Limits...
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 three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation 14]. Dash-dotted lines: Analytical results for the critical points [see equation 18]. 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
getmorefigures.php?uid=PMC4837380&req=5

f2: Steady state variance of n as a function of the noise parameter a, for three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation 14]. Dash-dotted lines: Analytical results for the critical points [see equation 18]. 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: The behavior of the variance as a function of the noise parameter a is shown in Fig. 2 for the three types of networks studied. As we can observe, despite a small but systematic overestimation for intermediate values of the noise parameter —attributable only to the annealed approximation for uncorrelated networks, the only one involved in its derivation—, the main features of the numerical steady state variance are correctly captured by the analytical expression in equation (14). In particular, both its dependence on a and the impact of the underlying network structure are well described by our approach. On the contrary, the mean-field solution proposed in the previous literature42, and included in Fig. 2 for comparison, fails to reproduce the behavior of the variance of n for large a and is, by definition, unable to explain its dependence on the network topology. It is, nonetheless, a good approximation for the Erdös-Rényi random network and for values of the noise parameter .


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 three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation 14]. Dash-dotted lines: Analytical results for the critical points [see equation 18]. 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

f2: Steady state variance of n as a function of the noise parameter a, for three different types of networks: Erdös-Rényi random network, Barabási-Albert scale-free network and dichotomous network.Symbols: Numerical results (averages over 20 networks, 10 realizations per network and 50000 time steps per realization). Solid lines: Analytical results [see equation 14]. Dash-dotted lines: Analytical results for the critical points [see equation 18]. 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: The behavior of the variance as a function of the noise parameter a is shown in Fig. 2 for the three types of networks studied. As we can observe, despite a small but systematic overestimation for intermediate values of the noise parameter —attributable only to the annealed approximation for uncorrelated networks, the only one involved in its derivation—, the main features of the numerical steady state variance are correctly captured by the analytical expression in equation (14). In particular, both its dependence on a and the impact of the underlying network structure are well described by our approach. On the contrary, the mean-field solution proposed in the previous literature42, and included in Fig. 2 for comparison, fails to reproduce the behavior of the variance of n for large a and is, by definition, unable to explain its dependence on the network topology. It is, nonetheless, a good approximation for the Erdös-Rényi random network and for values of the noise parameter .

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