With the advent of online networks, societies have become substantially more interconnected with individual members able to easily both maintain and modify their own social links. Here, we show that active network maintenance exposes agents to confirmation bias, the tendency to confirm one's beliefs, and we explore how this bias affects collective opinion formation. We introduce a model of binary opinion dynamics on a complex, fluctuating network with stochastic rewiring and we analyze these dynamics in the mean-field limit of large networks and fast link rewiring. We show that confirmation bias induces a segregation of individuals with different opinions and stabilizes the consensus state. We further show that bias can have an unusual, nonmonotonic effect on the time to consensus and this suggests a novel avenue for large-scale opinion manipulation.

While opinion dynamics models and language change models seem to be very different at first sight, we demonstrate in this paper that they are more similar than they look like. Here, we analyse the similarities and differences between the Social Influence with Recurrent Mobility model (Phys. Rev. Lett. 112, 158701) and the Utterance Selection Model of language change (Phys. Rev. E 73, 046118), two models that simulate a stochastic dynamics on a complex network, and show that their mesoscopic dynamic is strikingly similar. By drawing an analogy between the two models, we discuss possibilities of cross-fertilization of research between the two problems.

When agreement-dynamics models are placed on a directed random graph, a fraction of sites $\exp(-z)$, where $z$ is the average degree, becomes permanently fixed or flickering. In the Voter model, which has no surface tension, such zealots or flickers freely spread their opinions and that makes the system disordered. For models with a surface tension, like the Ising model or the Naming Game model, their role is limited and such systems are ordered at large~$z$. However, when $z$ decreases, the density of zealots or flickers increases, and below a certain threshold ($z\sim 1.9-2.0$) the system becomes disordered. Our results show that the agreement dynamics on directed networks is much different from their undirected analogues.

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

We study memory dependent binary-state dynamics, focusing on the noisy-voter model. This is a non-Markovian process if we consider the set of binary states of the population as the description variables, or Markovian if we incorporate "age", related to the time one has spent holding the same state, as a part of the description. We show that, in some cases, the model can be reduced to an effective Markovian process, where the age distribution of the population rapidly equilibrates to a quasi-steady state, while the global state of the system is out of equilibrium. This effective Markovian process shares the same phenomenology of the non-linear noisy-voter model and we establish a clear parallelism between these two extensions of the noisy-voter model.

We propose new Degroot-type social learning models with feedback in a continuous time, to investigate the effect of a noisy information source on consensus formation in a social network. Unlike the standard Degroot framework, noisy information models destroy consensus formation. On the other hand, the noisy opinion dynamics converge to the equilibrium distribution that encapsulates correlations among agents' opinions. Interestingly, such an equilibrium distribution is also a non-equilibrium steady state (NESS) with a non-zero probabilistic current loop. Thus, noisy information source leads to a NESS at long times that encodes persistent correlated opinion dynamics of learning agents.

We propose an analytical technique to study large fluctuations and switching from internal noise in complex networks. Using order-disorder kinetics as a generic example, we construct and analyze the most probable, or optimal path of fluctuations from one ordered state to another in real and synthetic networks. The method allows us to compute the distribution of large fluctuations and the time scale associated with switching between ordered states for networks consistent with mean-field assumptions. In general, we quantify how network heterogeneity influences the scaling patterns and probabilities of fluctuations. For instance, we find that the probability of a large fluctuation near an order-disorder transition decreases exponentially with the participation ratio of a network’s principle eigenvector – measuring how many nodes effectively contribute to an ordered state. Finally, the proposed theory is used to answer how and where a network should be targeted in order to optimize the time needed to observe a switch.

We propose an agent-based model of collective opinion formation to study the wisdom of crowds under social influence. The opinion of an agent is a continuous positive value, denoting its subjective answer to a factual question. The wisdom of crowds states that the average of all opinions is close to the truth, i.e. the correct answer. But if agents have the chance to adjust their opinion in response to the opinions of others, this effect can be destroyed. Our model investigates this scenario by evaluating two competing effects: (i) agents tend to keep their own opinion (individual conviction $\beta$), (ii) they tend to adjust their opinion if they have information about the opinions of others (social influence $\alpha$). For the latter, two different regimes (full information vs. aggregated information) are compared. Our simulations show that social influence only in rare cases enhances the wisdom of crowds. Most often, we find that agents converge to a collective opinion that is even farther away from the true answer. So, under social influence the wisdom of crowds can be systematically wrong.

We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity—variance of the underlying degree distribution—has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

We propose a method of detecting non-self-correcting information cascades in experiments in which subjects choose an option sequentially by observing the choices of previous subjects. The method uses the correlation function $C(t)$ between the first and the $t+1$-th subject's choices. $C(t)$ measures the strength of the domino effect, and the limit value $c\equiv \lim_{t\to \infty}C(t)$ determines whether the domino effect lasts forever $(c>0)$ or not $(c=0)$. The condition $c>0$ is an adequate condition for a non-self-correcting system, and the probability that the majority's choice remains wrong in the limit $t\to \infty$ is positive. We apply the method to data from two experiments in which $T$ subjects answered two-choice questions: (i) general knowledge questions ($T_{avg}=60$) and (ii) urn-choice questions ($T=63$). We find $c>0$ for difficult questions in (i) and all cases in (ii), and the systems are not self-correcting.

We propose a continuous process opinion formation model to study the dynamics of a multi-level relationship between voters, political parties, and facts in two-party democratic elections. In our model, opinions can take any real value between two extremes and an unaligned, moderate opinion state without a preference. Starting with a random opinion configuration, individual voter opinions evolve and change over time due to self-reflection, inter-personal communication, external media influence, and noise. Parties are influenced by their own ideologies, facts, and voters’ opinions. Elections are held periodically and the party that is closer in opinion to the majority of voters forms the new government. The government policy is then expected to be in proximity to the voter opinions and the policies of the currently ruling political party. We analyze the tension of opinions as a measure of how dramatically opinions can disagree within a given sample of voters and the success of the government and parties as the degree of coincidence between the policies and facts. Our model generates realistic quasi-periodic alternations between incumbents and challengers that are typical for two-party systems. Moreover, our model shows that relative to other voters’ strategies, conscious voting can lead to more successful governments of not only fact-oriented but also pragmatic and balanced political parties, irrespective of the strategies of the competing opposition parties. In addition, our simulations uncover several interesting features including less victories for strictly ideological or fact-oriented parties unless they include some aspects of populism or pragmatism. In this sense, our model can also describe situations where election outcomes are not necessarily based on votes for the current programs of competing parties and their placement on relevant issues, but instead result from voters’ dissatisfaction with the previous government and the votes against it.

We investigate the effects of update rules on the dynamics of an evolutionary game-theoretic model - the N-player evolutionary trust game - consisting of three types of players: investors, trustworthy trustees, and untrustworthy trustees. Interactions between players are limited to local neighborhoods determined by predefined spatial or social network topologies. We compare evolutionary update rules based on the payoffs obtained by their neighbors. Specifically, we investigate the dynamics generated when players use a deterministic strategic rule (i.e., unconditional imitation with and without using a noise process induced by a voter model), a stochastic pairwise payoff-based strategy (i.e., proportional imitation), and stochastic local Moran processes. We explore the system dynamics under these update rules based on different social networks and different levels of game difficulty. We observe that there are significant differences on the promoted trust and global net wealth depending on the update rule. If the game is harder, rules based on unconditional imitation achieve the highest global net wealth in the population. Besides a global perspective, we also study the spatial and temporal dynamics induced by the rules and we find important spatio-temporal correlations in the system for all rules. Indeed, the update rules lead to the formation of fractal structures on a lattice and, when the rules are stochastic, also the emergence of low frequencies in the output signal of the system (i.e., long-term memory).

We investigate a nonlinear version of coevolving voter models, in which node states and network structure update as a coupled stochastic process. Most prior work on coevolving voter models has focused on linear update rules with fixed and homogeneous rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" with the nodes in its neighborhood. To explore this idea, we incorporate a local-survey parameter σ_{i} that encodes the fraction of neighbors of an updating node i that share its opinion state. In an update, with probability σ_{i}^{q} (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1-σ_{i}^{q}, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes one of its discordant edges, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving voter model to several existing linear coevolving voter models on various network architectures. Relative to those models, we find in our model that initial network topology plays a larger role in the dynamics and that the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is held initially by a minority of the nodes can effectively spread to almost every node in a network if the minority nodes view themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

We introduce a new agent-based model of opinion dynamics in which binary opinions of each agent can be measured and described regarding both pre- and post-influence at both of two levels, public and private, vis-à-vis the influence source. The model combines ideas introduced within the q-voter model with noise, proposed by physicists, with the descriptive, four-dimensional model of social response, formulated by social psychologists. We investigate two versions of the same model that differ only by the updating order: an opinion on the public level is updated before an opinion on the private level or vice versa. We show how the results on the macroscopic scale depend on this order. The main finding of this paper is that both models produce the same outcome if one looks only at such a macroscopic variable as the total number of the individuals with positive opinions. However, if also the level of internal harmony (viz., dissonance) is measured, then significant, qualitative differences are seen between these two versions of the model. All results were obtained simultaneously within Monte Carlo simulations and analytical calculations. We discuss the importance of our studies and findings from three points of view: the theory of phase transitions, agent-based modeling of social systems, and social psychology.

We introduce a general framework for exploring the problem of selecting a committee of representatives with the aim of studying a networked voting rule based on a decentralized large-scale platform, which can assure a strong accountability of the elected. The results of our simulations suggest that this algorithm-based approach is able to obtain a high representativeness for relatively small committees, performing even better than a classical voting rule based on a closed list of candidates. We show that a general relation between committee size and representatives exists in the form of an inverse square root law and that the normalized committee size approximately scales with the inverse of the community size, allowing the scalability to very large populations. These findings are not strongly influenced by the different networks used to describe the individuals’ interactions, except for the presence of few individuals with very high connectivity which can have a marginal negative effect in the committee selection process.

We explain the anomaly of election results between large cities and rural areas in terms of urban scaling in the 1948–2016 US elections and in the 2016 EU referendum of the UK. The scaling curves are all universal and depend on a single parameter only, and one of the parties always shows superlinear scaling and drives the process, while the sublinear exponent of the other party is merely the consequence of probability conservation. Based on the recently developed model of urban scaling, we give a microscopic model of voter behavior in which we replace diversity characterizing humans in creative aspects with social diversity and tolerance. The model can also predict new political developments such as the fragmentation of the left and the immigration paradox.

We examine parliamentary presence data of the 2008-2012 and the 2012-2016 legislatures of Lithuanian parliament. We consider cumulative presence series of each individual representative in the data set. These series exhibit superdiffusive behavior. We propose a modified noisy voter model as a model for the parliamentary presence. We provide detailed analysis of anomalous diffusion of the individual agent trajectories and show that the modified model is able to reproduce empirical statistical properties.

We describe a simple model of heterogeneous, interacting agents making decisions between n≥2 discrete choices. For a special class of interactions, our model is the mean field description of random field Potts-like models and is effectively solved by finding the extrema of the average energy E per agent. In these cases, by studying the propagation of decision changes via avalanches, we argue that macroscopic dynamics is well captured by a gradient flow along E. We focus on the permutation symmetric case, where all n choices are (on average) the same, and spontaneous symmetry breaking (SSB) arises purely from cooperative social interactions. As examples, we show that bimodal heterogeneity naturally provides a mechanism for the spontaneous formation of hierarchies between decisions and that SSB is a preferred instability to discontinuous phase transitions between two symmetric points. Beyond the mean field limit, exponentially many stable equilibria emerge when we place this model on a graph of finite mean degree. We conclude with speculation on decision making with persistent collective oscillations. Throughout the paper, we emphasize analogies between methods of solution to our model and common intuition from diverse areas of physics, including statistical physics and electromagnetism.

We consider the question of determining how the topological structure influences a consensus dynamical processes taking place on a network. By considering a large data set of real-world networks we first determine that the removal of edges according to their communicability angle, an angle between position vectors of the nodes in an Euclidean communicability space, increases the average time of consensus by a factor of 5.68 in real-world networks. The edge betweenness centrality also identifies, in a smaller proportion, those critical edges for the consensus dynamics; i.e., its removal increases the time of consensus by a factor of 3.70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum, which is analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.

We consider the process of reaching the final state in the coevolving voter model. There is a coevolution of state dynamics, where a node can copy a state from a random neighbor with probabilty 1-p and link dynamics, where a node can rewire its link to another node of the same state with probability p. That exhibits an absorbing transition to a frozen phase above a critical value of rewiring probability. Our analytical and numerical studies show that in the active phase mean values of magnetization of nodes n and links m tend to the same value that depends on initial conditions. In a similar way mean degrees of spins up and spins down become equal. The system obeys a special statistical conservation law since a linear combination of both types magnetizations averaged over many realizations starting from the same initial conditions is a constant of motion: Λ≡(1-p)μm(t)+pn(t)=const., where μ is the mean node degree. The final mean magnetization of nodes and links in the active phase is proportional to Λ while the final density of active links is a square function of Λ. If the rewiring probability is above a critical value and the system separates into disconnected domains, then the values of nodes and links magnetizations are not the same and final mean degrees of spins up and spins down can be different.