Analytical solution of the voter model on uncorrelated networks

We present a mathematical description of the voter model dynamics on uncorrelated networks. When the average degree of the graph is µ 6 2 the system reaches complete order exponentially fast. For µ > 2, a finite system falls, before it fully orders, in a quasi-stationary state in which the average density of active links (links between opposite-state nodes) in surviving runs is constant and equal to (µ 2) 3(µ 1) , while an infinitely large system stays ad infinitum in a partially ordered stationary active state. The mean lifetime of the quasi- stationary state is proportional to the mean time to reach the fully ordered state T, which scales as T (µ 1)µ 2 N (µ 2)µ2 , where N is the number of nodes of the network, and µ2 is the second moment of the degree distribution. We find good agreement between these analytical results and numerical simulations on random networks with various degree distributions.

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

[2]  S. Redner,et al.  Voter model on heterogeneous graphs. , 2004, Physical review letters.

[3]  H. Hinrichsen,et al.  Critical coarsening without surface tension: the universality class of the voter model. , 2001, Physical review letters.

[4]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[5]  Luca Dall'Asta,et al.  Effective surface-tension in the noise-reduced voter model , 2006, cond-mat/0612186.

[6]  M. Mobilia,et al.  Voting and catalytic processes with inhomogeneities. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  S. Redner,et al.  On the role of zealotry in the voter model , 2007 .

[8]  S. Redner,et al.  Ultimate fate of constrained voters , 2004, cond-mat/0405652.

[9]  Nazareno G F Medeiros,et al.  Domain motion in the voter model with noise. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  S. Redner,et al.  Saturation transition in a monomer-monomer model of heterogeneous catalysis , 1990 .

[11]  D. Parisi,et al.  Comparison of voter and Glauber ordering dynamics on networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Rick Durrett,et al.  Spatial Models for Species-Area Curves , 1996 .

[13]  V. Eguíluz,et al.  Conservation laws for the voter model in complex networks , 2004, cond-mat/0408101.

[14]  F. Slanina,et al.  Analytical results for the Sznajd model of opinion formation , 2003, cond-mat/0305102.

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

[16]  Jayanth R Banavar,et al.  Spatial scaling in model plant communities. , 2005, Physical review letters.

[17]  P. Clifford,et al.  A model for spatial conflict , 1973 .

[18]  Claudio Castellano,et al.  Incomplete ordering of the voter model on small-world networks , 2003 .

[19]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

[20]  S Redner,et al.  Evolutionary dynamics on degree-heterogeneous graphs. , 2006, Physical review letters.

[21]  Voter model on Sierpinski fractals , 2005, cond-mat/0505350.

[22]  S. Redner,et al.  Constrained opinion dynamics: freezing and slow evolution , 2003 .

[23]  J. T. Cox,et al.  Diffusive Clustering in the Two Dimensional Voter Model , 1986 .

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

[25]  Daniele Vilone,et al.  Solution of voter model dynamics on annealed small-world networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Krapivsky.  Kinetics of monomer-monomer surface catalytic reactions. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[27]  M. A. Muñoz,et al.  Langevin description of critical phenomena with two symmetric absorbing states. , 2004, Physical review letters.

[28]  Maxi San Miguel,et al.  Ordering dynamics with two non-excluding options: bilingualism in language competition , 2006, physics/0609079.

[29]  Krapivsky,et al.  Exact results for kinetics of catalytic reactions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  K. Pearson.  Biometrika , 1902, The American Naturalist.

[31]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

[32]  M. Mobilia.  Does a single zealot affect an infinite group of voters? , 2003, Physical review letters.

[33]  Jérôme Chave,et al.  A Spatially Explicit Neutral Model of β-Diversity in Tropical Forests , 2002 .

[34]  T. Liggett.  Interacting Particle Systems , 1985 .

[35]  Maxi San Miguel,et al.  Microscopic Abrams-Strogatz model of language competition , 2006, physics/0603042.

[36]  K. Kaski,et al.  Anomalous lifetime distributions and topological traps in ordering dynamics , 2007, 0705.2560.

[37]  Krzysztof Suchecki,et al.  Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  M. Scheucher,et al.  A soluble kinetic model for spinodal decomposition , 1988 .

[39]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[40]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[41]  D. Stroock,et al.  A Martingale Approach to Infinite Systems of Interacting Processes , 1976 .