Mixing patterns in networks.

We study assortative mixing in networks, the tendency for vertices in networks to be connected to other vertices that are like (or unlike) them in some way. We consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age. As a special example of the latter we consider mixing according to vertex degree, i.e., according to the number of connections vertices have to other vertices: do gregarious people tend to associate with other gregarious people? We propose a number of measures of assortative mixing appropriate to the various mixing types, and apply them to a variety of real-world networks, showing that assortative mixing is a pervasive phenomenon found in many networks. We also propose several models of assortatively mixed networks, both analytic ones based on generating function methods, and numerical ones based on Monte Carlo graph generation techniques. We use these models to probe the properties of networks as their level of assortativity is varied. In the particular case of mixing by degree, we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices.

[1]  H. Spencer The structure of the nervous system. , 1870 .

[2]  D J PRICE,et al.  NETWORKS OF SCIENTIFIC PAPERS. , 1965, Science.

[3]  B. Everitt,et al.  Large sample standard errors of kappa and weighted kappa. , 1969 .

[4]  B. Everitt,et al.  Statistical methods for rates and proportions , 1973 .

[5]  Norman T. J. Bailey,et al.  The Mathematical Theory of Infectious Diseases , 1975 .

[6]  P. Giles,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[7]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[8]  N. Ling The Mathematical Theory of Infectious Diseases and its applications , 1978 .

[9]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[10]  B. Efron Computers and the Theory of Statistics: Thinking the Unthinkable , 1979 .

[11]  David Strauss On a general class of models for interaction , 1986 .

[12]  S. Brenner,et al.  The structure of the nervous system of the nematode Caenorhabditis elegans. , 1986, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[13]  R. May,et al.  Networks of sexual contacts: implications for the pattern of spread of HIV , 1989, AIDS.

[14]  Neo D. Martinez Artifacts or Attributes? Effects of Resolution on the Little Rock Lake Food Web , 1991 .

[15]  J. Catania,et al.  Condom use in multi-ethnic neighborhoods of San Francisco: the population-based AMEN (AIDS in Multi-Ethnic Neighborhoods) Study. , 1992, American journal of public health.

[16]  P. Gács,et al.  Algorithms , 1992 .

[17]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[18]  H. Bhadeshia Diffusion , 1995, Theory of Transformations in Steels.

[19]  Jerrold W. Grossman,et al.  A portion of the well-known collaboration graph , 1995 .

[20]  M. Huxham,et al.  Do Parasites Reduce the Chances of Triangulation in a Real Food Web , 1996 .

[21]  Herbert W. Hethcote,et al.  Epidemic models: Their structure and relation to data , 1996 .

[22]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[23]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

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

[25]  S. Redner How popular is your paper? An empirical study of the citation distribution , 1998, cond-mat/9804163.

[26]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[27]  R. Monasson Diffusion, localization and dispersion relations on “small-world” lattices , 1999 .

[28]  Gerard T. Barkema,et al.  Monte Carlo Methods in Statistical Physics , 1999 .

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

[30]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[31]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[32]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[33]  M. Newman,et al.  Epidemics and percolation in small-world networks. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[34]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[35]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[36]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[37]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[38]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[39]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[40]  Jon M. Kleinberg,et al.  The small-world phenomenon: an algorithmic perspective , 2000, STOC '00.

[41]  S. Havlin,et al.  Breakdown of the internet under intentional attack. , 2000, Physical review letters.

[42]  L. Amaral,et al.  The web of human sexual contacts , 2001, Nature.

[43]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  R. May,et al.  Infection dynamics on scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[47]  R Pastor-Satorras,et al.  Dynamical and correlation properties of the internet. , 2001, Physical review letters.

[48]  Lada A. Adamic,et al.  Search in Power-Law Networks , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[50]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[51]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[53]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[54]  K. Goh,et al.  Spectra and eigenvectors of scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  S. N. Dorogovtsev,et al.  Modern architecture of random graphs: Constructions and correlations , 2002 .

[56]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[57]  M E J Newman,et al.  Identity and Search in Social Networks , 2002, Science.

[58]  M. Newman Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[60]  S H Strogatz,et al.  Random graph models of social networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[61]  L. Sander,et al.  Percolation on heterogeneous networks as a model for epidemics. , 2002, Mathematical biosciences.

[62]  Stephanie Forrest,et al.  Email networks and the spread of computer viruses. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[63]  S. Bornholdt,et al.  Scale-free topology of e-mail networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  M E J Newman Assortative mixing in networks. , 2002, Physical review letters.

[65]  Walter Willinger,et al.  The origin of power laws in Internet topologies revisited , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[66]  Tom A. B. Snijders,et al.  Markov Chain Monte Carlo Estimation of Exponential Random Graph Models , 2002, J. Soc. Struct..

[67]  J. Montoya,et al.  Small world patterns in food webs. , 2002, Journal of theoretical biology.

[68]  Beom Jun Kim,et al.  Attack vulnerability of complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Y. Moreno,et al.  Disease spreading in structured scale-free networks , 2002, cond-mat/0210362.

[70]  Alessandro Vespignani,et al.  Absence of epidemic threshold in scale-free networks with degree correlations. , 2002, Physical review letters.

[71]  Petter Holme,et al.  Subnetwork hierarchies of biochemical pathways , 2002, Bioinform..

[72]  Y. Moreno,et al.  Resilience to damage of graphs with degree correlations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[73]  A Díaz-Guilera,et al.  Self-similar community structure in a network of human interactions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[74]  Alexei Vazquez,et al.  Computational complexity arising from degree correlations in networks , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  P. Bearman,et al.  Chains of Affection: The Structure of Adolescent Romantic and Sexual Networks1 , 2004, American Journal of Sociology.

[76]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]/Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .