Diffusion-annihilation processes in complex networks.

We present a detailed analytical study of the A+A --> 0 diffusion-annihilation process in complex networks. By means of microscopic arguments, we derive a set of rate equations for the density of A particles in vertices of a given degree, valid for any generic degree distribution, and which we solve for uncorrelated networks. For homogeneous networks (with bounded fluctuations), we recover the standard mean-field solution, i.e., a particle density decreasing as the inverse of time. For heterogeneous (scale-free networks) in the infinite network size limit, we obtain instead a density decreasing as a power law, with an exponent depending on the degree distribution. We also analyze the role of finite size effects, showing that any finite scale-free network leads to the mean-field behavior, with a prefactor depending on the network size. We check our analytical predictions with extensive numerical simulations on homogeneous networks with Poisson degree distribution and scale-free networks with different degree exponents.

[1]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

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

[3]  A. A. Ovchinnikov,et al.  Role of density fluctuations in bimolecular reaction kinetics , 1978 .

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

[5]  Benjamin P. Lee.  Renormalization group calculation for the reaction kA + 0 , 1994 .

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

[7]  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.

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

[9]  R. Dickman,et al.  Nonequilibrium Phase Transitions in Lattice Models , 1999 .

[10]  M. Weigt,et al.  On the properties of small-world network models , 1999, cond-mat/9903411.

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

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

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

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

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

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

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

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

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

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

[21]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  A. Wagner.  The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes. , 2001, Molecular biology and evolution.

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

[24]  R. May,et al.  How Viruses Spread Among Computers and People , 2001, Science.

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

[26]  Alessandro Vespignani,et al.  Epidemic dynamics in finite size scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  A. Barabasi,et al.  Evolution of the social network of scientific collaborations , 2001, cond-mat/0104162.

[28]  R. Pastor-Satorras,et al.  Epidemic spreading in correlated complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[30]  Alessandro Vespignani,et al.  Large-scale topological and dynamical properties of the Internet. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[32]  Guido Caldarelli,et al.  Universal scaling relations in food webs , 2003, Nature.

[33]  M. Newman,et al.  Origin of degree correlations in the Internet and other networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Duncan J. Watts,et al.  Six Degrees: The Science of a Connected Age , 2003 .

[35]  Sergey N. Dorogovtsev,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW (Physics) , 2003 .

[36]  Alessandro Vespignani,et al.  Cut-offs and finite size effects in scale-free networks , 2003, cond-mat/0311650.

[37]  L. Gallos,et al.  Absence of kinetic effects in reaction-diffusion processes in scale-free networks. , 2004, Physical review letters.

[38]  Alessandro Vespignani,et al.  Evolution and Structure of the Internet: A Statistical Physics Approach , 2004 .

[39]  K. Sneppen,et al.  Detection of topological patterns in complex networks: correlation profile of the internet , 2002, cond-mat/0205379.

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

[41]  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 .

[42]  R. Pastor-Satorras,et al.  Generation of uncorrelated random scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.