Steady-state dynamics of the forest fire model on complex networks

AbstractMany sociological networks, as well as biological and technological ones, can be represented in terms of complex networks with a heterogeneous connectivity pattern. Dynamical processes taking place on top of them can be very much influenced by this topological fact. In this paper we consider a paradigmatic model of non-equilibrium dynamics, namely the forest fire model, whose relevance lies in its capacity to represent several epidemic processes in a general parametrization. We study the behavior of this model in complex networks by developing the corresponding heterogeneous mean-field theory and solving it in its steady state. We provide exact and approximate expressions for homogeneous networks and several instances of heterogeneous networks. A comparison of our analytical results with extensive numerical simulations allows to draw the region of the parameter space in which heterogeneous mean-field theory provides an accurate description of the dynamics, and enlights the limits of validity of the mean-field theory in situations where dynamical correlations become important.

[1]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

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

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

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

[5]  Alessandro Vespignani,et al.  The role of the airline transportation network in the prediction and predictability of global epidemics , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[6]  P. Grassberger.  Critical behaviour of the Drossel-Schwabl forest fire model , 2002, cond-mat/0202022.

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

[8]  Henrik Jeldtoft Jensen,et al.  Self-Organized Criticality , 1998 .

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

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

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

[12]  M. A. Muñoz,et al.  Self-organization without conservation: true or just apparent scale-invariance? , 2009, 0905.1799.

[13]  Y. Moreno,et al.  Epidemic outbreaks in complex heterogeneous networks , 2001, cond-mat/0107267.

[14]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

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

[16]  Alain Barrat,et al.  Rate equation approach for correlations in growing network models. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  J. Gómez-Gardeñes,et al.  Spreading of sexually transmitted diseases in heterosexual populations , 2007, Proceedings of the National Academy of Sciences.

[18]  Marc-Thorsten Hütt,et al.  Topology regulates the distribution pattern of excitations in excitable dynamics on graphs. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Christensen,et al.  Self-organized critical forest-fire model: Mean-field theory and simulation results in 1 to 6 dimenisons. , 1993, Physical review letters.

[20]  Reuven Cohen,et al.  Efficient immunization strategies for computer networks and populations. , 2002, Physical review letters.

[21]  Drossel,et al.  Self-organized critical forest-fire model. , 1992, Physical review letters.

[22]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.

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

[24]  Robert M. Corless,et al.  A sequence of series for the Lambert W function , 1997, ISSAC.

[25]  O. Dunkel,et al.  A Course in Mathematical Analysis , 1904 .

[26]  R. Anderson,et al.  Power laws governing epidemics in isolated populations , 1996, Nature.

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

[28]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[29]  Yamir Moreno,et al.  The Bak-Sneppen model on scale-free networks , 2001, cond-mat/0108494.

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

[31]  R. Pastor-Satorras,et al.  Langevin approach for the dynamics of the contact process on annealed scale-free networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  P. Bak,et al.  A forest-fire model and some thoughts on turbulence , 1990 .

[33]  des lettres et des beaux-arts de Belgique.,et al.  Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles. , 1827 .

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

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

[36]  R. Pastor-Satorras,et al.  Diffusion-annihilation processes in complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[38]  J. Liu,et al.  The spread of disease with birth and death on networks , 2004, q-bio/0402042.

[39]  Bojin Zheng,et al.  Steady states and critical behavior of epidemic spreading on complex networks , 2008, 2008 7th World Congress on Intelligent Control and Automation.

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

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

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

[43]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[44]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[45]  R. Pastor-Satorras,et al.  Non-mean-field behavior of the contact process on scale-free networks. , 2005, Physical review letters.

[46]  M. Kuperman,et al.  Small world effect in an epidemiological model. , 2000, Physical review letters.

[47]  Alessandro Vespignani,et al.  Immunization of complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  B Kahng,et al.  Sandpile on scale-free networks. , 2003, Physical review letters.

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