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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 noise parameter a for a Barabási-Albert scale-free network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: 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: Crossover point between both asymptotic approximations (a* = 0.014157). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .
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f3: Steady state variance of n as a function of the noise parameter a for a Barabási-Albert scale-free network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: 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: Crossover point between both asymptotic approximations (a* = 0.014157). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .

Mentions: For a more precise characterization of the ranges of validity of these two asymptotic approximations with respect to the noise parameter a, we present in Fig. 3 the variance of n as a function of a for the numerical results and the three corresponding analytical expressions presented so far: the analytical result in equation (14), the asymptotic expression for small a in equation (16) and the asymptotic expression for large a in equation (17). Note the use a Barabási-Albert scale-free network as an example. Furthermore, we also show in this figure the crossover point a* between both approximations, that we define as the value of a that minimizes the distance between the logarithmic values of both functions (17) and (16).


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 noise parameter a for a Barabási-Albert scale-free network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: 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: Crossover point between both asymptotic approximations (a* = 0.014157). 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

f3: Steady state variance of n as a function of the noise parameter a for a Barabási-Albert scale-free network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid line: 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: Crossover point between both asymptotic approximations (a* = 0.014157). The interaction parameter is fixed as h = 1, the system size as N = 2500 and the mean degree as .
Mentions: For a more precise characterization of the ranges of validity of these two asymptotic approximations with respect to the noise parameter a, we present in Fig. 3 the variance of n as a function of a for the numerical results and the three corresponding analytical expressions presented so far: the analytical result in equation (14), the asymptotic expression for small a in equation (16) and the asymptotic expression for large a in equation (17). Note the use a Barabási-Albert scale-free network as an example. Furthermore, we also show in this figure the crossover point a* between both approximations, that we define as the value of a that minimizes the distance between the logarithmic values of both functions (17) and (16).

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