graph models (ERGMs) provide conceptually elegant recipes for generating soft-constrained random graphs. This chapter begins by explaining the theory and describing how to properly ... More


Generating Random Networks and Graphs

Ton Coolen, Alessia Annibale, and Ekaterina Roberts

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198709893
eISBN:
9780191780172
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198709893.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book supports researchers who need to generate random networks, or who are interested in the theoretical study of random graphs. The coverage includes exponential random graphs (where the ... More


Random models of networks

Ernesto Estrada

in The Structure of Complex Networks: Theory and Applications

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199591756
eISBN:
9780191774959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199591756.003.0012
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter describes random models frequently used for studying complex networks. These include the Erdös-Rényi, Barabási-Albert and its variations, small-world models of Watts-Strogatz and ... More


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