Frailty effects in networks: comparison and identification of individual heterogeneity versus preferential attachment in evolving networks

Preferential attachment is a proportionate growth process in networks, where nodes receive new links in proportion to their current degree. Preferential attachment is a popular generative mechanism to explain the widespread observation of power-law-distributed networks. An alternative explanation for the phenomenon is a randomly grown network with large individual variation in growth rates among the nodes (frailty). We derive analytically the distribution of individual rates, which will reproduce the connectivity distribution that is obtained from a general preferential attachment process (Yule process), and the structural differences between the two types of graphs are examined by simulations. We present a statistical test to distinguish the two generative mechanisms from each other and we apply the test to both simulated data and two real data sets of scientific citation and sexual partner networks. The findings from the latter analyses argue for frailty effects as an important mechanism underlying the dynamics of complex networks.

[1]  M. Newman.  Power laws, Pareto distributions and Zipf's law , 2005 .

[2]  M. Handcock,et al.  An assessment of preferential attachment as a mechanism for human sexual network formation , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[3]  Olivier Toubia,et al.  Explaining the Power-Law Degree Distribution in a Social Commerce Network , 2009, Soc. Networks.

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

[5]  Z. Neda,et al.  Measuring preferential attachment in evolving networks , 2001, cond-mat/0104131.

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

[7]  W. Reed,et al.  From gene families and genera to incomes and internet file sizes: why power laws are so common in nature. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Derek de Solla Price,et al.  A general theory of bibliometric and other cumulative advantage processes , 1976, J. Am. Soc. Inf. Sci..

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

[10]  G. Miller,et al.  Some effects of intermittent silence. , 1957, The American journal of psychology.

[11]  Martina Morris,et al.  Degree Distributions in Sexual Networks: A Framework for Evaluating Evidence , 2008, Sexually transmitted diseases.

[12]  M. Handcock,et al.  Likelihood-based inference for stochastic models of sexual network formation. , 2004, Theoretical population biology.

[13]  J M Hyman,et al.  Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[14]  W. N. Bailey.  Confluent Hypergeometric Functions , 1960, Nature.

[15]  W. Whitt,et al.  Modeling service-time distributions with non-exponential tails : Beta mixtures of exponentials , 1999 .

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

[17]  Ward Whitt,et al.  Computing Laplace Transforms for Numerical Inversion Via Continued Fractions , 1999, INFORMS J. Comput..

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

[19]  Fredrik Liljeros,et al.  Preferential attachment in sexual networks , 2007, Proceedings of the National Academy of Sciences.

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

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

[22]  Olivier Toubia,et al.  Explaining the Power-Law Degree Distribution in a Social Commerce Network , 2009 .

[23]  J. Willis.  Age and Area , 1926, The Quarterly Review of Biology.

[24]  Yuen Ren Chao,et al.  Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology , 1950 .

[25]  Luís A. Nunes Amaral,et al.  Sexual networks: implications for the transmission of sexually transmitted infections. , 2003, Microbes and infection.

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