Exchangeable Random Networks

We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that characterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. A particular case of directed networks with power-law out–degree is studied in more detail, as an example of the flexibility of the model in applications.

[1]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[2]  Eugenio Regazzini,et al.  A Central Limit Problem for Partially ExchangeableRandom Variables , 1997 .

[3]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[4]  Eugene P. Wigner,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1966 .

[5]  G. V. Balakin.  On Random Matrices , 1967 .

[6]  H. Cram'er,et al.  On asymptotic expansions for sums of independent random variables with a limiting stable distribution , 1994 .

[7]  Linyuan Lu,et al.  The Small World Phenomenon in Hybrid Power Law Graphs , 2004 .

[8]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[9]  Riccardo Zecchina,et al.  Alternative solutions to diluted p-spin models and XORSAT problems , 2002, ArXiv.

[10]  Stéphane Robin,et al.  Networks motifs: mean and variance for the count , 2006 .

[11]  Eugenio Regazzini,et al.  On the central limit problem for partially exchangeable random variables with values in a Hilbert space@@@On the central limit problem for partially exchangeable random variables with values in a Hilbert space , 1997 .

[12]  P. Bourgine,et al.  Topological and causal structure of the yeast transcriptional regulatory network , 2002, Nature Genetics.

[13]  M. Gerstein,et al.  Structure and evolution of transcriptional regulatory networks. , 2004, Current opinion in structural biology.

[14]  R. Milo,et al.  Network motifs in integrated cellular networks of transcription-regulation and protein-protein interaction. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[15]  B Bassetti,et al.  Random networks tossing biased coins. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  J. Angus.  The Asymptotic Theory of Extreme Order Statistics , 1990 .

[17]  Béla Bollobás,et al.  Mathematical results on scale‐free random graphs , 2005 .

[18]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[19]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[20]  Nicola J. Rinaldi,et al.  Transcriptional regulatory code of a eukaryotic genome , 2004, Nature.

[21]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[22]  M. Cosentino Lagomarsino,et al.  The large-scale logico-chemical structure of a transcriptional regulation network , 2005, q-bio/0502017.

[23]  Fan Chung Graham,et al.  A Random Graph Model for Power Law Graphs , 2001, Exp. Math..

[24]  A. Rao,et al.  A Markov chain Monte carol method for generating random (0, 1)-matrices with given marginals , 1996 .

[25]  S. Shen-Orr,et al.  Networks Network Motifs : Simple Building Blocks of Complex , 2002 .

[26]  Fan Chung Graham,et al.  The Average Distance in a Random Graph with Given Expected Degrees , 2004, Internet Math..

[27]  M. Mézard,et al.  Two Solutions to Diluted p-Spin Models and XORSAT Problems , 2003 .

[28]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[29]  Béla Bollobás,et al.  Directed scale-free graphs , 2003, SODA '03.

[30]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[31]  A. Arkin,et al.  Motifs, modules and games in bacteria. , 2003, Current opinion in microbiology.

[32]  R. Samworth,et al.  Central limit theorem and convergence to stable laws in Mallows distance , 2004, math/0406218.

[33]  Julio Collado-Vides,et al.  RegulonDB (version 3.2): transcriptional regulation and operon organization in Escherichia coli K-12 , 2001, Nucleic Acids Res..

[34]  Béla Bollobás,et al.  Robustness and Vulnerability of Scale-Free Random Graphs , 2004, Internet Math..

[35]  Nicola J. Rinaldi,et al.  Transcriptional Regulatory Networks in Saccharomyces cerevisiae , 2002, Science.

[36]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..

[37]  R. Milo,et al.  Subgraphs in random networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Eugenio Regazzini,et al.  On the Central Limit Problem for Partially Exchangeable Random Variables with Values in a Hilbert Space , 1998 .

[39]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[40]  Masato Okada,et al.  One step RSB scheme for the rate distortion function , 2002, cond-mat/0207637.

[41]  Charles M. Goldie,et al.  Poisson mixtures and quasi-infinite divisibility of distributions , 1979 .

[42]  M. Newman.  Random Graphs as Models of Networks , 2002, cond-mat/0202208.

[43]  Béla Bollobás,et al.  Coupling Scale-Free and Classical Random Graphs , 2004, Internet Math..

[44]  D. Aldous.  Exchangeability and related topics , 1985 .

[45]  Eugenio Regazzini,et al.  A central limit problem for partially exchangeable random variables@@@A central limit problem for partially exchangeable random variables , 1996 .

[46]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[47]  J. Grandell.  Mixed Poisson Processes , 1997 .

[48]  A. A. Levitskaya.  Systems of Random Equations over Finite Algebraic Structures , 2005 .

[49]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[50]  Fan Chung Graham,et al.  Random evolution in massive graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[51]  S. Shen-Orr,et al.  Network motifs in the transcriptional regulation network of Escherichia coli , 2002, Nature Genetics.

[52]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[53]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[54]  S. Shen-Orr,et al.  Superfamilies of Evolved and Designed Networks , 2004, Science.

[55]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[56]  Gordon E. Willmot,et al.  Asymptotic tail behaviour of Poisson mixtures by applications , 1990, Advances in Applied Probability.

[57]  Rick Durrett,et al.  Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics) , 2006 .

[58]  Yuguo Chen,et al.  Sequential Monte Carlo Methods for Statistical Analysis of Tables , 2005 .

[59]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[60]  F. Steutel,et al.  Infinite Divisibility of Probability Distributions on the Real Line , 2003 .

[61]  Fan Chung Graham,et al.  The Volume of the Giant Component of a Random Graph with Given Expected Degrees , 2006, SIAM J. Discret. Math..

[62]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[63]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[64]  A. Rí,et al.  On random matrices II , 1968 .