Zealots in the mean-field noisy voter model

Coarsening and metastability of the long-range voter model in three dimensions

We study analytically the ordering kinetics and the final metastable states in the three-dimensional long-range voter model where N agents described by a Boolean spin variable S_{i} can be found in two states (or opinion) ±1. The kinetics is such that each agent copies the opinion of another at distance r chosen with probability P(r)∝r^{-α} (α>0). In the thermodynamic limit N→∞ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function C(r)=〈S_{i}S_{j}〉 (where r is the i-j distance) decreases algebraically in a slow, nonintegrable way. Specifically, we find C(r)∼r^{-1}, or C(r)∼r^{-(6-α)}, or C(r)∼r^{-α} for α>5, 3<α≤5, and 0≤α≤3, respectively. In a finite system metastability is escaped after a time of order N and full ordering is eventually achieved. The dynamics leading to metastability is of the coarsening type, with an ever-increasing correlation length L(t) (for N→∞). We find L(t)∼t^{1/2} for α>5, L(t)∼t^{5/2α} for 4<α≤5, and L(t)∼t^{5/8} for 3≤α≤4. For 0≤α<3 there is not macroscopic coarsening because stationarity is reached in a microscopic time. Such results allow us to conjecture the behavior of the model for generic spatial dimension.

Cross-border political competition

Individuals are increasingly exposed to news and opinion from beyond national borders. This news and opinion are often concentrated in clusters of ideological homophily, such as political parties, factions, or interest groups. But how does exposure to cross-border information affect the diffusion of ideas across national and ideological borders? Here, we develop a non-linear mathematical model for the cross-border spread of two ideologies. First, we describe the standard deterministic model where the populations of each country are assumed to be constant and homogeneously mixed. We solve the system of differential equations numerically by the Runge-Kutta method and show how small changes in the influence of a minority ideology can trigger shifts in the global political equilibrium. Second, we simulate recruitment as a stochastic differential process for each political affiliation and fit model solutions to population growth rates and voting populations in US presidential elections from 1932 to 2020. We also project the dynamics of several possible scenarios from 2020 to the end of the century. We show that cross-border influence plays a fundamental role in determining election outcomes. An increase in foreign support for a national party's ideas could change the election outcome, independent of domestic recruitment capacity. One key finding of our study suggests that voter turnout in the US will grow at a faster rate than non-voters in the coming decades. This trend is attributed to the enhanced recruitment capabilities of both major parties among non-partisans over time, making political disaffection less prominent. This phenomenon holds true across all simulated scenarios.

Ordering kinetics of the two-dimensional voter model with long-range interactions

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance r with probability P(r)∝r^{-α}. The model is characterized by different regimes, as α is varied. For α>4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as L(t)∝sqrt[t], until consensus is reached in a time of the order of NlnN, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as ρ(t)∝1/lnt. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0<α≤4, standard scaling is reinstated and the correlation length increases algebraically as L(t)∝t^{1/z}, with 1/z=2/α for 3<α<4 and 1/z=2/3 for 0<α<3. In addition, for α≤3, L(t) depends on N at any time t>0. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the N→∞ limit. In finite systems, consensus is reached in a time of the order of N for any α<4.

Discord in the voter model for complex networks

Online social networks have become primary means of communication. As they often exhibit undesirable effects such as hostility, polarization, or echo chambers, it is crucial to develop analytical tools that help us better understand them. In this paper we are interested in the evolution of discord in social networks. Formally, we introduce a method to calculate the probability of discord between any two agents in the multistate voter model with and without zealots. Our work applies to any directed, weighted graph with any finite number of possible opinions, allows for various update rates across agents, and does not imply any approximation. Under certain topological conditions, the opinions are independent and the joint distribution can be decoupled. Otherwise, the evolution of discord probabilities is described by a linear system of ordinary differential equations. We prove the existence of a unique equilibrium solution, which can be computed via an iterative algorithm. The classical definition of active links density is generalized to take into account long-range, weighted interactions. We illustrate our findings on real-life and synthetic networks. In particular, we investigate the impact of clustering on discord and uncover a rich landscape of varied behaviors in polarized networks. This sheds lights on the evolution of discord between, and within, antagonistic communities.

First-passage distributions of an asymmetric noisy voter model

This paper explores the first-passage times in an asymmetric noisy voter model through analytical methods. The noise in the model leads to bistable behavior, and the asymmetry arises from heterogeneous rates for spontaneous switching. We obtain exact analytical expressions for the probability distribution for two different initial conditions, first-passage times for switching transitions and first return times to a stable state for all system sizes, offering a deeper understanding of the model's dynamics. Additionally, we derive exact expressions for the mean switching time, mean return time, and their mean square variants. The findings are verified through numerical simulations. To enhance clarity regarding the model's behavior, we also provide approximate solutions, emphasizing the parameter dependence of first-passage times in the small switching parameter regime. An interesting result in this regime is that while the mean switching time in the leading order is independent of system size, the mean return time depends inversely on system size. This study not only advances our analytical understanding of the asymmetric noisy voter model but also establishes a framework for exploring similar phenomena in social and biological systems where the noisy voter model is applicable.

Partisan voter model: Stochastic description and noise-induced transitions

We give a comprehensive mean-field analysis of the partisan voter model (PVM) and report analytical results for exit probabilities, fixation times, and the quasistationary distribution. In addition, and similarly to the noisy voter model, we introduce a noisy version of the PVM, named the noisy partisan voter model (NPVM), which accounts for the preferences of each agent for the two possible states, as well as for idiosyncratic spontaneous changes of state. We find that the finite-size noise-induced transition of the noisy voter model is modified in the NPVM leading to the emergence of intermediate phases that were absent in the standard version of the noisy voter model, as well as to both continuous and discontinuous transitions.

Noisy voter models in switching environments

We study the stationary states of variants of the noisy voter model, subject to fluctuating parameters or external environments. Specifically, we consider scenarios in which the herding-to-noise ratio switches randomly and on different timescales between two values. We show that this can lead to a phase in which polarized and heterogeneous states exist. Second, we analyze a population of noisy voters subject to groups of external influencers, and show how multipeak stationary distributions emerge. Our work is based on a combination of individual-based simulations, analytical approximations in terms of a piecewise-deterministic Markov processes (PDMP), and on corrections to this process capturing intrinsic stochasticity in the linear-noise approximation. We also propose a numerical scheme to obtain the stationary distribution of PDMPs with three environmental states and linear velocity fields.

Local and global ordering dynamics in multistate voter models

We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multistate voter models with all-to-all interaction and on uncorrelated networks. Individual realizations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a metastable state. The density of active links and the entropy in these states varies from realization to realization. Making some simple assumptions we are able to analytically calculate the average density of active links and the average entropy in each of these states. We also show that, averaged over realizations, the density of active links decays exponentially, with a timescale set by the size and geometry of the graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential only at long times when there are at most two opinions left in the population. Finally, we show how metastable states comprising only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions.

Bounded Confidence Evolution of Opinions and Actions in Social Networks

Inspired by the continuous opinion and discrete action (CODA) model, bounded confidence and social networks, the bounded confidence evolution of opinions and actions in social networks is investigated and a social network opinions and actions evolutions (SNOAEs) model is proposed. In the SNOAE model, it is assumed that each agent has a CODA for a certain issue. Agents' opinions are private and invisible, that is, an individual agent only knows its own opinion and cannot obtain other agents' opinions unless there is a social network connection edge that allows their communication; agents' actions are public and visible to all agents and impact other agents' actions. Opinions and actions evolve in a directed social network. In the limitation of the bounded confidence, other agents' actions or agents' opinions noticed or obtained by network communication, respectively, are used by agents to update their opinions. Based on the SNOAE model, the evolution of the opinions and actions with bounded confidence is investigated in social networks both theoretically and experimentally with a detailed simulation analysis. Theoretical research results show that discrete actions can attract agents who trust the discrete action, and make agents to express extreme opinions. Simulation experiments results show that social network connection probability, bounded confidence, and the opinion threshold of action choice parameters have strong impacts on the evolution of opinions and actions. However, the number of agents in the social network has no obvious influence on the evolution of opinions and actions.

Adversarial attacks on voter model dynamics in complex networks

This paper investigates adversarial attacks conducted to distort voter model dynamics in complex networks. Specifically, a simple adversarial attack method is proposed to hold the state of opinions of an individual closer to the target state in the voter model dynamics. This indicates that even when one opinion is the majority the vote outcome can be inverted (i.e., the outcome can lean toward the other opinion) by adding extremely small (hard-to-detect) perturbations strategically generated in social networks. Adversarial attacks are relatively more effective in complex (large and dense) networks. These results indicate that opinion dynamics can be unknowingly distorted.

Shannon information criterion for low-high diversity transition in Moran and voter models

Mutation and drift play opposite roles in genetics. While mutation creates diversity, drift can cause gene variants to disappear, especially when they are rare. In the absence of natural selection and migration, the balance between the drift and mutation in a well-mixed population defines its diversity. The Moran model captures the effects of these two evolutionary forces and has a counterpart in social dynamics, known as the voter model with external opinion influencers. Two extreme outcomes of the voter model dynamics are consensus and coexistence of opinions, which correspond to low and high diversity in the Moran model. Here we use a Shannon's information-theoretic approach to characterize the smooth transition between the states of consensus and coexistence of opinions in the voter model. Mapping the Moran into the voter model, we extend the results to the mutation-drift balance and characterize the transition between low and high diversity in finite populations. Describing the population as a network of connected individuals, we show that the transition between the two regimes depends on the network topology of the population and on the possible asymmetries in the mutation rates.

Anomalous diffusion in nonlinear transformations of the noisy voter model

Voter models are well known in the interdisciplinary community, yet they have not been studied from the perspective of anomalous diffusion. In this paper, we show that the original voter model exhibits a ballistic regime. Nonlinear transformations of the observation variable and time scale allow us to observe other regimes of anomalous diffusion as well as normal diffusion. We show that numerical simulation results coincide with derived analytical approximations describing the temporal evolution of the raw moments.

Non-Markovian majority-vote model

Non-Markovian dynamics pervades human activity and social networks and it induces memory effects and burstiness in a wide range of processes including interevent time distributions, duration of interactions in temporal networks, and human mobility. Here, we propose a non-Markovian majority-vote model (NMMV) that introduces non-Markovian effects in the standard (Markovian) majority-vote model (SMV). The SMV model is one of the simplest two-state stochastic models for studying opinion dynamics, and displays a continuous order-disorder phase transition at a critical noise. In the NMMV model we assume that the probability that an agent changes state is not only dependent on the majority state of his neighbors but it also depends on his age, i.e., how long the agent has been in his current state. The NMMV model has two regimes: the aging regime implies that the probability that an agent changes state is decreasing with his age, while in the antiaging regime the probability that an agent changes state is increasing with his age. Interestingly, we find that the critical noise at which we observe the order-disorder phase transition is a nonmonotonic function of the rate β of the aging (antiaging) process. In particular the critical noise in the aging regime displays a maximum as a function of β while in the antiaging regime displays a minimum. This implies that the aging/antiaging dynamics can retard/anticipate the transition and that there is an optimal rate β for maximally perturbing the value of the critical noise. The analytical results obtained in the framework of the heterogeneous mean-field approach are validated by extensive numerical simulations on a large variety of network topologies.

Zealots in multistate noisy voter models

The noisy voter model is a stylized representation of opinion dynamics. Individuals copy opinions from other individuals, and are subject to spontaneous state changes. In the case of two opinion states this model is known to have a noise-driven transition between a unimodal phase, in which both opinions are present, and a bimodal phase, in which one of the opinions dominates. The presence of zealots can remove the unimodal and bimodal phases in the model with two opinion states. Here we study the effects of zealots in noisy voter models with M>2 opinion states on complete interaction graphs. We find that the phase behavior diversifies, with up to six possible qualitatively different types of stationary states. The presence of zealots removes some of these phases, but not all. We analyze situations in which zealots affect the entire population, or only a fraction of agents, and show that this situation corresponds to a single-community model with a fractional number of zealots, further enriching the phase diagram. Our study is conducted analytically based on effective birth-death dynamics for the number of individuals holding a given opinion. Results are confirmed in numerical simulations.

A Heterogeneous Network Modeling Method Based on Public Goods Game Theory to Explore Cooperative Behavior in VANETs

Cooperative vehicular networking has been widely studied in recent years. Existing evolution game theoretic approaches to study cooperative behavior in Vehicular Ad hoc Network (VANET) are mainly based on the assumption that VANET is constructed as a homogeneous network. This modeling method only extracts part attributes of vehicles and does not distinguish the differences between strategy and attribute. In this paper, we focus on the heterogeneous network model based on the public goods game theory for VANET. Then we propose a Dynamic Altruism Public Goods Game (DAPGG) model consisting of rational nodes, altruistic nodes, and zealots to more realistically characterize the real VANET. Rational nodes only care about their own benefits, altruistic nodes comprehensively consider the payoffs in the neighborhood, while zealots insist on behaving cooperatively. Finally, we explore the impacts of these attributes on the evolution of cooperation under different network conditions. The simulation results show that only adding altruistic nodes can effectively improve the proportion of cooperators, but it may cause conflicts between individual benefits and neighborhood benefits. Altruistic nodes together with zealots can better improve the proportion of cooperators, even if the network conditions are not suitable for the spread of cooperative behavior.

Generalized Independence in the q-Voter Model: How Do Parameters Influence the Phase Transition?

We study the q-voter model with flexibility, which allows for describing a broad spectrum of independence from zealots, inflexibility, or stubbornness through noisy voters to self-anticonformity. Analyzing the model within the pair approximation allows us to derive the analytical formula for the critical point, below which an ordered (agreement) phase is stable. We determine the role of flexibility, which can be understood as an amount of variability associated with an independent behavior, as well as the role of the average network degree in shaping the character of the phase transition. We check the existence of the scaling relation, which previously was derived for the Sznajd model. We show that the scaling is universal, in a sense that it does not depend neither on the size of the group of influence nor on the average network degree. Analyzing the model in terms of the rescaled parameter, we determine the critical point, the jump of the order parameter, as well as the width of the hysteresis as a function of the average network degree 〈k〉 and the size of the group of influence q.

Nonlinear q-voter model from the quenched perspective

We compare two versions of the nonlinear q-voter model: the original one, with annealed randomness, and the modified one, with quenched randomness. In the original model, each voter changes its opinion with a certain probability ϵ if the group of influence is not unanimous. In contrast, the modified version introduces two types of voters that act in a deterministic way in the case of disagreement in the influence group: the fraction ϵ of voters always change their current opinion, whereas the rest of them always maintain it. Although both concepts of randomness lead to the same average number of opinion changes in the system on the microscopic level, they cause qualitatively distinct results on the macroscopic level. We focus on the mean-field description of these models. Our approach relies on the stability analysis by the linearization technique developed within dynamical system theory. This approach allows us to derive complete, exact phase diagrams for both models. The results obtained in this paper indicate that quenched randomness promotes continuous phase transitions to a greater extent, whereas annealed randomness favors discontinuous ones. The quenched model also creates combinations of continuous and discontinuous phase transitions unobserved in the annealed model, in which the up-down symmetry may be spontaneously broken inside or outside the hysteresis loop. The analytical results are confirmed by Monte Carlo simulations carried out on a complete graph.

Nonuniversal opinion dynamics driven by opposing external influences

We study the opinion dynamics of a generalized voter model in which N voters are additionally influenced by two opposing news sources whose effect is to promote political polarization. As the influence of these news sources is increased, the mean time to reach consensus scales nonuniversally as N^{α}. The parameter α quantifies the influence of the news sources and increases without bound as the news sources become increasingly influential. The time to reach a politically polarized state, in which roughly equal fractions of the populations are in each opinion state, is generally short, and the steady-state opinion distribution exhibits a transition from near consensus to a politically polarized state as a function of α.

Multistate voter model with imperfect copying

The voter model with multiple states has found applications in areas as diverse as population genetics, opinion formation, species competition, and language dynamics, among others. In a single step of the dynamics, an individual chosen at random copies the state of a random neighbor in the population. In this basic formulation, it is assumed that the copying is perfect, and thus an exact copy of an individual is generated at each time step. Here, we introduce and study a variant of the multistate voter model in mean field that incorporates a degree of imperfection or error in the copying process, which leaves the states of the two interacting individuals similar but not exactly equal. This dynamics can also be interpreted as a perfect copying with the addition of noise: a minimalistic model for flocking. We found that the ordering properties of this multistate noisy voter model, measured by a parameter ψ in [0,1], depend on the amplitude η of the copying error or noise and the population size N. In the case of perfect copying η=0, the system reaches an absorbing configuration with complete order (ψ=1) for all values of N. However, for any degree of imperfection η>0, we show that the average value of ψ at the stationary state decreases with N as 〈ψ〉≃6/(π^{2}η^{2}N) for η≪1 and η^{2}N≳1, and thus the system becomes totally disordered in the thermodynamic limit N→∞. We also show that 〈ψ〉≃1-π^{2}/6η^{2}N in the vanishing small error limit η→0, which implies that complete order is never achieved for η>0. These results are supported by Monte Carlo simulations of the model, which allow to study other scenarios as well.

Conformity in numbers-Does criticality in social responses exist?

Within this paper we explore the idea of a critical value representing the proportion of majority members within a group that affects dramatic changes in influence targets’ conformity. We consider the threshold q-voter model when the responses of the Willis-Nail model, a well-established two-dimensional model of social response, are used as a foundation. Specifically, we study a generalized threshold q-voter model when all basic types of social response described by Willis-Nail model are considered, i.e. conformity, anticonformity, independence, and uniformity/congruence. These responses occur in our model with complementary probabilities. We introduce independently two thresholds: one needed for conformity, as well as a second one for anticonformity. In the case of conformity, at least r individuals among q neighbors have to share the same opinion in order to persuade a voter to follow majority’s opinion, whereas in the case of anticonformity, at least w individuals among q neighbors have to share the same opinion in order to influence voters to take an opinion that goes against that of their own reference group. We solve the model on a complete graph and show that the threshold for conformity significantly influences the results. For example, there is a critical threshold for conformity above which the system behaves as in the case of unanimity, i.e. displays continuous and discontinuous phase transitions. On the other hand, the threshold for anticonformity is almost irrelevant. We discuss our results from the perspective of theories of social psychology, as well as the philosophy of agent-based modeling.

Social influence with recurrent mobility and multiple options

In this paper, we discuss the possible generalizations of the social influence with recurrent mobility (SIRM) model [Phys. Rev. Lett. 112, 158701 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.158701]. Although the SIRM model worked approximately satisfying when U.S. election was modeled, it has its limits: it has been developed only for two-party systems and can lead to unphysical behavior when one of the parties has extreme vote share close to 0 or 1. We propose here generalizations to the SIRM model by its extension for multiparty systems that are mathematically well-posed in case of extreme vote shares, too, by handling the noise term in a different way. In addition, we show that our method opens alternative applications for the study of elections by using an alternative calibration procedure and makes it possible to analyze the influence of the "free will" (creating a new party) and other local effects for different commuting network topologies.

Analytical and numerical study of the non-linear noisy voter model on complex networks

We study the noisy voter model using a specific non-linear dependence of the rates that takes into account collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random network environments. In the all-to-all setup, we find that the non-linear interactions induce bona fide phase transitions that, contrary to the linear version of the model, survive in the thermodynamic limit. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. While a non-trivial finite-size dependence of the moments of the probability distribution is derived from our treatment, mean-field exponents are nevertheless obtained in the thermodynamic limit. These theoretical predictions are well confirmed by numerical simulations of the stochastic process.

Resisting Influence: How the Strength of Predispositions to Resist Control Can Change Strategies for Optimal Opinion Control in the Voter Model

In this paper, we investigate influence maximization, or optimal opinion control, in a modified version of the two-state voter dynamics in which a native state and a controlled or influenced state are accounted for. We include agent predispositions to resist influence in the form of a probability q with which agents spontaneously switch back to the native state when in the controlled state. We argue that in contrast to the original voter model, optimal control in this setting depends on q: For low strength of predispositions q, optimal control should focus on hub nodes, but for large q, optimal control can be achieved by focusing on the lowest degree nodes. We investigate this transition between hub and low-degree node control for heterogeneous undirected networks and give analytical and numerical arguments for the existence of two control regimes.