Table Of ContentGENERATING RANDOM NETWORKS AND GRAPHS
Generating Random Networks and Graphs
A.C.C. Coolen
DepartmentofMathematics,King’sCollegeLondon,UK
A. Annibale
DepartmentofMathematics,King’sCollegeLondon,UK
E.S. Roberts
DepartmentofMathematics,King’sCollegeLondon,UK
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Preface
Networks have become popular tools for characterizing and visualizing complex sys-
tems with many interacting variables. In the study of such systems, it is often helpful
to be able to generate random networks with statistical features that mimic those of
the true network at hand. These random networks can serve as ‘null hypotheses’ in
hypothesis testing, or as proxies in simulation studies. How to generate random net-
workswithcontrolledtopologicalcharacteristicsaccuratelyandefficientlyhasbecome
an important practical challenge, and is in itself an interesting theoretical question.
Randomnetworksresult,bydefinition,fromstochasticprocesses.Thenaturallan-
guage for graph generation protocols is therefore that of network or graph ensembles.
These ensembles are constrained by the statistical features that one chooses to im-
pose. The core of the network generation problem is to construct stochastic processes
or algorithms that, upon equilibration, sample the space of all networks that have
thedesiredfeatures,butareotherwise unbiased.Moreover,thesealgorithmsshouldbe
able to generate large networks within practical time scales.
We set out to write a concise but self-contained text on the generation of random
networks with controlled topological properties, including the mathematical deriva-
tionsofrelevantalgorithmsandtheirnumericalimplementations.Suchdetailsappear
to be under-represented in the presently available books. They tend to be found in
discipline-specific journals that are not easily accessible, and research articles devoted
toonealgorithmatatime.Wehavetriedtobringtogetherthemostimportantresults
that are scattered over the physics, combinatorial mathematics and computer science
literature, and explain with uniform notation the rationale, the potential and limita-
tions, and application domains of each approach. We seek to explain the theoretical
issues and the practical algorithms resulting from analysis of the problem in such a
way that the reader is able to focus on either aspect if he or she so wishes. To achieve
this, we have separated (where possible) practical recipes from theoretical arguments,
to aid selective navigation driven by personal and domain-specific preferences and
needs. More extensive mathematical arguments are delegated to appendices, and we
have supplemented the main text with examples and exercises with solutions.
This book is aimed at established or junior researchers and their graduate or ad-
vanced undergraduate students in computer science, quantitative biology, social sci-
ence,ecology,bioinformatics,appliedmathematicsandtheoreticalphysics,whosework
involvesnetworksorgraphs.Wehopethatitmayserveasausefulreference,asatext-
book for postgraduate lecture courses and as an inspiration for further quantitative
research.
London, November 2016 Ton Coolen, Alessia Annibale and Kate Roberts
Acknowledgements
Itisourgreatpleasuretothankthemanycolleaguesandstudentswithwhomoverthe
yearswehaveenjoyeddiscussingthemodellingandgenerationofgraphsandnetworks.
Inparticular,wewouldliketomention(inalphabeticalorder)ElenaAgliari,Ginestra
Bianconi, Zdisl(cid:32)aw Burda, Sun Sook Chung, Owen Courtney, Andrea De Martino,
LuisFernandes,ThomasFink,FrancaFraternali,ClaraGr´acio,AlexanderHartmann,
JensKleinjung,ReimerKu¨hn,AlexanderMozeika,AlessandroPandini,NuriaPlanell-
Morell, Conrad Perez-Vicente and Peter Sollich.
Contents
PART I THE BASICS
1 Introduction 3
2 Definitions and concepts 9
2.1 Definitions of graphs and their local characteristics 9
2.2 Macroscopic characterizations of graphs 11
2.3 Solutions of exercises 17
3 Random graph ensembles 19
3.1 The Erd¨os–R´enyi graph ensemble 20
3.2 Graph ensembles with hard or soft topological constraints 22
3.3 The link between ensembles and algorithms 25
3.4 Solutions of exercises 28
PART II RANDOM GRAPH ENSEMBLES
4 Soft constraints: exponential random graph models 33
4.1 Definitions and basic properties of ERGMs 33
4.2 ERGMs that can be solved exactly 36
4.3 ERGMs with phase transitions: the two-star model 40
4.4 ERGMs with phase transitions: the Strauss (triangle) model 47
4.5 Stochastic block models for graphs with community structure 57
4.6 Strengths and weaknesses of ERGMs as null models 59
4.7 Solutions of exercises 62
5 Ensembles with hard constraints 64
5.1 Basic properties and tools 64
5.2 Nondirected graphs with hard-constrained number of links 68
5.3 Prescribeddegreestatisticsandhard-constrainednumberoflinks 70
5.4 Ensembles with prescribed numbers of links and two-stars 73
5.5 Ensembles with constrained degrees and short loops 75
5.6 Solutions of exercises 79
PART III GENERATING GRAPHS FROM GRAPH ENSEMBLES
6 Markov Chain Monte Carlo sampling of graphs 83
6.1 The Markov Chain Monte Carlo sampling method 83
6.2 MCMC sampling for exponential random graph models 89
6.3 MCMC sampling for graph ensembles with hard constraints 94
6.4 Properties of move families – hinge flips and edge swaps 97
6.5 Solutions of exercises 102
viii Contents
7 Graphs with hard constraints: further applications and ex-
tensions 105
7.1 Uniform versus non-uniform sampling of candidate moves 105
7.2 Graphswithaprescribeddegreedistributionandnumberoflinks 112
7.3 Ensembles with controlled degrees and degree correlations 117
7.4 Generating graphs with prescribed degrees and correlations 121
7.5 Edge swaps revisited 128
7.6 Non-uniform sampling of allowed edge swaps 131
7.7 Non-uniform sampling from a restricted set of moves: a hybrid
MCMC algorithm 135
7.8 Extensions to directed graphs 139
7.9 Solutions of exercises 141
PART IV GRAPHS DEFINED BY ALGORITHMS
8 Network growth algorithms 153
8.1 Configuration model 153
8.2 Preferential attachment and scale-free networks 157
8.3 Analyzing growth algorithms 163
8.4 Solutions of exercises 166
9 Specific constructions 169
9.1 Watts–Strogatz model and the ‘small world’ property 169
9.2 Geometric graphs 173
9.3 Planar graphs 177
9.4 Weighted graphs 179
9.5 Solutions of exercises 181
PART V FURTHER TOPICS
10 Graphs on structured spaces 187
10.1 Temporal networks 187
10.2 Multiplex graphs 193
10.3 Networks with labelled nodes 197
10.4 Relations and connections between models 202
10.5 Solutions of exercises 204
11 Applications of random graphs 207
11.1 Power grids 207
11.2 Social networks 208
11.3 Food webs 212
11.4 World Wide Web 215
11.5 Protein–protein interaction networks 216
Key symbols and terminology 221
Appendix A The delta distribution 229
AppendixB ClusteringandcorrelationsinErd¨os–R´enyigraphs 231
Contents ix
B.1 Clustering coefficients 231
B.2 Degree correlations in the sparse regime 233
AppendixC Solutionofthetwo-starexponentialrandomgraph
model (ERGM) in the sparse regime 236
Appendix D Steepest descent integration 240
Appendix E Number of sparse graphs with prescribed average
degree and degree variance 242
Appendix F Evolution of graph mobilities for edge-swap dy-
namics 244
AppendixG Numberofgraphswithprescribeddegreesequence 247
Appendix H Degree correlations in ensembles with constrained
degrees 249
Appendix I Evolution of triangle and square counters due to
ordered edge swaps 253
Appendix J Algorithms 256
J.1 Algorithms based on link flips 259
J.2 MCMC algorithms sampling from graphs with hard constraints 268
J.3 Algorithms based on hinge flips 271
J.4 Algorithms based on edge swaps 277
J.5 Growth Algorithms 282
J.6 Auxiliary routines 289
References 297
Index 309
