Highly dispersed networks generated by enhanced redirection

We analyze growing networks that are built by enhanced redirection. Nodes are sequentially added and each incoming node attaches to a randomly chosen 'target' node with probability 1 − r, or to the parent of the target node with probability r. When the redirection probability r is an increasing function of the degree of the parent node, with r → 1 as the parent degree diverges, networks grown via this enhanced redirection mechanism exhibit unusual properties, including (i) multiple macrohubs, i.e., nodes with degrees proportional to the number of network nodes N; (ii) non-extensivity of the degree distribution in which the number of nodes of degree k, Nk, scales as Nν−1/kν, with 1 < ν < 2; (iii) lack of self-averaging, with large fluctuations between individual network realizations. These features are robust and continue to hold when the incoming node has out-degree greater than 1 so that networks contain closed loops. The latter networks are strongly clustered; for the specific case of double attachment, the average local clustering coefficient is 〈Ci〉 = 4ln2 − 2 = 0.772 58... .

[1]  Wolfgang Woess,et al.  Isotropic Markov semigroups on ultra-metric spaces , 2013, 1304.6271.

[2]  Ginestra Bianconi,et al.  Competition and multiscaling in evolving networks , 2001 .

[3]  P. Krapivsky,et al.  Random ancestor trees , 2010, 1004.1690.

[4]  Ben-Naim,et al.  Scale invariance and lack of self-averaging in fragmentation , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[6]  A. Barabasi,et al.  Bose-Einstein condensation in complex networks. , 2000, Physical review letters.

[7]  S Redner,et al.  Statistics of changes in lead node in connectivity-driven networks. , 2002, Physical review letters.

[8]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[9]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[10]  N. L. Johnson,et al.  Urn models and their application : an approach to modern discrete probability theory , 1978 .

[11]  Albert-László Barabási,et al.  Hierarchical organization in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Andrea Rinaldo,et al.  Network structures from selection principles. , 2004, Physical review letters.

[13]  Bernard Derrida,et al.  Statistical properties of valleys in the annealed random map model , 1988 .

[14]  Samuel Kotz,et al.  Urn Models and Their Applications: An Approach to Modern Discrete Probability Theory , 1978, The Mathematical Gazette.

[15]  A. Vázquez.  Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Christos H. Papadimitriou,et al.  Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet , 2002, ICALP.

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

[18]  K. Pearson.  Biometrika , 1902, The American Naturalist.

[19]  Bernard Derrida,et al.  Statistical properties of randomly broken objects and of multivalley structures in disordered systems , 1987 .

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

[21]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[22]  Paul L. Krapivsky,et al.  A statistical physics perspective on Web growth , 2002, Comput. Networks.

[23]  Hernán D Rozenfeld,et al.  Designer nets from local strategies. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Long Wang,et al.  Empirical analysis of online social networks in the age of Web 2.0 , 2008 .

[25]  Alan Gabel,et al.  Highly dispersed networks by enhanced redirection. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  S. Redner,et al.  Network growth by copying. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  R. Guimerà,et al.  The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[28]  K. Dean,et al.  A 20 By , 2009 .

[29]  Morton E. O'Kelly,et al.  Hub‐and‐Spoke Networks in Air Transportation: An Analytical Review , 1999 .

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

[31]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[32]  Francisco J. Acedo,et al.  Co-Authorship in Management and Organizational Studies: An Empirical and Network Analysis , 2006 .

[33]  S. Redner,et al.  Connectivity of growing random networks. , 2000, Physical review letters.

[34]  Anthony C. Davison,et al.  Statistics of Extremes , 2015, International Encyclopedia of Statistical Science.

[35]  Lan V. Zhang,et al.  Evidence for dynamically organized modularity in the yeast protein–protein interaction network , 2004, Nature.

[36]  R. Solé,et al.  Optimization in Complex Networks , 2001, cond-mat/0111222.

[37]  Bernard Derrida,et al.  The random map model: a disordered model with deterministic dynamics , 1987 .

[38]  Marc Barthelemy,et al.  Spatial Networks , 2010, Encyclopedia of Social Network Analysis and Mining.

[39]  P. Krapivsky,et al.  Stable distributions in stochastic fragmentation , 2001, cond-mat/0108547.

[40]  M. Ausloos,et al.  Growing network with j-redirection , 2007 .

[41]  Sidney Redner,et al.  Sublinear but never superlinear preferential attachment by local network growth , 2012, ArXiv.

[42]  S. N. Dorogovtsev,et al.  Structure of growing networks with preferential linking. , 2000, Physical review letters.

[43]  Dmitri V. Krioukov,et al.  Scale-free networks as pre-asymptotic regimes of super-linear preferential attachment , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[45]  Higgs.  Frequency distributions in population genetics parallel those in statistical physics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  Marián Boguñá,et al.  Popularity versus similarity in growing networks , 2011, Nature.

[47]  B. M. Fulk.  MATH , 1992 .

[48]  G. Pólya,et al.  Über die Statistik verketteter Vorgänge , 1923 .

[49]  S. Redner,et al.  Finiteness and fluctuations in growing networks , 2002, cond-mat/0207107.

[50]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[51]  S. Redner,et al.  Organization of growing random networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.