Nonlinear q-voter model with inflexible zealots

Modeling biases in binary decision-making within the generalized nonlinear q-voter model

Collective decision-making is a process by which a group of individuals determines a shared outcome that shapes societal dynamics; from innovation diffusion to organizational choices. A common approach to model these processes is using binary dynamics, where the choices are reduced to two alternatives. One of the most popular models in this context is the q-voter model, which assumes that opinion changes are driven by peer pressure from a unanimous group. However, real-world decisions are also shaped by prior personal choices and external influences, such as mass media, which introduce biases that can favor certain options over others. To address this, we propose a generalized q-voter model that incorporates these biases. In our model, when the influence group is not unanimous, the probability that an individual changes its opinion depends on its current state, breaking the symmetry between opinions. In limiting cases, our model recovers both the original q-voter model and several recently introduced modifications of the q-voter model, while extending the framework to capture a broader range of scenarios. We analyze the model on a complete graph using analytical methods and Monte Carlo simulations. Our results highlight two key findings: (1) for larger influence groups (q>3), a phase emerges where both adopted and partially adopted states coexist, (2) in small systems, greater initial support for an opinion does not necessarily increase its likelihood of widespread adoption, as reflected in the unique form of the exit probability. These results point to one of the key issues in social science, the importance of group size in collective action.

How social network structure impacts the ability of zealots to promote weak opinions

Social networks are often permeated by agents who promote their opinions without allowing for their own mind to be changed. Understanding how these so-called "zealots" act to increase the prevalence of their promoted opinion over the network is important for understanding opinion dynamics. In this work, we consider these promoted opinions to be "weak" and therefore less likely to be accepted relative to the default opinion in the network. We show how the proportion of zealots in the network, the relative strength of the weak opinion, and the structure of the network impact the long-term proportion of those in the network who subscribe to the weak opinion.

Social influence and consensus building: Introducing a q-voter model with weighted influence

We present a model of opinion formation where an individual's opinion is influenced by interactions with a group of agents. The model introduces a novel bias mechanism that favors one opinion, a feature not previously explored. In the absence of bias, the system reduces to a mean field voter model. We identify three regimes: favoring negative opinions, favoring positive opinions, and a neutral case. In large systems, equilibrium outcomes become independent of group size, with only the bias influencing the final consensus. For smaller groups, however, the time to reach equilibrium depends on group size. Our results show that even a small initial bias leads to a consensus, with all agents eventually sharing the same opinion if the bias is not zero. The system also exhibits critical slowing down near the neutral bias, which acts as a dynamical threshold. The time to reach consensus scales logarithmically for non-neutral biases and linearly with system size for the neutral case. While short-term dynamics are influenced by group size, long-term behavior is determined solely by the bias.

Ordering dynamics of nonlinear voter models

We study the ordering dynamics of nonlinear voter models with multiple states, also providing a discussion of the two-state model. The rate with which an individual adopts an opinion scales as the qth power of the number of the individual's neighbors in that state. For q>1 the dynamics favor the opinion held by the most agents. The ordering to consensus is driven by deterministic drift, and noise plays only a minor role. For q<1 the dynamics favors minority opinions, and for multistate models the ordering proceeds through a noise-driven succession of metastable states. Unlike linear multistate systems, the nonlinear model cannot be reduced to an effective two-state model. We find that the average density of active interfaces in the model with multiple opinion states does not show a single exponential decay in time for q<1, again at variance with the linear model. This highlights the special character of the conventional (linear) voter model, in which deterministic drift is absent. As part of our analysis, we develop a pair approximation for the multistate model on graphs, valid for any positive real value of q, improving on previous approximations for nonlinear two-state voter models.

Composition of the Influence Group in the q-Voter Model and Its Impact on the Dynamics of Opinions

Despite ample research devoted to the non-linear q-voter model and its extensions, little or no attention has been paid to the relationship between the composition of the influence group and the resulting dynamics of opinions. In this paper, we investigate two variants of the q-voter model with independence. Following the original q-voter model, in the first one, among the q members of the influence group, each given agent can be selected more than once. In the other variant, the repetitions of agents are explicitly forbidden. The models are analyzed by means of Monte Carlo simulations and via analytical approximations. The impact of repetitions on the dynamics of the model for different parameter ranges is discussed.

Critical exponents of master-node network model

The dynamics of competing opinions in social network plays an important role in society, with many applications in diverse social contexts such as consensus, election, morality, and so on. Here, we study a model of interacting agents connected in networks in order to analyze their decision stochastic process. We consider a first-neighbor interaction between agents in a one-dimensional network with the shape of ring topology. Moreover, some agents are also connected to a hub, or master node, who has preferential choice or bias. Such connections are quenched. As the main results, we observed a continuous nonequilibrium phase transition to an absorbing state as a function of control parameters. By using the finite-size scaling method we analyzed the static and dynamic critical exponents to show that this model probably cannot match any universality class already known.

Social Depolarization and Diversity of Opinions-Unified ABM Framework

Most sociophysics opinion dynamics simulations assume that contacts between agents lead to greater similarity of opinions, and that there is a tendency for agents having similar opinions to group together. These mechanisms result, in many types of models, in significant polarization, understood as separation between groups of agents having conflicting opinions. The addition of inflexible agents (zealots) or mechanisms, which drive conflicting opinions even further apart, only exacerbates these polarizing processes. Using a universal mathematical framework, formulated in the language of utility functions, we present novel simulation results. They combine polarizing tendencies with mechanisms potentially favoring diverse, non-polarized environments. The simulations are aimed at answering the following question: How can non-polarized systems exist in stable configurations? The framework enables easy introduction, and study, of the effects of external "pro-diversity", and its contribution to the utility function. Specific examples presented in this paper include an extension of the classic square geometry Ising-like model, in which agents modify their opinions, and a dynamic scale-free network system with two different mechanisms promoting local diversity, where agents modify the structure of the connecting network while keeping their opinions stable. Despite the differences between these models, they show fundamental similarities in results in terms of the existence of low temperature, stable, locally and globally diverse states, i.e., states in which agents with differing opinions remain closely linked. While these results do not answer the socially relevant question of how to combat the growing polarization observed in many modern democratic societies, they open a path towards modeling polarization diminishing activities. These, in turn, could act as guidance for implementing actual depolarization social strategies.

How does homophily shape the topology of a dynamic network?

We consider a dynamic network of individuals that may hold one of two different opinions in a two-party society. As a dynamical model, agents can endlessly create and delete links to satisfy a preferred degree, and the network is shaped by homophily, a form of social interaction. Characterized by the parameter J∈[-1,1], the latter plays a role similar to Ising spins: agents create links to others of the same opinion with probability (1+J)/2 and delete them with probability (1-J)/2. Using Monte Carlo simulations and mean-field theory, we focus on the network structure in the steady state. We study the effects of J on degree distributions and the fraction of cross-party links. While the extreme cases of homophily or heterophily (J=±1) are easily understood to result in complete polarization or anti-polarization, intermediate values of J lead to interesting features of the network. Our model exhibits the intriguing feature of an "overwhelming transition" occurring when communities of different sizes are subject to sufficient heterophily: agents of the minority group are oversubscribed and their average degree greatly exceeds that of the majority group. In addition, we introduce an original measure of polarization which displays distinct advantages over the commonly used average edge homogeneity.

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.

Agent Based Model of Anti-Vaccination Movements: Simulations and Comparison with Empirical Data

Background: A realistic description of the social processes leading to the increasing reluctance to various forms of vaccination is a very challenging task. This is due to the complexity of the psychological and social mechanisms determining the positioning of individuals and groups against vaccination and associated activities. Understanding the role played by social media and the Internet in the current spread of the anti-vaccination (AV) movement is of crucial importance. Methods: We present novel, long-term Big Data analyses of Internet activity connected with the AV movement for such different societies as the US and Poland. The datasets we analyzed cover multiyear periods preceding the COVID-19 pandemic, documenting the behavior of vaccine related Internet activity with high temporal resolution. To understand the empirical observations, in particular the mechanism driving the peaks of AV activity, we propose an Agent Based Model (ABM) of the AV movement. The model includes the interplay between multiple driving factors: contacts with medical practitioners and public vaccination campaigns, interpersonal communication, and the influence of the infosphere (social networks, WEB pages, user comments, etc.). The model takes into account the difference between the rational approach of the pro-vaccination information providers and the largely emotional appeal of anti-vaccination propaganda. Results: The datasets studied show the presence of short-lived, high intensity activity peaks, much higher than the low activity background. The peaks are seemingly random in size and time separation. Such behavior strongly suggests a nonlinear nature for the social interactions driving the AV movement instead of the slow, gradual growth typical of linear processes. The ABM simulations reproduce the observed temporal behavior of the AV interest very closely. For a range of parameters, the simulations result in a relatively small fraction of people refusing vaccination, but a slight change in critical parameters (such as willingness to post anti-vaccination information) may lead to a catastrophic breakdown of vaccination support in the model society, due to nonlinear feedback effects. The model allows the effectiveness of strategies combating the anti-vaccination movement to be studied. An increase in intensity of standard pro-vaccination communications by government agencies and medical personnel is found to have little effect. On the other hand, focused campaigns using the Internet and social media and copying the highly emotional and narrative-focused format used by the anti-vaccination activists can diminish the AV influence. Similar effects result from censoring and taking down anti-vaccination communications by social media platforms. The benefit of such tactics might, however, be offset by their social cost, for example, the increased polarization and potential to exploit it for political goals, or increased 'persecution' and 'martyrdom' tropes.

External fields, independence, and disorder in q-voter models

Among the many social influences expressed in q-voter models, independent agents are responsible for disordered behavior in an otherwise consensus-prone scheme. Despite some parametrizations allowing the model to converge to any given stationary concentration, small perturbations in its parameters cause the model to suffer great variations in its outcome. This paper proposes that an external field may explain less unstable outcomes in the q-voter model. We soften independence to become skepticism, a phenomenon induced by an unreliable external field interference in social processes. The external field, analogous to mass media in real settings, leads to both quicker convergence to a fairly ordered state when independence is low, and to higher disorder whenever it is under moderate perceived unreliability of the external field.

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.

Competing local and global interactions in social dynamics: How important is the friendship network?

Motivated by the empirical study that identifies a correlation between particular social responses and different interaction ranges, we study the q-voter model with various combinations of local and global sources of conformity and anticonformity. The models are investigated by means of the pair approximation and Monte Carlo simulations on Watts-Strogatz and Barabási-Albert networks. We show that within the model with local conformity and global anticonformity, the agreement in the system is the most difficult one to achieve and the role of the network structure is the most significant. Interestingly, the model with swapped interaction ranges, namely, with global conformity and local anticonformity, becomes almost insensitive to the changes in the network structure. The obtained results may have far reaching consequences for marketing strategies conducted via social media channels.

Symmetrical threshold model with independence on random graphs

We study the homogeneous symmetrical threshold model with independence (noise) by pair approximation and Monte Carlo simulations on Erdős-Rényi and Watts-Strogatz graphs. The model is a modified version of the famous Granovetter's threshold model: with probability p a voter acts independently, i.e., takes randomly one of two states ±1; with complementary probability 1-p, a voter takes a given state, if a sufficiently large fraction (above a given threshold r) of individuals in its neighborhood is in this state. We show that the character of the phase transition, induced by the noise parameter p, depends on the threshold r, as well as graph's parameters. For r=0.5 only continuous phase transitions are observed, whereas for r>0.5 discontinuous phase transitions also are possible. The hysteresis increases with the average degree 〈k〉 and the rewriting parameter β. On the other hand, the dependence between the width of the hysteresis and the threshold r is nonmonotonic. The value of r, for which the maximum hysteresis is observed, overlaps pretty well with the size of the majority used for the descriptive norms in order to manipulate people within social experiments. We put the results obtained within this paper into a broader picture and discuss them in the context of two other models of binary opinions: the majority-vote and the q-voter model. Finally, we discuss why the appearance of social hysteresis in models of opinion dynamics is desirable.

Pair approximation for the noisy threshold q-voter model

In the standard q-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size q. A more realistic extension is the threshold q voter, where a minimal agreement (at least 0<q_{0}≤q opposite opinions) is sufficient to flip the central agent's opinion, including also the possibility of independent (nonconformist) choices. Variants of this model including nonconformist behavior have been previously studied in fully connected networks (mean-field limit). Here we investigate its dynamics in random networks. Particularly, while in the mean-field case it is irrelevant whether repetitions in the influence group are allowed, we show that this is not the case in networks, and we study the impact of both cases, with or without repetition. Furthermore, the results of computer simulations are compared with the predictions of the pair approximation derived for uncorrelated networks of arbitrary degree distributions.

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.

Is Independence Necessary for a Discontinuous Phase Transition within the q-Voter Model?

We ask a question about the possibility of a discontinuous phase transition and the related social hysteresis within the q-voter model with anticonformity. Previously, it was claimed that within the q-voter model the social hysteresis can emerge only because of an independent behavior, and for the model with anticonformity only continuous phase transitions are possible. However, this claim was derived from the model, in which the size of the influence group needed for the conformity was the same as the size of the group needed for the anticonformity. Here, we abandon this assumption on the equality of two types of social response and introduce the generalized model, in which the size of the influence group needed for the conformity q c and the size of the influence group needed for the anticonformity q a are independent variables and in general q c ≠ q a . We investigate the model on the complete graph, similarly as it was done for the original q-voter model with anticonformity, and we show that such a generalized model displays both types of phase transitions depending on parameters q c and q a .

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.

Threshold q-voter model

We introduce the threshold q-voter opinion dynamics where an agent, facing a binary choice, can change its mind when at least q_{0} among q neighbors share the opposite opinion. Otherwise, the agent can still change its mind with a certain probability ɛ. This threshold dynamics contemplates the possibility of persuasion by an influence group even when there is not full agreement among its members. In fact, individuals can follow their peers not only when there is unanimity (q_{0}=q) in the lobby group, as assumed in the q-voter model, but also, depending on the circumstances, when there is simple majority (q_{0}>q/2), Byzantine consensus (q_{0}>2q/3), or any minimal number q_{0} among q. This realistic threshold gives place to emerging collective states and phase transitions which are not observed in the standard q voter. The threshold q_{0}, together with the stochasticity introduced by ɛ, yields a phenomenology that mimics as particular cases the q voter with stochastic drivings such as nonconformity and independence. In particular, nonconsensus majority states are possible, as well as mixed phases. Continuous and discontinuous phase transitions can occur, but also transitions from fluctuating phases into absorbing states.

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.

Consensus and clustering in opinion formation on networks

Ideas that challenge the status quo either evaporate or dominate. The study of opinion dynamics in the socio-physics literature treats space as uniform and considers individuals in an isolated community, using ordinary differential equation (ODE) models. We extend these ODE models to include multiple communities and their interactions. These extended ODE models can be thought of as being ODEs on directed graphs. We study in detail these models to determine conditions under which there will be consensus and pluralism within the system. Most of the consensus/pluralism analysis is done for the case of one and two cities. However, we numerically show for the case of a symmetric cycle graph that an elementary bifurcation analysis provides insight into the phenomena of clustering. Moreover, for the case of a cycle graph with a hub, we discuss how having a sufficient proportion of zealots in the hub leads to the entire network sharing the opinion of the zealots.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Spatial evolution of regularization in learned behavior of animals

Stochastic population dynamics of learned traits are studied, where individual learners behave according to a reinforcement learner model, which is a nonlinear version of the Bush-Mosteller model. Depending on a regularization parameter (parameter a), the learners may possess different degrees of overmatching (regularization behavior, 0 ≤ a < 1), frequency matching (corresponding to a=1), or undermatching behavior (a > 1). Both non-spatial and spatial models are considered, to study the interplay of individual heterogeneity of behavior, spatial and temporal effects of learning, and the possibility of emergence of regional culture. In non-spatial models, we observe that populations of individuals learning from each other converge to a universally shared, deterministic rule (either rule "1" or rule "0"), only if they to some extent possess the ability to generalize (a < 1). Otherwise, a low-coherence solution where both rules are used intermittently by everyone, is achieved. If the evolution of the regularization ability is included, then we find that a initially evolves toward lower values, and a shared solution is established when everyone reliably uses the same rule. The spatial (2D) model has two well known limiting cases: if a=0 (the strongest degree of regularization), the model converges to a threshold voter model, and if a=1 (frequency matching), it is equivalent to the discrete diffusion equation. If 0 < a < 1 (the case where individuals regularize), spatial patterns emerge, where patches of different usage of the rule are formed. Smaller values of a lead to sharper and longer lived patches. Values of a < 1 close to unity result in probabilistic outcomes where patches only survive if they are attached to the boundary. Analytical treatment of the 1D case reveals the existence of approximate equilibria that have front structure, where spatially intermittent deterministic usage of one and the other rule are separated by interfaces whose analytical form is derived.

Zealots in the mean-field noisy voter model

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: Apart from the original herding processes, voters may change their states because of an intrinsic noisy-in-origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasiconsensus state, where most of the voters share the same opinion, to one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding or zealotry acting together in the voter model yields a nontrivial mixture of the scenarios with the two mechanisms acting alone: It represents a situation where the global-local (noise-herding) competition is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasiconsensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases. In the most general case we find two different transitions: a discontinuous transition from an asymmetric bimodal phase to an extreme asymmetric phase and a second continuous transition from the extreme asymmetric phase to an asymmetric unimodal phase.

Tricriticality in the q-neighbor Ising model on a partially duplex clique

We analyze a modified kinetic Ising model, a so-called q-neighbor Ising model, with Metropolis dynamics [Phys. Rev. E 92, 052105 (2015)PLEEE81539-375510.1103/PhysRevE.92.052105] on a duplex clique and a partially duplex clique. In the q-neighbor Ising model each spin interacts only with q spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field-like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the q-neighbor Ising model exhibits a continuous phase transition for q=3, discontinuous phase transition for q≥4, and for q=1 and q=2 the phase transition is not observed. On the other hand, in the case of the duplex clique continuous phase transitions are observed for all values of q, even for q=1 and q=2. Subsequently we introduce a partially duplex clique, parametrized by r∈[0,1], which allows us to tune the network from monoplex (r=0) to duplex (r=1). Such a generalized topology, in which a fraction r of all nodes appear on both levels, allows us to obtain the critical value of r=r^{*}(q) at which a tricriticality (switch from continuous to discontinuous phase transition) appears.

Modeling and numerical simulations of the influenced Sznajd model

This paper investigates the effects of independent nonconformists or influencers on the behavioral dynamic of a population of agents interacting with each other based on the Sznajd model. The system is modeled on a complete graph using the master equation. The acquired equation has been numerically solved. Accuracy of the mathematical model and its corresponding assumptions have been validated by numerical simulations. Regions of initial magnetization have been found from where the system converges to one of two unique steady-state PDFs, depending on the distribution of influencers. The scaling property and entropy of the stationary system in presence of varying level of influence have been presented and discussed.

Person-Situation Debate Revisited: Phase Transitions with Quenched and Annealed Disorders

We study the q-voter model driven by stochastic noise arising from one out of two types of nonconformity: anticonformity or independence. We compare two approaches that were inspired by the famous psychological controversy known as the person–situation debate. We relate the person approach with the quenched disorder and the situation approach with the annealed disorder, and investigate how these two approaches influence order–disorder phase transitions observed in the q-voter model with noise. We show that under a quenched disorder, differences between models with independence and anticonformity are weaker and only quantitative. In contrast, annealing has a much more profound impact on the system and leads to qualitative differences between models on a macroscopic level. Furthermore, only under an annealed disorder may the discontinuous phase transitions appear. It seems that freezing the agents’ behavior at the beginning of simulation—introducing quenched disorder—supports second-order phase transitions, whereas allowing agents to reverse their attitude in time—incorporating annealed disorder—supports discontinuous ones. We show that anticonformity is insensitive to the type of disorder, and in all cases it gives the same result. We precede our study with a short insight from statistical physics into annealed vs. quenched disorder and a brief review of these two approaches in models of opinion dynamics.

Kinetic Ising models with various single-spin-flip dynamics on quenched and annealed random regular graphs

We investigate a kinetic Ising model with several single-spin-flip dynamics (including Metropolis and heat bath) on quenched and annealed random regular graphs. As expected, on the quenched structures all proposed algorithms reproduce the same results since the conditions for the detailed balance and the Boltzmann distribution in an equilibrium are satisfied. However, on the annealed graphs the situation is far less clear-the network annealing disturbs the equilibrium moving the system away from it. Consequently, distinct dynamics lead to different steady states. We show that some algorithms are more resistant to the annealed disorder, which causes only small quantitative changes in the model behavior. On the other hand, there are dynamics for which the influence of annealing on the system is significant, and qualitative changes arise like switching the type of phase transition from a continuous to a discontinuous one. We try to identify features of the proposed dynamics which are responsible for the above phenomenon.

Q-voter model with nonconformity in freely forming groups: Does the size distribution matter?

We study a q-voter model with stochastic driving on a complete graph with q being a random variable described by probability density function P(q), instead of a constant value. We investigate two types of P(q): (1) artificial with the fixed expected value 〈q〉, but a changing variance and (2) empirical of freely forming groups in informal places. We investigate also two types of stochasticity that can be interpreted as different kinds of nonconformity (anticonformity or independence) to answer the question about differences observed at the macroscopic level between these two types of nonconformity in real social systems. Moreover, we ask the question if the behavior of a system depends on the average value of the group size q or rather on probability distribution function P(q).

Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry

We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in Mellor et al. [Europhys. Lett. 113, 48001 (2016)EULEEJ0295-507510.1209/0295-5075/113/48001]. In this model, each individual supports one of two parties and is either a susceptible voter of type q_{1} or q_{2}, or is an inflexible zealot. At each time step, a q_{i}-susceptible voter (i=1,2) consults a group of q_{i} neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever q_{1}≠q_{2} and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the nonequilibrium stationary state of the system in terms of its probability distribution, nonvanishing currents, and unequal-time two-point correlation functions. We also study the switching time properties of the model by exploiting an approximate mapping onto the model of Mobilia [Phys. Rev. E 92, 012803 (2015)PLEEE81539-375510.1103/PhysRevE.92.012803] that satisfies the detailed balance, and we outline some properties of the model near criticality.

Pair approximation for the q-voter model with independence on complex networks

We investigate the q-voter model with stochastic noise arising from independence on complex networks. Using the pair approximation, we provide a comprehensive, mathematical description of its behavior and derive a formula for the critical point. The analytical results are validated by carrying out Monte Carlo experiments. The pair approximation prediction exhibits substantial agreement with simulations, especially for networks with weak clustering and large average degree. Nonetheless, for the average degree close to q, some discrepancies originate. It is the first time we are aware of that the presented approach has been applied to the nonlinear voter dynamics with noise. Up till now, the analytical results have been obtained only for a complete graph. We show that in the limiting case the prediction of pair approximation coincides with the known solution on a fully connected network.

The Hunt Opinion Model-An Agent Based Approach to Recurring Fashion Cycles

We study a simple agent-based model of the recurring fashion cycles in the society that consists of two interacting communities: “snobs” and “followers” (or “opinion hunters”, hence the name of the model). Followers conform to all other individuals, whereas snobs conform only to their own group and anticonform to the other. The model allows to examine the role of the social structure, i.e. the influence of the number of inter-links between the two communities, as well as the role of the stability of links. The latter is accomplished by considering two versions of the same model—quenched (parameterized by fraction L of fixed inter-links) and annealed (parameterized by probability p that a given inter-link exists). Using Monte Carlo simulations and analytical treatment (the latter only for the annealed model), we show that there is a critical fraction of inter-links, above which recurring cycles occur. For p ≤ 0.5 we derive a relation between parameters L and p that allows to compare both models and show that the critical value of inter-connections, p*, is the same for both versions of the model (annealed and quenched) but the period of a fashion cycle is shorter for the quenched model. Near the critical point, the cycles are irregular and a change of fashion is difficult to predict. For the annealed model we also provide a deeper theoretical analysis. We conjecture on topological grounds that the so-called saddle node heteroclinic bifurcation appears at p*. For p ≥ 0.5 we show analytically the existence of the second critical value of p, for which the system undergoes Hopf’s bifurcation.

Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity

Human history has been marked by social instability and conflict, often driven by the irreconcilability of opposing sets of beliefs, ideologies, and religious dogmas. The dynamics of belief systems has been studied mainly from two distinct perspectives, namely how cognitive biases lead to individual belief rigidity and how social influence leads to social conformity. Here we propose a unifying framework that connects cognitive and social forces together in order to study the dynamics of societal belief evolution. Each individual is endowed with a network of interacting beliefs that evolves through interaction with other individuals in a social network. The adoption of beliefs is affected by both internal coherence and social conformity. Our framework may offer explanations for how social transitions can arise in otherwise homogeneous populations, how small numbers of zealots with highly coherent beliefs can overturn societal consensus, and how belief rigidity protects fringe groups and cults against invasion from mainstream beliefs, allowing them to persist and even thrive in larger societies. Our results suggest that strong consensus may be insufficient to guarantee social stability, that the cognitive coherence of belief-systems is vital in determining their ability to spread, and that coherent belief-systems may pose a serious problem for resolving social polarization, due to their ability to prevent consensus even under high levels of social exposure. We argue that the inclusion of cognitive factors into a social model could provide a more complete picture of collective human dynamics.

Phase transitions in the q-voter model with noise on a duplex clique

We study a nonlinear q-voter model with stochastic noise, interpreted in the social context as independence, on a duplex network. To study the role of the multilevelness in this model we propose three methods of transferring the model from a mono- to a multiplex network. They take into account two criteria: one related to the status of independence (LOCAL vs GLOBAL) and one related to peer pressure (AND vs OR). In order to examine the influence of the presence of more than one level in the social network, we perform simulations on a particularly simple multiplex: a duplex clique, which consists of two fully overlapped complete graphs (cliques). Solving numerically the rate equation and simultaneously conducting Monte Carlo simulations, we provide evidence that even a simple rearrangement into a duplex topology may lead to significant changes in the observed behavior. However, qualitative changes in the phase transitions can be observed for only one of the considered rules: LOCAL&AND. For this rule the phase transition becomes discontinuous for q=5, whereas for a monoplex such behavior is observed for q=6. Interestingly, only this rule admits construction of realistic variants of the model, in line with recent social experiments.

Oscillating hysteresis in the q-neighbor Ising model

We modify the kinetic Ising model with Metropolis dynamics, allowing each spin to interact only with q spins randomly chosen from the whole system, which corresponds to the topology of a complete graph. We show that the model with q≥3 exhibits a phase transition between ferromagnetic and paramagnetic phases at temperature T*, which linearly increases with q. Moreover, we show that for q=3 the phase transition is continuous and that it is discontinuous for larger values of q. For q>3, the hysteresis exhibits oscillatory behavior-expanding for even values of q and shrinking for odd values of q. Due to the mean-field-like nature of the model, we are able to derive the analytical form of transition probabilities and, therefore, calculate not only the probability density function of the order parameter but also precisely determine the hysteresis and the effective potential showing stable, unstable, and metastable steady states. Our results show that a seemingly small modification of the kinetic Ising model leads not only to the switch from a continuous to a discontinuous phase transition, but also to an unexpected oscillating behavior of the hysteresis and a puzzling phenomenon for q=5, which might be taken as evidence for the so-called mixed-order phase transition.