Q-voter Model Research Articles

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 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 and the size of the influence group needed for the anticonformity are independent variables and in general . 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 and .

Homogeneous Symmetrical Threshold Model with Nonconformity: Independence versus Anticonformity

We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so‐called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and possesses an up‐down symmetry, which makes it similar to other binary opinion models with a single‐flip dynamics, such as the majority‐vote and the q‐voter models. Because within the majority‐vote model with independence only continuous phase transitions are observed, whereas within the q‐voter model with independence also discontinuous phase transitions are possible, we ask the question about the factor, which could be responsible for discontinuity of the order parameter. We investigate the model via the mean‐field approach, which gives the exact result in the case of a complete graph, as well as via Monte Carlo simulations. Additionally, we provide a heuristic reasoning, which explains observed phenomena. We show that indeed if the threshold r = 0.5, which corresponds to the majority‐vote model, an order‐disorder transition is continuous. Moreover, results obtained for both versions of the model (one with independence and the second one with anticonformity) give the same results, only rescaled by the factor of 2. However, for r &gt; 0.5 the jump of the order parameter and the hysteresis is observed for the model with independence, and both versions of the model give qualitatively different results.

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.

Impact of memory on opinion dynamics

We investigate an agent-based model of opinion dynamics with two types of social response: conformity and independence. Conformity is introduced to the model analogously as in the Sznajd model or q-voter model, which means that only unanimous group exerts peer pressure on individuals. The novelty, in relation to previous versions of the q-voter model, is memory possessed by each agent and external noise T, which plays the role of social temperature. Each agent has its own memories of past experiences related to the social costs and benefits of being independent or conformist. If an agent was awarded in past more for being independent, it will have a greater tendency to be independent than conformist and vice versa. We will show that depending on the social temperature T the system spontaneously organizes into one of two regimes. Below a certain critical social temperature Tc, all agents in the society acquire personal traits (so called person state). Some of them become permanent conformists and others start to behave forever independently. This means that initially homogeneous population becomes heterogeneous, and agents respond differently to social influence. For T>Tc, all agents with equal probabilities behave independently or conform to peer pressure (so called situation state). This regime change between person and situation state, which reminds the idea of an annealed vs. quenched disorder, affects also public opinion. Particularly interesting results are obtained for individualistic societies, in which public opinion is non-monotonic function of T, which means that there is an optimal social temperature for which an agreement in the society is the highest.

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.

Conformity, Anticonformity and Polarization of Opinions: Insights from a Mathematical Model of Opinion Dynamics

Understanding and quantifying polarization in social systems is important because of many reasons. It could for instance help to avoid segregation and conflicts in the society or to control polarized debates and predict their outcomes. In this paper, we present a version of the q-voter model of opinion dynamics with two types of responses to social influence: conformity (like in the original q-voter model) and anticonformity. We put the model on a social network with the double-clique topology in order to check how the interplay between those responses impacts the opinion dynamics in a population divided into two antagonistic segments. The model is analyzed analytically, numerically and by means of Monte Carlo simulations. Our results show that the system undergoes two bifurcations as the number of cross-links between cliques changes. Below the first critical point, consensus in the entire system is possible. Thus, two antagonistic cliques may share the same opinion only if they are loosely connected. Above that point, the system ends up in a polarized state.

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).

Pair approximation for theq-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.

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 etal. [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.

Characterization of the nonequilibrium steady state of a heterogeneous nonlinear q-voter model with zealotry

We introduce a heterogeneous nonlinear q-voter model with zealots and two types of susceptible voters, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a zealot or a susceptible voter of type q1 or q2. While here zealots never change their opinion, a qi-susceptible voter consults a group of qi neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever and has surprisingly rich properties. Here, we focus on the characterization of the model's non-equilibrium stationary state (NESS) in terms of its probability distribution and currents in the distinct regimes of low and high density of zealotry. We unveil the NESS properties in each of these phases by computing the opinion distribution and the circulation of probability currents, as well as the two-point correlation functions at unequal times (formally related to a “probability angular momentum”). Our analytical calculations obtained in the realm of a linear Gaussian approximation are compared with numerical results.

The Interplay Between Conformity and Anticonformity and its Polarizing Effect on Society

Simmering debates leading to polarization are observed in many domains. Although empirical findings show a strong correlation between this phenomenon and modularity of a social network, still little is known about the actual mechanisms driving communities to conflicting opinions. In this paper, we used an agent-based model to check if the polarization may be induced by a competition between two types of social response: conformity and anticonformity. The proposed model builds on the q-voter model (Castellano et al. 2009b) and uses a double-clique topology in order to capture segmentation of a community. Our results indicate that the interplay between intra-clique conformity and inter-clique anticonformity may indeed lead to a polarized state of the entire system. We have found a dynamic phase transition controlled by the fraction $L$ of cross-links between cliques. In the regime of small values of $L$ system is able to reach the total positive consensus. If the values of $L$ are large enough, anticonformity takes over and the system always ends up in a polarized stated. Putting it the other way around, the segmentation of the network is not a sufficient condition for the polarization to appear. A suitable level of antagonistic interactions between segments is namely required to arrive at a polarized steady state within our model.

Mapping the [formula omitted]-voter model: From a single chain to complex networks

We propose and compare six different ways of mapping the modified q-voter model to complex networks. Considering square lattices, Barabási–Albert, Watts–Strogatz and real Twitter networks, we ask the question if always a particular choice of the group of influence of a fixed size q leads to different behavior at the macroscopic level. Using Monte Carlo simulations we show that the answer depends on the relative average path length of the network and for real-life topologies the differences between the considered mappings may be negligible.

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.

Conformism-driven phases of opinion formation on heterogeneous networks: the q-voter model case

The q-voter model, a variant of the classic voter model, has been analyzed by several authors. While allowing us to study opinion dynamics, this model is also believed to be one of the most representative among the many defined in the wide field of sociophysics. Here, we investigate the consequences of conformity on the consensus reaching process, by numerically simulating a q-voter model with agents behaving either as conformists or nonconformists, embedded on heterogeneous network topologies (as small-world and scale-free). In fact, although it is already known that conformity enhances the reaching of consensus, the related process is often studied only on fully-connected networks, thus strongly limiting our full understanding of it. This paper represents a first step in the direction of analyzing more realistic social models, showing that different opinion formation phases, driven by the conformist agents density, are observable. As a result, we identify threshold values of the density of conformist agents, varying across different topologies and separating different regimes of our system, ranging from a disordered phase, where different opinions coexist, to a gradually more ordered phase, where consensus is eventually reached.

Analytical expression for the exit probability of theq-voter model in one dimension

We present in this paper an approximation that is able to give an analytical expression for the exit probability of the q-voter model in one dimension. This expression gives a better fit for the more recent data about simulations in large networks [A. M. Timpanaro and C. P. C. do Prado, Phys. Rev. E 89, 052808 (2014)] and as such departs from the expression ρ(q)/ρ(q)+(1-ρ)(q) found in papers that investigated small networks only [R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008); P. Przybyła et al., Phys. Rev. E 84, 031117 (2011); F. Slanina et al., Europhys. Lett. 82, 18006 (2008)]. The approximation consists in assuming a large separation on the time scales at which active groups of agents convince inactive ones and the time taken in the competition between active groups. Some interesting findings are that for q=2 we still have ρ(2)/ρ(2)+(1-ρ)(2) as the exit probability and for q>2 we can obtain a lower-order approximation of the form ρ(s)/ρ(s)+(1-ρ)(s) with s varying from q for low values of q to q-1/2 for large values of q. As such, this work can also be seen as a deduction for why the exit probability ρ(q)/ρ(q)+(1-ρ)(q) gives a good fit, without relying on mean-field arguments or on the assumption that only the first step is nondeterministic, as q and q-1/2 will give very similar results when q→∞.

Nonlinear q-voter model with inflexible zealots.

We study the dynamics of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time step, a susceptible adopts the opinion of a neighbor if this belongs to a group of q≥2 neighbors all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties, the model is characterized by a fluctuating stationary state and, below a zealotry density threshold, the distribution of opinions is bimodal. After a characteristic time, most susceptibles become supporters of the party having more zealots and the opinion distribution is asymmetric. When the number of zealots of both parties is the same, the opinion distribution is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the zealotry density threshold, when there is an unequal number of zealots of each type, the probability distribution is single-peaked and non-Gaussian. These properties are investigated analytically and with stochastic simulations. We also study the mean time to reach a consensus when zealots support only one party.