Finiteness and fluctuations in growing networks

We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behaviour and they are also sensitive to the initial conditions. We argue that fluctuations in the number of nodes of degree k become Gaussian for fixed degree as the size of the network diverges. We also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution.

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

[2]  Herbert A. Simon,et al.  Some Further Notes on a Class of Skew Distribution Functions , 1960, Inf. Control..

[3]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[4]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[5]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[6]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.

[7]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[8]  T. Haaf,et al.  Genome evolution: Global methylation in eutherian hybrids , 1999, Nature.

[9]  Lada A. Adamic,et al.  Internet: Growth dynamics of the World-Wide Web , 1999, Nature.

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

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

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

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

[14]  S. N. Dorogovtsev,et al.  Size-dependent degree distribution of a scale-free growing network. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Z. Burda,et al.  Statistical ensemble of scale-free random graphs. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  J. Kertész,et al.  Preferential growth: exact solution of the time-dependent distributions. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  D. Zanette,et al.  Vertical transmission of culture and the distribution of family names. , 2000, nlin/0009046.

[18]  S. N. Dorogovtsev,et al.  Anomalous percolation properties of growing networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[20]  J. Hopcroft,et al.  Are randomly grown graphs really random? , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[22]  Walter Willinger,et al.  Scaling phenomena in the Internet: Critically examining criticality , 2002, Proceedings of the National Academy of Sciences of the United States of America.