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

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.

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.

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.

Q-voter model with independence on signed random graphs: Homogeneous approximations

The q-voter model with independence is generalized to signed random graphs and studied by means of Monte Carlo simulations and theoretically using the mean-field approximation and different forms of the pair approximation. In the signed network with quenched disorder, positive and negative signs associated randomly with the links correspond to reinforcing and antagonistic interactions, promoting, respectively, the same or opposite orientations of two-state spins representing agents' opinions; otherwise, the opinions are called mismatched. With probability 1-p, the agents change their opinions if the opinions of all members of a randomly selected q neighborhood are mismatched, and with probability p, they choose an opinion randomly. The model on networks with finite mean degree 〈k〉 and fixed fraction of the antagonistic interactions r exhibits ferromagnetic transition with varying the independence parameter p, which can be first or second order, depending on q and r, and disappears for large r. Besides, numerical evidence is provided for the occurrence of the spin-glass-like transition for large r. The order and critical lines for the ferromagnetic transition on the p vs r phase diagram obtained in Monte Carlo simulations are reproduced qualitatively by the mean-field approximation. Within the range of applicability of the pair approximation, for the model with 〈k〉 finite but 〈k〉≫q, predictions of the homogeneous pair approximation concerning the ferromagnetic transition show much better quantitative agreement with numerical results for small r but fail for larger r. A more advanced signed homogeneous pair approximation is formulated which distinguishes between classes of active links with a given sign connecting nodes occupied by agents with mismatched opinions; for the model with 〈k〉≫q its predictions agree quantitatively with numerical results in a whole range of r where the ferromagnetic transition occurs.

Q-neighbor Ising model on multiplex networks with partial overlap of nodes

The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate master equations as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition, and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate master equations are usually quantitatively more accurate but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition.

Consensus, Polarization and Hysteresis in the Three-State Noisy q-Voter Model with Bounded Confidence

In this work, we address the question of the role of the influence of group size on the emergence of various collective social phenomena, such as consensus, polarization and social hysteresis. To answer this question, we study the three-state noisy q-voter model with bounded confidence, in which agents can be in one of three states: two extremes (leftist and rightist) and centrist. We study the model on a complete graph within the mean-field approach and show that, depending on the size q of the influence group, saddle-node bifurcation cascades of different length appear and different collective phenomena are possible. In particular, for all values of q>1, social hysteresis is observed. Furthermore, for small values of q∈(1,4), disagreement, polarization and domination of centrists (a consensus understood as the general agreement, not unanimity) can be achieved but not the domination of extremists. The latter is possible only for larger groups of influence. Finally, by comparing our model to others, we discuss how a small change in the rules at the microscopic level can dramatically change the macroscopic behavior of the model.

Switching from a continuous to a discontinuous phase transition under quenched disorder

Discontinuous phase transitions are particularly interesting from a social point of view because of their relationship to social hysteresis and critical mass. In this paper, we show that the replacement of a time-varying (annealed, situation-based) disorder by a static (quenched, personality-based) one can lead to a change from a continuous to a discontinuous phase transition. This is a result beyond the state of the art, because so far numerous studies on various complex systems (physical, biological, and social) have indicated that the quenched disorder can round or destroy the existence of a discontinuous phase transition. To show the possibility of the opposite behavior, we study a multistate q-voter model, with two types of disorder related to random competing interactions (conformity and anticonformity). We confirm, both analytically and through Monte Carlo simulations, that indeed discontinuous phase transitions can be induced by a static disorder.

Pair approximation for the q-voter models with quenched disorder on networks

Using two models of opinion dynamics, the q-voter model with independence and the q-voter model with anticonformity, we discuss how the change of disorder from annealed to quenched affects phase transitions on networks. To derive phase diagrams on networks, we develop the pair approximation for the quenched versions of the models. This formalism can be also applied to other quenched dynamics of similar kind. The results indicate that such a change of disorder eliminates all discontinuous phase transitions and broadens ordered phases. We show that although the annealed and quenched types of disorder lead to the same result in the q-voter model with anticonformity at the mean-field level, they do lead to distinct phase diagrams on networks. These phase diagrams shift towards each other as the average node degree of a network increases, and eventually, they coincide in the mean-field limit. In contrast, for the q-voter model with independence, the phase diagrams move towards the same direction regardless of the disorder type, and they do not coincide even in the mean-field limit. To validate our results, we carry out Monte Carlo simulations on random regular graphs and Barabási-Albert networks. Although the pair approximation may incorrectly predict the type of phase transitions for the annealed models, we have not observed such errors for their quenched counterparts.

Opinion dynamics: public and private

We study here the dynamics of opinion formation in a society where we take into account the internally held beliefs and externally expressed opinions of the individuals, which are not necessarily the same at all times. While these two components can influence one another, their difference, both in dynamics and in the steady state, poses interesting scenarios in terms of the transition to consensus in the society and characterizations of such consensus. Here we study this public and private opinion dynamics and the critical behaviour of the consensus forming transitions, using a kinetic exchange model. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Discontinuous phase transitions in the q-voter model with generalized anticonformity on random graphs

We study the binary q-voter model with generalized anticonformity on random Erdős–Rényi graphs. In such a model, two types of social responses, conformity and anticonformity, occur with complementary probabilities and the size of the source of influence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c$$\end{document}qc in case of conformity is independent from the size of the source of influence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document}qa in case of anticonformity. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c=q_a=q$$\end{document}qc=qa=q the model reduces to the original q-voter model with anticonformity. Previously, such a generalized model was studied only on the complete graph, which corresponds to the mean-field approach. It was shown that it can display discontinuous phase transitions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c \ge q_a + \Delta q$$\end{document}qc≥qa+Δq, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta q=4$$\end{document}Δq=4 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a \le 3$$\end{document}qa≤3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta q=3$$\end{document}Δq=3 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a>3$$\end{document}qa>3. In this paper, we pose the question if discontinuous phase transitions survive on random graphs with an average node degree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle k\rangle \le 150$$\end{document}⟨k⟩≤150 observed empirically in social networks. Using the pair approximation, as well as Monte Carlo simulations, we show that discontinuous phase transitions indeed can survive, even for relatively small values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle k\rangle$$\end{document}⟨k⟩. Moreover, we show that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a < q_c - 1$$\end{document}qa<qc-1 pair approximation results overlap the Monte Carlo ones. On the other hand, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a \ge q_c - 1$$\end{document}qa≥qc-1 pair approximation gives qualitatively wrong results indicating discontinuous phase transitions neither observed in the simulations nor within the mean-field approach. Finally, we report an intriguing result showing that the difference between the spinodals obtained within the pair approximation and the mean-field approach follows a power law with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle k\rangle$$\end{document}⟨k⟩, as long as the pair approximation indicates correctly the type of the phase transition.

Discontinuous phase transitions in the multi-state noisy q-voter model: quenched vs. annealed disorder

We introduce a generalized version of the noisy q-voter model, one of the most popular opinion dynamics models, in which voters can be in one of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 2$$\end{document}s≥2 states. As in the original binary q-voter model, which corresponds to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=2$$\end{document}s=2, at each update randomly selected voter can conform to its q randomly chosen neighbors only if they are all in the same state. Additionally, a voter can act independently, taking a randomly chosen state, which introduces disorder to the system. We consider two types of disorder: (1) annealed, which means that each voter can act independently with probability p and with complementary probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-p$$\end{document}1-p conform to others, and (2) quenched, which means that there is a fraction p of all voters, which are permanently independent and the rest of them are conformists. We analyze the model on the complete graph analytically and via Monte Carlo simulations. We show that for the number of states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>2$$\end{document}s>2 the model displays discontinuous phase transitions for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>1$$\end{document}q>1, on contrary to the model with binary opinions, in which discontinuous phase transitions are observed only for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>5$$\end{document}q>5. Moreover, unlike the case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=2$$\end{document}s=2, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>2$$\end{document}s>2 discontinuous phase transitions survive under the quenched disorder, although they are less sharp than under the annealed one.

A Veritable Zoology of Successive Phase Transitions in the Asymmetric q-Voter Model on Multiplex Networks

We analyze a nonlinear q-voter model with stochastic noise, interpreted in the social context as independence, on a duplex network. The size of the lobby q (i.e., the pressure group) is a crucial parameter that changes the behavior of the system. The q-voter model has been applied on multiplex networks, and it has been shown that the character of the phase transition depends on the number of levels in the multiplex network as well as on the value of q. The primary aim of this study is to examine phase transition character in the case when on each level of the network the lobby size is different, resulting in two parameters q1 and q2. In a system of a duplex clique (i.e., two fully overlapped complete graphs) we find evidence of successive phase transitions when a continuous phase transition is followed by a discontinuous one or two consecutive discontinuous phase transitions appear, depending on the parameter. When analyzing this system, we even encounter mixed-order (or hybrid) phase transition. The observation of successive phase transitions is a new quantity in binary state opinion formation models and we show that our analytical considerations are fully supported by Monte-Carlo simulations.