Fast fixation with a generic network structure.

We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model), and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may covary, and that the results are robust to correlations in the network or the initial condition.

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

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

[3]  John Wakeley,et al.  A diffusion approximation for selection and drift in a subdivided population. , 2003, Genetics.

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

[5]  G J Baxter,et al.  Fixation and consensus times on a network: a unified approach. , 2008, Physical review letters.

[6]  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.

[7]  William Croft,et al.  Explaining language change : an evolutionary approach , 2000 .

[8]  Maxi San Miguel,et al.  Generic absorbing transition in coevolution dynamics. , 2007, Physical review letters.

[9]  Kenneth C. W. Kammeyer,et al.  An introduction to population , 1974 .

[10]  R. A. Blythe,et al.  Ordering in voter models on networks: exact reduction to a single-coordinate diffusion , 2010, 1006.1557.

[11]  Alain Barrat,et al.  Who's talking first? Consensus or lack thereof in coevolving opinion formation models. , 2007, Physical review letters.

[12]  S. Wright.  Evolution in mendelian populations , 1931 .

[13]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

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

[15]  A. J. McKane,et al.  Stochastic models of evolution in genetics, ecology and linguistics , 2007, cond-mat/0703478.

[16]  J. T. Cox,et al.  Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$ , 1989 .

[17]  W. Bruce Croft,et al.  Modeling language change: An evaluation of Trudgill's theory of the emergence of New Zealand English , 2009, Language Variation and Change.

[18]  S. Redner,et al.  Voter models on heterogeneous networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Leonard M. Sander,et al.  A Generalized Voter Model on Complex Networks , 2009 .

[20]  G. J. Baxter,et al.  A voter model with time dependent flip rates , 2011, 1104.2840.

[21]  W. Bruce Croft,et al.  Selection Model of , 2022 .

[22]  R. A. Fisher,et al.  The Genetical Theory of Natural Selection , 1931 .

[23]  L. Gasser,et al.  Centers and peripheries: Network roles in language change , 2010 .

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

[25]  S. Redner,et al.  Heterogeneous voter models. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  R. Pastor-Satorras,et al.  Heterogenous mean-field analysis of a generalized voter-like model on networks , 2011, 1106.4215.