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The noisy voter model on complex networks.

Carro A, Toral R, San Miguel M - Sci Rep (2016)

Bottom Line: Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity.In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations.Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

View Article: PubMed Central - PubMed

Affiliation: IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, Campus Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain.

ABSTRACT
We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

No MeSH data available.


Related in: MedlinePlus

Steady state variance of n as a function of the variance of the degree distribution  for two values of the noise parameter a.In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Circles with error bars: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line and squares: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .
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f4: Steady state variance of n as a function of the variance of the degree distribution for two values of the noise parameter a.In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Circles with error bars: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line and squares: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Mentions: Noticing that, for both asymptotic approximations, the variance becomes an explicit function of the variance of the underlying degree distribution , we present in Fig. 4 a comparison between these analytical functional relationships and the corresponding numerical results for two different values of the noise parameter a. In particular, taking into account the ranges of validity of the asymptotic approximations characterized above (see Fig. 3), we chose values of the noise parameter respectively before [panel a)] and after [panel b)] the crossover point a*, and both of them in the region of a leading to significant differences between network types (see Fig. 2).


The noisy voter model on complex networks.

Carro A, Toral R, San Miguel M - Sci Rep (2016)

Steady state variance of n as a function of the variance of the degree distribution  for two values of the noise parameter a.In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Circles with error bars: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line and squares: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4837380&req=5

f4: Steady state variance of n as a function of the variance of the degree distribution for two values of the noise parameter a.In order to keep all parameters constant except the variance of the degree distribution, a different network type is used for each point (in order of increasing : Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network). Circles with error bars: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line and squares: Analytical results [see equation (14)]. Dotted line: asymptotic approximation for small a [see equation (16)]. Dash-dotted line: asymptotic approximation for large a [see equation (17)]. Dashed line: Mean-field approximation (see42). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .
Mentions: Noticing that, for both asymptotic approximations, the variance becomes an explicit function of the variance of the underlying degree distribution , we present in Fig. 4 a comparison between these analytical functional relationships and the corresponding numerical results for two different values of the noise parameter a. In particular, taking into account the ranges of validity of the asymptotic approximations characterized above (see Fig. 3), we chose values of the noise parameter respectively before [panel a)] and after [panel b)] the crossover point a*, and both of them in the region of a leading to significant differences between network types (see Fig. 2).

Bottom Line: Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity.In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations.Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

View Article: PubMed Central - PubMed

Affiliation: IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, Campus Universitat de les Illes Balears, E-07122, Palma de Mallorca, Spain.

ABSTRACT
We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity--variance of the underlying degree distribution--has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

No MeSH data available.


Related in: MedlinePlus