Finite-size scaling in complex networks.

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.

[1]  Michael E. Fisher,et al.  Scaling Theory for Finite-Size Effects in the Critical Region , 1972 .

[2]  T. E. Harris.  Contact Interactions on a Lattice , 1974 .

[3]  É. Brézin.  An Investigation of Finite Size Scaling , 1982 .

[4]  R. Botet,et al.  Size Scaling for Infinitely Coordinated Systems , 1982 .

[5]  K. Binder,et al.  Finite-Size Tests of Hyperscaling , 1985 .

[6]  Browne,et al.  Critical behavior of an autocatalytic reaction model. , 1990, Physical review. A, Atomic, molecular, and optical physics.

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

[8]  M. Newman,et al.  Exact solution of site and bond percolation on small-world networks. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

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

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

[12]  P Minnhagen,et al.  XY model in small-world networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.

[14]  Reuven Cohen,et al.  Percolation critical exponents in scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  F. Iglói,et al.  First- and second-order phase transitions in scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Beom Jun Kim,et al.  Synchronization on small-world networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  C. Herrero.  Ising model in small-world networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Beom Jun Kim,et al.  Comment on "Ising model on a small world network". , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Sergey N. Dorogovtsev,et al.  Ising Model on Networks with an Arbitrary Distribution of Connections , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  R. Zecchina,et al.  Ferromagnetic ordering in graphs with arbitrary degree distribution , 2002, cond-mat/0203416.

[21]  S. N. Dorogovtsev,et al.  Critical phenomena in networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  C. Herrero,et al.  Ising model in scale-free networks: a Monte Carlo simulation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  G. Ódor.  Universality classes in nonequilibrium lattice systems , 2002, cond-mat/0205644.

[24]  S. N. Dorogovtsev,et al.  Potts model on complex networks , 2004 .

[25]  S Lübeck,et al.  Finite-size scaling of directed percolation above the upper critical dimension. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[27]  Hyunggyu Park,et al.  Collective synchronization in spatially extended systems of coupled oscillators with random frequencies. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Hyunggyu Park,et al.  Asymmetrically coupled directed percolation systems. , 2005, Physical review letters.

[29]  Deok-Sun Lee.  Synchronization transition in scale-free networks: clusters of synchrony. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[31]  Márton Karsai,et al.  Nonequilibrium phase transitions and finite-size scaling in weighted scale-free networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  R. Pastor-Satorras,et al.  Castellano and Pastor-Satorras Reply: , 2007 .

[33]  Hyunggyu Park,et al.  Comment on "non-mean-field behavior of the contact process on scale-free networks". , 2006, Physical review letters.