arXiv subjects

Katarzyna Sznajd-Weron

Publications and source records attributed to Katarzyna Sznajd-Weron.

At least 19 recordsLinked to original sources

arXiv2026

Targeting Behavior or Intention? Intervention Strategies in Expressed-Private Opinion Dynamics

Exogenous interventions, including economic incentives, can be used to promote behavioral change and the adoption of new products or practices. They may act on different levels, either directly affecting customers' behavior (expressed level) or changing intentions (private level). The key question is which intervention strategy is more effective: targeting the expressed or the private level. Although expressed-private opinion (EPO) models are especially suitable for addressing this problem, they have not yet been used in this context. To fill this gap, we extend previous agent-based EPO models by incorporating exogenous interventions and compare different intervention strategies. We analyze the resulting model using a mean-field approximation, complemented by Monte Carlo simulations on a complete graph. Our results show that there is no universal answer to whether an intervention should target the expressed or the private opinion. The effectiveness of an intervention depends on the adoption level at the moment of its introduction. Interventions acting on the expressed level are more effective when adoption is still low, whereas interventions acting on the private level are more effective when adoption is already high and the goal is to support internal acceptance and reduce dissonance. Moreover, the model shows that high adoption is accompanied by low dissonance between expressed and private opinions, which is consistent with self-congruity theory. This agreement provides theoretical support for the model and strengthens its policy implications.

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arXiv2026

Opinion dynamics: Statistical physics and beyond

Opinion dynamics, the study of how individual beliefs and collective public opinion evolve, is a fertile domain for applying statistical physics to complex social phenomena. Like physical systems, societies exhibit macroscopic regularities from localized interactions, leading to outcomes such as consensus or fragmentation. This field has grown significantly, attracting interdisciplinary methods and driven by a surge in large-scale behavioral data. This review covers its rapid progress, bridging the literature dispersion. We begin with essential concepts and definitions, encompassing the nature of opinions, microscopic and macroscopic dynamics. This foundation leads to an overview of empirical research, from lab experiments to large-scale data analysis, which informs and validates models of opinion dynamics. We then present individual-based models, categorized by their macroscopic phenomena (e.g., consensus, polarization, echo chambers) and microscopic mechanisms (e.g., homophily, assimilation). Furthermore, the review covers common analytical and computational tools, including stochastic processes, treatments, simulations, and optimization. Finally, we explore emerging frontiers, such as connecting empirical data to models and using AI agents as testbeds for novel social phenomena. By systematizing terminology and emphasizing analogies with traditional physics, this review aims to consolidate knowledge, provide a robust theoretical foundation, and shape future research in opinion dynamics.

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arXiv2026

Competition between private and expressed opinions in binary choice: the $α$-EPO $q$-voter model

People often express opinions that differ from their privately held views, a phenomenon known in economy as preference falsification. Expressed-private opinion (EPO) models capture this by assigning each agent two dynamical variables: a private (internal) and an expressed (external) opinion. Within the nonlinear $q$-voter model, two EPO variants have been studied so far: with and without self-anticonformity. In both formulations, agents update private and expressed binary opinions, one after another and at the same rate, which has led to two update schemes studied previously: AT (act then think), in which an agent first updates its expressed and then its private opinion, and TA (think then act), in which the order is reversed. To eliminate this ad hoc distinction and quantify the interplay between private and expressed opinions, we introduce the $α$-EPO $q$-voter model with asynchronous updating -- in each elementary step, an agent updates its private opinion with probability $α$ or its expressed opinion with complementary probability $1-α$. We derive mean-field theory and, for the first time for EPO $q$-voter dynamics, a pair approximation, and validate them with Monte Carlo simulations on artificial and real organizational networks. Comparing the two model variants, we show that the collective outcome controlled by $α$ strongly depends on self-anticonformity: with self-anticonformity the results are robust to $α$, whereas without it $α$ shifts the agreement-disagreement threshold and can change the type of phase transition. In the mean-field limit this change occurs only for $q=3$, but the pair approximation reveals an additional low-connectivity regime in which both $α$ and the average degree $k$ control the emergence and width of hysteresis also for larger influence groups.

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arXiv2025

When heterogeneity drives hysteresis: Anticonformity in the multistate $q$-voter model on networks

Discontinuous phase transitions are closely linked to tipping points, critical mass effects, and hysteresis, phenomena that have been confirmed empirically and recognized as highly important in social systems. The multistate $q$-voter model, an agent-based approach to simulate discrete decision-making and opinion dynamics, is particularly relevant in this context. Previous studies of the $q$-voter model with anticonformity on complete graphs uncovered a counterintuitive result. Changing the model formulation from the annealed (homogeneous agents with varying behavior) to quenched (heterogeneous agents with fixed behavior) produces discontinuous phase transitions. This is contrary to the common expectation that quenched heterogeneity smooths transitions. To test whether this effect is merely a mean-field artifact, we extend the analysis to random graphs. Using pair approximation and Monte Carlo simulations, we show that the phenomenon persists beyond the complete graph, specifically on random graphs and Barabási-Albert scale-free networks. The novelty of our work is twofold: (i) we demonstrate for the first time that replacing the annealed with the quenched approach can change the type of phase transitions from continuous to discontinuous not only on complete graphs but also on sparser networks, and (ii) we provide pair-approximation results for the multistate $q$-voter model with competing conformity and anticonformity mechanisms, covering both quenched and annealed cases, which had previously been studied only in binary models.

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arXiv2025

Decision-Making under Negativity Bias: Double Hysteresis in the Opinion-Dependent $q$-Voter Model

Negative information often exerts a disproportionately strong impact on human decision-making, a phenomenon known as the negativity bias. In behavioral economics, this effect is formally captured by Prospect Theory, which posits that losses loom larger than equivalent gains. For example, a single negative product review can outweigh numerous positive ones, reflecting this principle of loss aversion in consumer behavior. While this psychological effect has been widely documented, its implications for collective opinion dynamics, critical for understanding market stability and reputation dynamics, remain poorly understood. Here, we generalize the $q$-voter model with independence by introducing opinion-dependent influence group sizes, $q_+$ and $q_-$, which represent the social reinforcement needed to change an opinion from negative to positive and from positive to negative, respectively. We study two versions of this asymmetric model: a baseline model that reduces to the standard $q$-voter model when $q_+ = q_- = q$, and an extended model that incorporates an additional asymmetry expressed as a preference for one opinion. In its reduced version, this represents a minimal model in terms of non-linearity within the $q$-voter framework that allows for discontinuous phase transitions and hysteresis. Using mean-field analysis and computer simulations, we show that these modifications lead to rich collective behaviors, including double hysteresis, one form of which is irreversible, providing a mechanism for path-dependence and the sustained, irrecoverable damage to collective sentiment, brand equity, or market confidence.

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arXiv2025

Pair Approximation Meets Reality: Diffusion of Innovation in Organizational Networks within the biased-independence q-Voter Model

Collective adaptation, whether in innovation adoption, pro-environmental or organizational change, emerges from the interplay between individual decisions and social influence. Agent-based modeling provides a useful tool for studying such processes. Here, we introduce the biased-independence $q$-voter model, a generalization of the $q$-voter model with independence, one of the most popular agent-based models of opinion dynamics. In our model, individuals choose between two options, adopt or not adopt, under the competing influences of conformity and independent choice. Independent choice between two options is determined by an engagement parameter, inspired by earlier agent-based model of eco-innovation diffusion. When the engagement parameter equals $0.5$, the model reduces to the original $q$-voter model with independence; values different from $0.5$ break the symmetry between the two options. To place our study in a broader context, we briefly review asymmetric versions of the $q$-voter model proposed to date. The novelty of this work goes beyond introducing a generalized model: we develop the pair approximation (PA) for an asymmetric $q$-voter model and, for the first time, validate it on empirical organizational networks. Our results show that the interplay of social influence, independence, and option preference generates discontinuous phase transitions and irreversible hysteresis, reflecting path-dependent adoption dynamics. Surprisingly, the PA agrees well with Monte Carlo simulations on some empirical networks, even small ones, highlighting its potential as a computationally efficient bridge between individual decision-making and collective actions.

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arXiv2025

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.

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arXiv2022

Polarization in the three-state $q$-voter model with anticonformity and bounded confidence

Engaging with dissenting views, fostering productive disagreements or strategic anticonformity can benefit organizations as it challenges the status quo. The question arises, however, whether such strategic anticonformity ultimately leads to social polarization, which is not a desirable phenomenon. We address this question within an agent-based model of discrete choices. Using the way of modeling social responses in continuous opinion models, we propose a three-state $q$-voter model with anticonformity and bounded confidence. We analyze the model on a complete graph using the mean-field approach and Monte Carlo simulations. We show that strong polarization appears only for a small probability of anticonformity, which means that conformity combined with homophily enhances polarization. Our findings agree with results obtained previously in the group discussion experiment and within various continuous opinion models.

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arXiv2022

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.

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arXiv2022

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

Discontinuous phase transitions occurs to be 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 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.

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arXiv2022

The threshold model with anticonformity under random sequential updating

We study an asymmetric version of the threshold model with anticonformity under asynchronous update mode that mimics continuous time. We study this model on a complete graph using three different approaches: mean-field approximation, Monte Carlo simulation, and the Markov chain approach. The latter approach yields analytical results for arbitrarily small systems, in contrast to the mean-field approach, which is strictly correct only for an infinite system. We show that for sufficiently large systems, all three approaches produce the same results, as expected. We consider two cases: (1) homogeneous, in which all agents have the same tolerance threshold, and (2) heterogeneous, in which the thresholds are given by a beta distribution parametrized by two positive shape parameters $α$ and $β$. The heterogeneous case can be treated as a generalized model that reduces to a homogeneous model in special cases. We show that particularly interesting behaviors, including social hysteresis and critical mass, arise only for values of $α$ and $β$ that yield the shape of the distribution observed in real social systems.

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arXiv2021

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 $q_c$ in case of conformity is independent from the size of the source of influence $q_a$ in case of anticonformity. For $q_c=q_a=q$ 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 $q_c \ge q_a + Δq$, where $Δq=4$ for $q_a \le 3$ and $Δq=3$ for $q_a>3$. In this paper, we pose the question if discontinuous phase transitions survive on random graphs with an average node degree $\langle k\rangle \le 150$ 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 $\langle k\rangle$. Moreover, we show that for $q_a < q_c - 1$ pair approximation results overlap the Monte Carlo ones. On the other hand, for $q_a \ge q_c - 1$ 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 $\langle k\rangle$, as long as the pair approximation indicates correctly the type of the phase transition.

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arXiv2021

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 $s \ge 2$ states. As in the original binary $q$-voter model, which corresponds to $s=2$, at each update randomly selected voter can conform to its $q$ randomly chosen neighbors (copy their state) only if all $q$ neighbors are 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 $1-p$ 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 $s>2$ model displays discontinuous phase transitions for any $q>1$, on contrary to the model with binary opinions, in which discontinuous phase transitions are observed only for $q>5$. Moreover, unlike the case of $s=2$, for $s>2$ discontinuous phase transitions survive under the quenched disorder, although they are less sharp than under the annealed one.

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arXiv2020

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 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.

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arXiv2020

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 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.

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arXiv2020

In search for the social hysteresis -- the symmetrical threshold model with independence on Watts-Strogatz graphs

We study the homogeneous symmetrical threshold model with independence (noise) by pair approximation and Monte Carlo simulations on 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 +/-; with complementary probability $1-p$, a voter takes a given state, if 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 also discontinuous phase transitions are possible. The hysteresis increases with the average degree and the rewriting parameter beta. On the other hand, the dependence between the width of the hysteresis and the threshold $r$ is non-monotonic. The value of r, for which the maximum hysteresis is observed, overlaps pretty well the size of the majority used for the descriptive norms in order to manipulate people within social experiments. We put results obtained within this paper in a broader picture and discuss them in the context of two other models of binary opinions, namely the majority-vote and the q-voter model.

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arXiv2019

Statistical Physics Of Opinion Formation: is it a SPOOF?

We present a short review based on the nonlinear $q$-voter model about problems and methods raised within statistical physics of opinion formation (SPOOF). We describe relations between models of opinion formation, developed by physicists, and theoretical models of social response, known in social psychology. We draw attention to issues that are interesting for social psychologists and physicists. We show examples of studies directly inspired by social psychology like: "independence vs. anticonformity" or "personality vs. situation". We summarize the results that have been already obtained and point out what else can be done, also with respect to other models in SPOOF. Finally, we demonstrate several analytical methods useful in SPOOF, such as the concept of effective force and potential, Landau's approach to phase transitions, or mean-field and pair approximations.

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arXiv2019

Homogeneous symmetrical threshold model with nonconformity: independence vs. 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 posses 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 > 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.

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