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The noisy voter model on complex networks

Adrián Carro, Raúl Toral, Maxi San Miguel

arXiv:1602.06935v3physics.soc-phcond-mat.stat-mechcs.SIq-fin.GN

TL;DR

The noisy voter model’s behavior depends on network topology, but it has received less attention than the voter model, and quantifying topology’s effects is practically important. The paper develops an annealed-network analytical approach and finds that degree heterogeneity substantially influences the transition’s critical point.

  • Problem

    The noisy voter model has received less attention than the voter model, while quantifying how network topology affects these systems is practically important.

    Source: §1 · pp. 1–2

  • Method

    The authors analyze stochastic binary-state models using an annealed approximation for uncorrelated networks that represents connections as weights in a fully connected network.

    Source: §2.3 · p. 4, Discussion · p. 14

  • Results

    Degree heterogeneity substantially influences the critical point of the noisy voter model’s noise-induced, finite-size transition.

    Source: Discussion · p. 14

  • Takeaways & Limitations

    The transition shift suggests that changing the underlying interaction network can produce different behavioral regimes.

    Source: Discussion · p. 14

    Equations (14), (16), and (17) do not allow the model parameters a and h and the normalized degree variance to be inferred for a given system.

    Source: §3.3 · p. 13

Abstract

from arXiv · show

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.

1 Introduction

Network topology can shape the dynamics and critical behavior of binary-state systems, but the noisy voter model on complex networks has received comparatively little attention. The paper introduces an annealed analytical approach that captures effects of degree heterogeneity beyond mean-field predictions.

1 Introduction

  • Network topology can influence system dynamics and critical behavior across applications ranging from statistical physics to biological, ecological, social, and epidemiological systems.Quantifying topology’s effect is presented as practically important.Source: §1 · p. 1
  • Pairwise interactions promote order, while random state changes disrupt homogeneous states and restore ergodicity; their competition produces a noise-induced, finite-size transition.The transition separates mostly ordered and mostly disordered regimes.Source: §1 · pp. 1–2
  • Prior noisy-voter studies largely focused on regular lattices or fully connected networks, while mean-field studies of complex topologies captured only size and mean degree.This leaves effects of other network properties outside those mean-field results.Source: §1 · p. 2
  • Degree heterogeneity affects the noisy voter model’s critical point, local ordering, and temporal-correlation form, beyond what mean-field predictions capture.The approach also enables inference of network degree heterogeneity from aggregate system behavior.Source: §1 · p. 2

2 Model and methods

The paper formulates the noisy voter model as a stochastic process and derives equations for individual-node moments. It then uses an annealed approximation for uncorrelated networks to make the network sums analytically tractable while retaining degree structure.

2.1 Model definition

  • Each node copies a randomly chosen neighbor or changes state randomly; the noise and interaction parameters reduce to the effective ratio a/h.For nonzero noise the process has no absorbing states and is ergodic; at a = 0 it recovers the voter model’s two absorbing states.Source: §2.1 · pp. 2–3

2.2 General formulation

  • The master equation yields first- and second-order moment equations; the noisy voter model’s rate form closes them, but second-order steady-state equations require network connections.Neighbor sums depend on specific node pairs, motivating an approximation that rewrites them as sums over the whole network.Source: §2.2 · p. 3

2.3 Annealed approximation for uncorrelated networks

  • The annealed approximation replaces the network with a weighted, fully connected graph using connection probabilities, enabling whole-network sums while preserving the degree sequence and total links.The weights represent the probabilities that node pairs are connected in an uncorrelated network.Source: §2.3 · p. 4

3 Results

The results examine how network degree heterogeneity shapes the noisy voter model’s finite-size transition, local ordering, and aggregate temporal correlations. They also consider how autocorrelation patterns can help infer degree heterogeneity.

3.1 Noise-induced, finite-size transition

  • The annealed analytical expression captures the steady-state variance’s noise and network dependence, whereas mean field misses topology and fails at large noise.The analytical expression systematically overestimates numerical results at intermediate noise; mean field remains a good approximation for Erdős–Rényi networks when a ≲ 10^-1.Source: Figure 2 · p. 6, §3.1 · p. 6
  • For finite noise, greater degree-distribution variance increases the variance of n, while the relevant network measure is degree variance relative to squared mean degree.The asymptotic approximations match the analytical expression within their respective validity ranges; the annealed approximation systematically overestimates numerical results and cannot explain topologies with large structural correlations.Source: §3.1 · p. 6, Asymptotic behavior of the variance of n · p. 7
  • The bimodal–unimodal transition persists on networks, and its critical noise depends on topology; the dichotomous network’s critical point is almost three times the Erdős–Rényi value.Both the analytical expression and mean-field approximation capture the finite-size shift ac → 0 as N → ∞, but only the proposed approach reproduces network dependence, approximately linear in degree-distribution variance.Source: §3.1 · p. 7, Critical point · p. 8, Figure 5 · p. 10

3.2 Local order

  • Interface-density realizations remain active at nonzero noise, fluctuating near full order below the critical point and around high disorder above it.Below the transition there are sporadic excursions toward disorder; above it, large excursions toward full order occur.Source: §3.2 · p. 10, Figure 6 · p. 11
  • The mean-field pair approximation fits intermediate and large noise but fails at small noise and cannot explain network effects beyond mean degree.Its assumptions capture short-range order but not the long-range order characteristic of small a; at a = 0, it has distinct nonzero and zero solutions, with the zero solution correct for finite systems.Source: §3.2 · pp. 11–12
  • The annealed approximation captures interface-density limits but misses intermediate-noise values by removing local effects; greater degree variance predicts lower interface density at a ≲ 10^-2.It reproduces convergence toward 0 for small a and 1/2 for large a, but fails for 10^-3 ≲ a ≲ 3; it fits fully connected networks, where local effects are absent.Source: §3.2 · p. 11, Figure 7 · p. 12

3.3 Inference of network properties from the autocorrelation of n

  • The section turns to the steady-state autocorrelation function of n to characterize the dynamic nature of the system’s active steady state.The supplied passage introduces this focus but does not include the autocorrelation findings.Source: §3.3 · p. 12
  • With no degree heterogeneity, the autocorrelation follows a single exponential, but with nonzero heterogeneity it generally has two exponential decay components.The homogeneous-network result matches the fully connected and mean-field prediction; for heterogeneous networks the additional component decays as e^(-(2a+h)τ).Source: §3.3 · p. 13
  • For any network, at long time-lags, normalized autocorrelation is parallel to e^(-2aτ) in log-linear scale, with a vertical shift proportional to degree heterogeneity due to initial deviation from single-exponential behavior.This description is confirmed numerically for the dichotomous network in Fig. 8.Source: §3.3 · p. 13
  • Equations (14), (16), and (17) alone cannot identify the model parameters and degree variance, or establish heterogeneity's contribution to fluctuations, without prior knowledge of a and h.Source: §3.3 · p. 13
  • Measuring autocorrelation of aggregate n, with system size N known, allows a, h, and normalized degree variance σ2 to be inferred using equation (23) with equation (16) or (17).The choice between equations (16) and (17) is determined by self-consistency, depending on the inferred value of a.Source: §3.3 · pp. 13–14
  • For the dichotomous network, fitting equation (23) yielded a = 0.0099, h = 0.94, and σ2 = 2.539, close to actual a = 0.01, h = 1, and σ2 = 2.625.The fit used aggregate autocorrelation data and equation (16).Source: §3.3 · p. 14

Discussion

The proposed annealed approximation treats network structure as parametric heterogeneity, revealing how degree-distribution properties shape noisy-voter behavior beyond mean degree. Simulations generally agree with the analysis, with a stated exception for local order at intermediate noise.

Discussion

  • The method uses an annealed approximation for uncorrelated networks to account for network structure as parametric heterogeneity.Source: Discussion · p. 14
  • Degree heterogeneity shifts the critical point, raises steady-state local order, and affects temporal correlations; those correlations can support inference of normalized degree heterogeneity from aggregate behavior.The analysis also finds dependence on averages over the degree distribution beyond mean degree.Source: Discussion · p. 14
  • Simulations show generally good agreement with the analytical results, except for local order at intermediate noise, where the discrepancy is attributed to the annealed approximation's limits for local effects.The numerical networks were not constrained to avoid correlations, although the method assumes uncorrelated networks.Source: Discussion · p. 14

A Master equation

The supplement derives a master equation for the full binary-state distribution, then uses it to obtain time-evolution equations for individual-node first moments.

A Master equation

  • The node transition rates specify changes from state 0 to 1 and from state 1 to 0, and step operators are introduced to rewrite the probability equations.Source: A Master equation · pp. 14–15
  • The master equation for the N-node distribution is formed by summing each node's contribution to state-change probabilities.Source: A Master equation · pp. 14–15

B Equation for the time evolution of the first-order moments ⟨si⟩

C Equation for the time evolution of the second-order cross-moments ⟨sisj⟩

The supplement derives the time evolution of second-order cross-moments from the master equation, treating distinct-node and same-node cases separately.

C Equation for the time evolution of the second-order cross-moments ⟨sisj⟩

D Variance of n

The supplement derives the steady-state variance of the global variable n by reducing covariance equations to a recurrence and applying the annealed approximation for uncorrelated networks.

D Variance of n

  • Applying the annealed approximation for uncorrelated networks replaces sums over neighbors with sums over the whole system to solve for S1.Source: D Variance of n · pp. 18, 20
  • The covariance matrix is used to derive the variance, which is rewritten using a sequence of degree-weighted variables Sx and a recurrence relating them.The recurrence solution expresses the variables in terms of S1.Source: D Variance of n · pp. 18–19
  • The derivation takes the recurrence to the limit x → ∞ under a convergence condition stated to hold generally and always for h > 0 and k ≥ 2.Source: D Variance of n · pp. 19–20
  • The variance is then expressed as a function of S1 before the final expression is obtained.Source: D Variance of n · p. 21
  • The derivation arrives at the final expression for the steady-state variance of the global variable n.Source: D Variance of n · p. 21

E Asymptotic approximations for the variance of n

The steady-state variance approximation requires separate small- and large-noise regimes because its form depends on the product aN. The derivation applies the annealed approximation’s degree constraint and gives first-order-in-N approximations in both regimes.

E Asymptotic approximations for the variance of n

F Critical point approximation

The critical point is defined by the parameter relationship at which the steady-state variance reaches the value for a uniform distribution on 0 to N. The small-a asymptotic approximation gives the critical-point solution discussed in the main text, consistent with its expected O(N^-1) scale.

F Critical point approximation

  • The critical point is the relationship between a and h at which the steady-state variance reaches N(N+2)/12.This variance corresponds to a uniform distribution between 0 and N.Source: F Critical point approximation · p. 23
  • The small-a approximation yields the critical-point value discussed in the main text and is consistent with a critical value of order O(N^-1).The variance threshold is N(N+2)/12, corresponding to a uniform distribution between 0 and N.Source: F Critical point approximation · p. 23
  • Assuming instead an O(N^0) critical value and using the large-a approximation gives an O(N^-1) result, inconsistent with that assumption.The expected O(N^-1) scale follows from the fully connected critical value and the stated O(N^0) network-structure change.Source: F Critical point approximation · p. 23

G Order parameter: the interface density ρ

The section derives an analytical expression for the interface density in the steady state. It uses the annealed approximation and expresses the density in terms of the variance of n.

G Order parameter: the interface density ρ

H Autocorrelation function of n

The section derives an analytical expression for the steady-state autocorrelation function of n. It treats the system state at time t as the initial condition for evaluating the later state at time t + τ, then expresses the correlation using the variance of n and S1.

H Autocorrelation function of n

  • The time lag τ separates the initial state n(t) from the later state n(t + τ), with n(t) used as the initial condition.The state n(t) is represented in terms of the individual variables {si} and the steady-state mean ⟨n⟩st = N/2.Source: H Autocorrelation function of n · p. 24
  • The calculation obtains the conditional first moments by integrating their temporal evolution under the noisy voter model’s transition rates.It first derives a differential equation for an intermediate quantity dependent on the initial condition b(t).Source: H Autocorrelation function of n · pp. 24–25
  • The steady-state calculation uses ⟨si⟩st = 1/2 and covariance σij = ⟨sisj⟩st − ⟨si⟩².The autocorrelation function is defined and evaluated in the steady state.Source: H Autocorrelation function of n · pp. 24–25
  • The derivation produces the steady-state autocorrelation function of n, with terms ordered by their exponential decay.The final expression identifies the variance of n and S1.Source: H Autocorrelation function of n · p. 26

I Suppementary Figure S1

Figure S9 compares steady-state average interface density across network types as noise varies, alongside analytical and mean-field pair-approximation results.

  • The figure plots steady-state average interface density against noise parameter a on a linear-logarithmic scale.Source: Figure S9 · p. 26
  • At mean degree k = 8, it includes Erdős–Rényi, Barabási–Albert scale-free, dichotomous, and fully connected topologies.Source: Figure S9 · p. 26
  • Symbols show numerical results, solid lines analytical results, and dashed and dash-dotted lines mean-field pair approximations at k = 8 and k = 2499, respectively.The interaction parameter is h = 1 and system size N = 2500.Source: Figure S9 · p. 26

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