We study the effect of latency on binary-choice opinion formation models. Latency is introduced into the models as an additional dynamic rule: after a voter changes its opinion, it enters a waiting period of stochastic length where no further changes take place. We first focus on the voter model and show that as a result of introducing latency, the average magnetization is not conserved, and the system is driven toward zero magnetization, independently of initial conditions. The model is studied analytically in the mean-field case and by simulations in one dimension. We also address the behavior of the Majority Rule model with added latency, and show that the competition between imitation and latency leads to a rich phenomenology.
The voter model rules are simple, with agents copying the state of a random neighbor, but they lead to non-trivial dynamics. Besides opinion processes, the model has also applications for catalysis and species competition. Inspired by the temporal inhomogeneities found in human interactions, one can introduce ageing in the agents: the probability to update decreases with the time elapsed since the last change. This modified dynamics induces an approach to consensus via coarsening in complex networks. Additionally, multilayer networks produce profound changes in the dynamics of models. In this …
Collective leadership and herding may arise in standard models of opinion dynamics as an interplay of a strong separation of time scales within the population and its hierarchical organization. Using the voter model as a simple opinion formation model, we show that, in the herding phase, a group of agents become effectively the leaders of the dynamics while the rest of the population follow blindly their opinion. Interestingly, in some cases such herding dynamics accelerates the time to consensus, which then become size independent or, on the contrary, makes the consensus nearly impossible. Th…
We study the behavior of a generalized consensus dynamics on a temporal network of interactions, the activity driven network with attractiveness. In this temporal network model, agents are endowed with an intrinsic activity $a$, ruling the rate at which they generate connections, and an intrinsic attractiveness $b$, modulating the rate at which they receive connections. The consensus dynamics considered is a mixed voter/Moran dynamics. Each agent, either in state $0$ or $1$, modifies his/her state when connecting with a peer. Thus, an active agent copies his/her state from the peer (with proba…
We present a full stochastic description of the pair approximation scheme to study binary-state dynamics on heterogeneous networks. Within this general approach, we obtain a set of equations for the dynamical correlations, fluctuations and finite-size effects, as well as for the temporal evolution of all relevant variables. We test this scheme for a prototypical model of opinion dynamics known as the noisy voter model that has a finite-size critical point. Using a closure approach based on a system size expansion around a stochastic dynamical attractor we obtain very accurate results, as compa…
The adaptive voter model has been widely studied as a conceptual model for opinion formation processes on time-evolving social networks. Past studies on the effect of zealots, i.e. nodes aiming to spread their fixed opinion throughout the system, only considered the voter model on a static network. Here, we extend the study of zealotry to the case of an adaptive network topology co-evolving with the state of the nodes and investigate opinion spreading induced by zealots depending on their initial density and connectedness. Numerical simulations reveal that below the fragmentation threshold a l…
This study investigates adversarial attacks conducted to distort voter model dynamics in complex networks. Specifically, a simple adversarial attack method is proposed to hold the state of opinions of an individual closer to the target state in the voter model dynamics. This indicates that even when one opinion is the majority, the vote outcome can be inverted (i.e., the outcome can lean toward the other opinion) by adding extremely small (hard-to-detect) perturbations strategically generated in social networks. Adversarial attacks are relatively more effective in complex (large and dense) net…
Numerous models in opinion dynamics focus on the temporal dynamics within a single electoral unit (e.g., country). The empirical observations, on the other hand, are often made across multiple electoral units (e.g., polling stations) at a single point in time (e.g., elections). Aggregates of these observations, while quite useful in many applications, neglect the underlying heterogeneity in opinions. To address this issue we build a simple agent-based model in which all agents have fixed opinions, but are able to change their electoral units. We demonstrate that this model is able to generate …
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 proce…
We introduce the reputational voter model (RVM) to account for the time-varying abilities of individuals to influence their neighbors. To understand of the RVM, we first discuss the fitness voter model (FVM), in which each voter has a fixed and distinct fitness. In a voting event where voter $i$ is fitter than voter $j$, only $j$ changes opinion. We show that the dynamics of the FVM and the voter model are identical. We next discuss the adaptive voter model (AVM), in which the influencing voter in a voting event increases its fitness by a fixed amount. The dynamics of the AVM is non-stationary…
One of the fundamental structural properties of many networks is triangle closure. Whereas the influence of this transitivity on a variety of contagion dynamics has been previously explored, existing models of coevolving or adaptive network systems use rewiring rules that randomize away this important property. In contrast, we study here a modified coevolving voter model dynamics that explicitly reinforces and maintains such clustering. Employing extensive numerical simulations, we establish that the transitions and dynamical states observed in coevolving voter model networks without clusterin…
We study the dynamics of opinion formation in a heterogeneous voter model on a complete graph, in which each agent is endowed with an integer fitness parameter $k \ge 0$, in addition to its $+$ or $-$ opinion state. The evolution of the distribution of $k$--values and the opinion dynamics are coupled together, so as to allow the system to dynamically develop heterogeneity and memory in a simple way. When two agents with different opinions interact, their $k$--values are compared and, with probability $p$ the agent with the lower value adopts the opinion of the one with the higher value, while …
The diffusion of opinions in Social Networks is a relevant process for adopting positions and attracting potential voters in political campaigns. Opinion polarization, bias, targeted diffusion, and the radicalization of postures are key elements for understanding voting dynamics. In particular, social bots are a new element that can have a pronounced effect on the formation of opinions during elections by, for instance, creating fake accounts in social networks to manipulate elections. Here we propose a voter model incorporating bots and radical or intolerant individuals in the decision-making…
The voter model with multiple states has found applications in areas as diverse as population genetics, opinion formation, species competition and language dynamics, among others. In a single step of the dynamics, an individual chosen at random copies the state of a random neighbor in the population. In this basic formulation it is assumed that the copying is perfect, and thus an exact copy of an individual is generated at each time step. Here we introduce and study a variant of the multi-state voter model in mean-field that incorporates a degree of imperfection or error in the copying process…
Stochastic Dynamics of the Multi-State Voter Model over a Network based on Interacting Cliques and Zealot Candidates
在一类由非重叠集团组成的复杂网络上研究了多状态选民模型的随机动力学，每个集团都有一个政治候选人，并通过Erdens-Rényi链路与其他成员相互作用.该模型的数值模拟解释的一个特设的平均场论，专门调整，以解决间/内集团的相互作用。在热力学极限的适当定义下（在增加网络大小的同时，代理的平均度保持固定），该模型被发现显示了由Alberto和Castellano发现的经验标度（Phys Rev Lett 99（13）：138701，2007），而选票分布大致类似于在巴西选举中观察到的。
We study the contrarian voter model for opinion formation in a society under the influence of an external oscillating propaganda and stochastic noise. Each agent of the population can hold one of two possible opinions on a given issue --against or in favor, and interacts with its neighbors following either an imitation dynamics (voter behavior) or an anti-alignment dynamics (contrarian behavior): each agent adopts the opinion of a random neighbor with a time-dependent probability $p(t)$, or takes the opposite opinion with probability $1-p(t)$. The imitation probability $p(t)$ is controlled by …
Social media influence online activity by recommending to users content strongly correlated with what they have preferred in the past. In this way they constrain users within filter bubbles that strongly limit their exposure to new or alternative content. We investigate this type of dynamics by considering a multistate voter model where, with a given probability $λ$, a user interacts with a "personalized information" suggesting the opinion most frequently held in the past. By means of theoretical arguments and numerical simulations, we show the existence of a nontrivial transition between a re…
Reducibility of higher-order to pairwise interactions: Social impact models on hypergraphs
Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（跨学科物理与复杂系统研究所 IFISC (CSIC-UIB)）；Instituto de Cálculo, FCEN, Universidad de Buenos Aires and CONICET（布宜诺斯艾利斯大学物理与数学学院计算研究所及阿根廷国家科学研究委员会）
We study an adaptive network model driven by a nonlinear voter dynamics. Each node in the network represents a voter and can be in one of two states that correspond to different opinions shared by the voters. A voter disagreeing with its neighbor's opinion may either adopt it or rewire its link to another randomly chosen voter with any opinion. The system is studied by means of the pair approximation in which a distinction between the average degrees of nodes in different states is made. This approach allows us to identify two dynamically active phases, a symmetric and an asymmetric one. The a…
The inference of outcomes in dynamic processes from structural features of systems is a crucial endeavor in network science. Recent research has suggested a machine learning-based approach for the interpretation of dynamic patterns emerging in complex network systems. The hypothesis is applied in this study towards showing opinions can be classified in the voter model with the use of topological features from complex networks. An analysis was first performed with 2 (0 or 1) opinions from a voter model and extended for 3 and then 4 opinions from the same model. A breakdown of the key network to…
We study a coevolving nonlinear voter model on a two-layer network. Coevolution stands for coupled dynamics of the state of the nodes and of the topology of the network in each layer. The plasticity parameter p measures the relative time scale of the evolution of the states of the nodes and the evolution of the network by link rewiring. Nonlinearity of the interactions is taken into account through a parameter q that describes the nonlinear effect of local majorities, being q = 1 the marginal situation of the ordinary voter model. Finally the connection between the two layers is measured by a …
We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multi-state voter models with all-to-all interaction and on uncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a meta-stable state. The density of active links and the entropy in these states varies from realisation to realisation. Making some simple assumptions we are able to analytically calculate the average density of active links and the ave…
The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: apart from the original herding processes, voters may change their states because of an intrinsic, noisy in origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasi-consensus state, where most of the voters share the same opinion, to a one with coexistence. Upon intr…
We introduce the Multiscale Voter Model (MVM) to investigate clan influence at multiple scale -- family, neighborhood, political party... -- in opinion formation on real complex networks. Clans, consisting of similar nodes, are constructed using a coarse-graining procedure on network embeddings that allows us to control for the length scale of interactions. We ran numerical simulations to monitor the evolution of MVM dynamics in real and synthetic networks, and identified a transition between a final stage of full consensus and one with mixed binary opinions. The transition depends on the scal…
The voter model with memory-dependent dynamics is theoretically and numerically studied at the mean-field level. The `internal age', or time an individual spends holding the same state, is added to the set of binary states of the population, such that the probability of changing state (or activation probability $p_i$) depends on this age. A closed set of integro-differential equations describing the time evolution of the fraction of individuals with a given state and age is derived, and from it analytical results are obtained characterizing the behavior of the system close to the absorbing sta…
