Consensus formation on coevolving networks: groups' formation and structure
Balazs Kozma, Alain Barrat
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10 pages, 3 figures,to appear in a special proceedings issue of J. Phys. A covering the "Complex Networks: from Biology to Information Technology" conference (Pula, Italy, 2007)
We study the effect of adaptivity on a social model of opinion dynamics and consensus formation. We analyze how the adaptivity of the network of contacts between agents to the underlying social dynamics affects the size and topological properties of groups and the convergence time to the stable final state. We find that, while on static networks these properties are determined by percolation phenomena, on adaptive networks the rewiring process leads to different behaviors: Adaptive rewiring fosters group formation by enhancing communication between agents of similar opinion, though it also makes possible the division of clusters. We show how the convergence time is determined by the characteristic time of link rearrangement. We finally investigate how the adaptivity yields nontrivial correlations between the internal topology and the size of the groups of agreeing agents.
We investigate different opinion formation models on adaptive network topologies. Depending on the dynamical process, rewiring can either (i) lead to the elimination of interactions between agents in different states, and accelerate the convergence to a consensus state or break the network in non-interacting groups or (ii) counter-intuitively, favor the existence of diverse interacting groups for exponentially long times. The mean-field analysis allows to elucidate the mechanisms at play. Strikingly, allowing the interacting agents to bear more than one opinion at the same time drastically cha…
We study a nonlinear coevolving voter model with triadic closure local rewiring. We find three phases with different topological properties and configuration in the steady state: absorbing consensus phase with a single component, absorbing fragmented phase with two components in opposite consensus states, and a dynamically active shattered phase with many isolated nodes. This shattered phase, which does not exist for a coevolving model with global rewiring, has a lifetime that scale exponentially with system size. We characterize the transitions between these phases in terms of the size of the…
We study the joint effect of the non-linearity of interactions and noise on coevolutionary dynamics. We choose the coevolving voter model as a prototype framework for this problem. By numerical simulations and analytical approximations we find three main phases that differ in the absolute magnetization and the size of the largest component: a consensus phase, a coexistence phase, and a dynamical fragmentation phase. More detailed analysis reveals inner differences in these phases, allowing us to divide two of them further. In the consensus phase we can distinguish between a weak or alternating…
We study a coevolving nonlinear voter model describing the coupled evolution of the states of the nodes and the network topology. Nonlinearity of the interaction is measured by a parameter q. The network topology changes by rewiring links at a rate p. By analytical and numerical analysis we obtain a phase diagram in p, q parameter space with three different phases: Dynamically active coexistence phase in a single component network, absorbing consensus phase in a single component network, and absorbing phase in a fragmented network. For finite systems the active phase has a lifetime that grows …
In human societies opinion formation is mediated by social interactions, consequently taking place on a network of relationships and at the same time influencing the structure of the network and its evolution. To investigate this coevolution of opinions and social interaction structure we develop a dynamic agent-based network model, by taking into account short range interactions like discussions between individuals, long range interactions like a sense for overall mood modulated by the attitudes of individuals, and external field corresponding to outside influence. Moreover, individual biases…
We study an opinion formation model by the means of a co-evolving complex network where the vertices represent the individuals, characterised by their evolving opinions, and the edges represent the interactions among them. The network adapts to the spreading of opinions in two ways: not only connected agents interact and eventually change their thinking but an agent may also rewire one of its links to a neighborhood holding the same opinion as his. The dynamics depends on an external parameter Φ, which controls the plasticity of the network. We show how the information entropy associated to th…
We analyze Axelrod's model of social interactions on coevolving complex networks. We introduce four extensions with different mechanisms of edge rewiring. The models are intended to catch two kinds of interactions - preferential attachment, which can be observed in scientists or actors collaborations, and local rewiring, which can be observed in friendship formation in everyday relations. Numerical simulations show that proposed dynamics can lead to the power-law distribution of nodes' degree and high value of the clustering coefficient, while still retaining the small-world effect in three mo…
The dynamical origin of complex networks, i.e., the underlying principles governing network evolution, is a crucial issue in network study. In this paper, by carrying out analysis to the temporal data of Flickr and Epinions--two typical social media networks, we found that the dynamical pattern in neighborhood, especially the formation of triadic links, plays a dominant role in the evolution of networks. We thus proposed a coevolving dynamical model for such networks, in which the evolution is only driven by the local dynamics--the preferential triadic closure. Numerical experiments verified t…
We study the evolution of cooperation in the prisoner's dilemma game, whereby a coevolutionary rule is introduced that molds the random topology of the interaction network in two ways. First, existing links are deleted whenever a player adopts a new strategy or its degree exceeds a threshold value, and second, new links are added randomly after a given number of game iterations. These coevolutionary processes correspond to the generic formation of new and deletion of existing links that, especially in human societies, appear frequently as a consequence of ongoing socialization, change of lifes…
Several models (including the widely used Sznajd model) have been proposed in order to describe the social phenomenon of consensus formation. The objective of the present paper is to supplement the simulations based on these models with a ``real world simulation''; it considers a situation which can be considered as an ideal laboratory for analyzing consensus formation, namely the authorization or prohibition of using cell phones while driving. This is a convenient laboratory for several reasons (i) The issue was raised in similar terms in all industrialized countries, a circumstance that faci…
We investigate an extension of the voter model in which voters are equipped with an individual inertia to change their opinion. This inertia depends on the persistence time of a voter's current opinion (ageing). We focus on the case of only two different inertia values: zero if a voter just changed towards a new opinion and $ν$ otherwise. We are interested in the average time to reach consensus, i.e. the state in which all voters have adopted the same opinion. Adding inertia to the system means to slow down the dynamics at the voter's level, which should presumably lead to a slower consensus f…
A current paradigm in computer simulation studies of social sciences problems by physicists is the emergence of consensus. The question is to establish when the dynamics of a set of interacting agents that can choose among several options (political vote, opinion, cultural features, etc.) leads to a consensus in one of these options, or when a state with several coexisting social options prevail. We consider here stochastic dynamic models naturally studied by computer simulations. We will first review some basic results for the voter model. This is a binary option stochastic model, and probabl…
Among the different disciplines in the social, behavioral and economic sciences, a fundamental class of problems is related to the prediction of the final state of the presence of individual opinions in a large population. The main aspects investigated in the opinion dynamics are related to the possibility of reaching an agreement and the time needed. In general a consensus model allows to understand if a set of interacting agents can reach a consensus when choosing among several options: political vote, opinions, cultural features are well-known examples of them. Opinion dynamics models then …
Formation of consensus, in binary yes/no type of voting, is a well defined process. However, even in presence of clear incentives, the dynamics involved can be incredibly complex. Specifically, formations of large groups of similarly opinionated individuals could create a condition of `support-bubbles' or spontaneous polarization that renders consensus virtually unattainable (e.g., the question of the UK exiting the EU). There have been earlier attempts in capturing the dynamics of consensus formation in societies through simple $Z_2$-symmetric models hoping to capture the essential dynamics o…
We introduce a statistical physics model for opinion dynamics on random networks where agents adopt the opinion held by the majority of their direct neighbors only if the fraction of these neighbors exceeds a certain threshold, p_u. We find a transition from total final consensus to a mixed phase where opinions coexist amongst the agents. The relevant parameters are the relative sizes in the initial opinion distribution within the population and the connectivity of the underlying network. As the order parameter we define the asymptotic state of opinions. In the phase diagram we find regions of…
We study how groups reach consensus by varying communication network structure and individual incentives. In 342 networks of seven individuals, single opinionated "leaders" can drive decision outcomes, but do not accelerate consensus formation, whereas conflicting opinions slow consensus. While networks with more links reach consensus faster, this advantage disappears under conflict. Unopinionated individuals make choices consistent with a local majority rule combined with "inertia" favouring their previous choice, while opinionated individuals favour their preferred option but yield under hig…
We study opinion dynamics over multiplex networks where agents interact with bounded confidence. Namely, two neighbouring individuals exchange opinions and compromise if their opinions do not differ by more than a given threshold. In literature, agents are generally assumed to have a homogeneous confidence bound. Here, we study analytically and numerically opinion evolution over structured networks characterised by multiple layers with respective confidence thresholds and general initial opinion distributions. Through rigorous probability analysis, we show analytically the critical thresholds …
Reaching consensus -- a macroscopic state where the system constituents display the same microscopic state -- is a necessity in multiple complex socio-technical and techno-economic systems: their correct functioning ultimately depends on it. In many distributed systems -- of which blockchain-based applications are a paradigmatic example -- the process of consensus formation is crucial not only for the emergence of a leading majority but for the very functioning of the system. We build a minimalistic network model of consensus formation on blockchain systems for quantifying how central nodes --…
Opinion dynamics on networks has wide applications to empirical and engineered systems and profound prospects in the general study of complex systems. Many efforts have been devoted to understanding how opinion dynamics is affected by network topology. However, human social interactions are best characterized as temporal networks in which ordering of interactions cannot be ignored. Temporal activity patterns including heterogeneous contact strength and interevent times, turnover edge/node dynamics and daily patterns could have significant effects that would not be captured by static aggregate …
Geometric realization of opinion is considered as a simplex and the opinion space of a group of individuals is a simplicial complex whose topological features are monitored in the process of opinion formation. The agents are physically located on the nodes of the scale-free network. Social interactions include all concepts of social dynamics present in the mainstream models augmented by four additional interaction mechanisms which depend on the local properties of opinions and their overlapping properties. The results pertaining to the formation of consensus are of particular interest. An anal…
Consensus formation and difference of opinion have long been the subject of research. However, relevant laws and systems within society are being updated to reflect the changes in information networks. Online environment has come to fulfill a major role as a real and concrete place of opposing opinions and consensus formation. In the future, quantitative findings on consensus formation, and findings on relevant trends, must be summarized, and quantitative research related to trends likely to give rise to social and economic risk is required. Thus, the potential for comparing research related t…
In the commented paper, the authors declare an analogy with the quantum mechanical pure states established through the application of the high-dimensional combinatorial Laplacian considering that the simplicial complex is in the pure state when it is formed by collection of the pure states. In their work, besides giving a completely erroneous analogy for the pure quantum mechanical state, which contradicts to very basic postulates of quantum mechanic, the authors clearly fail to provide and explain mathematical formalism behind their claims. Their intention is to rigorously prove that the exis…
We here discuss the process of opinion formation in an open community where agents are made to interact and consequently update their beliefs. New actors (birth) are assumed to replace individuals that abandon the community (deaths). This dynamics is simulated in the framework of a simplified model that accounts for mutual affinity between agents. A rich phenomenology is presented and discussed with reference to the original (closed group) setting. Numerical findings are supported by analytical calculations.
We developed a statistical mechanics approach to the problem of opinion formation in interacting agents, constrained by a set of social rules, $B$. To provide the agents with an adaptive quality, we represented both the social agents and the social rule by perceptrons. For fully connected societies we find that if the agents' interaction is weak, all agents adapt to the social rule $B$, with which they form a consensus; but if the interaction is sufficiently strong a consensus is built against the established $status$ $quo$. This behavior is observed for all temperatures $T$ and for all values…
With the advent of online networks, societies are substantially more connected with individual members able to easily modify and maintain 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 affects collective opinion formation. We introduce a model of binary opinion dynamics on a complex network with fast, stochastic rewiring and show that confirmation bias induces a segregation of individuals with different opinions. We use the dynamics of global opinion to generally categoriz…
Models of the consensus of the individual state in social systems have been the subject of recent researches in the physics literature. We investigate how network structures coevolve with the individual state under the framework of social identity theory. And we propose an adaptive network model to achieve state consensus or local structural adjustment of individuals by evaluating the homogeneity among them. Specifically, the similarity threshold significantly affects the evolution of the network with different initial conditions, and thus there emerges obvious community structure and polariza…
We study the role of hierarchical structures in a simple model of collective consensus formation based on the bounded confidence model with continuous individual opinions. For the particular variation of this model considered in this paper, we assume that a bias towards an extreme opinion is introduced whenever two individuals interact and form a common decision. As a simple proxy for hierarchical social structures, we introduce a two-step decision making process in which in the second step groups of like-minded individuals are replaced by representatives once they have reached local consensus…
The dynamical process of opinion formation within a model using a local majority opinion updating rule is studied numerically in networks with the small-world geometrical property. The network is one in which shortcuts are added to randomly chosen pairs of nodes in an underlying regular lattice. The presence of a small number of shortcuts is found to shorten the time to reach a consensus significantly. The effects of having shortcuts in a lattice of fixed spatial dimension are shown to be analogous to that of increasing the spatial dimension in regular lattices. The shortening of the consensus…
The structure of a network can significantly influence the properties of the dynamical processes which take place on them. While many studies have been devoted to this influence, much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks. Adaptive rewiring of links can happen in real life systems such as acquaintance networks where people are more likely to maintain a social connection if their views and values are similar. In our study, we consider different variants of a model for consensus formation. Ou…
Despite the widespread use of Barabasi's scale-free networks and Erdos-Renyi networks of which degree correlation (assortativity) is neutral, numerous studies demonstrated that online social networks tend to show assortative mixing (positive degree correlation), while non-social networks show a disassortative mixing (negative degree correlation). First, we analyzed the variability in the assortativity coefficients of different groups of the same platform by using three different subreddits in Reddit. Our data analysis results showed that Reddit is disassortative, and assortativity coefficients…
Convex Regions of Opinion Dynamics, Approaches to the Complexity of Binary Consensus with Reference to Addiction and Obliviousness: Integrated Dimer Model Perspective
We study here the dynamics of opinion formation in a society where we take into account of 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 behavior of the consensus forming transitions, using a kinetic excha…
We consider a dual model of decision making, in which an individual forms its opinion based on contrasting mechanisms of imitation and rational calculation. The decision making model (DMM) implements imitating behavior by means of a network of coupled two-state master equations that undergoes a phase transition at a critical value of a control parameter. The evolutionary spatial game (EGM), being a generalization of the Prisoner's dilemma game, is used to determine in objective fashion the cooperative or anti-cooperative strategy adopted by individuals. Interactions between two sources of dyna…
Opinion formation and disease spreading are among the most studied dynamical processes on complex networks. In real societies, it is expected that these two processes depend on and affect each other. However, little is known about the effects of opinion dynamics over disease dynamics and vice versa, since most studies treat them separately. In this work we study the dynamics of the voter model for opinion formation intertwined with that of the contact process for disease spreading, in a population of agents that interact via two types of connections, social and contact. These two interacting d…
In this chapter, we derive and analyse models for consensus dynamics on hypergraphs. As we discuss, unless there are nonlinear node interaction functions, it is always possible to rewrite the system in terms of a new network of effective pairwise node interactions, regardless of the initially underlying multi-way interaction structure. We thus focus on dynamics based on a certain class of non-linear interaction functions, which can model different sociological phenomena such as peer pressure and stubbornness. Unlike for linear consensus dynamics on networks, we show how our nonlinear model dyn…
It is interesting and of significant importance to investigate how network structures co-evolve with opinions. The existing models of such co-evolution typically lead to the final states where network nodes either reach a global consensus or break into separated communities, each of which holding its own community consensus. Such results, however, can hardly explain the richness of real-life observations that opinions are always diversified with no global or even community consensus, and people seldom, if not never, totally cut off themselves from dissenters. In this article, we show that, a s…
Current models for opinion dynamics typically utilize a Poisson process for speaker selection, making the waiting time between events exponentially distributed. Human interaction tends to be bursty, though, having higher probabilities of either extremely short waiting times or long periods of silence. To quantify the burstiness effects on the dynamics of social models, we place in competition two groups exhibiting different speakers' waiting-time distributions. These competitions are implemented in the binary Naming Game, and show that the relevant aspect of the waiting-time distribution is th…
We study the dynamics of the Naming Game [Baronchelli et al., (2006) J. Stat. Mech.: Theory Exp. P06014] in empirical social networks. This stylized agent-based model captures essential features of agreement dynamics in a network of autonomous agents, corresponding to the development of shared classification schemes in a network of artificial agents or opinion spreading and social dynamics in social networks. Our study focuses on the impact that communities in the underlying social graphs have on the outcome of the agreement process. We find that networks with strong community structure hinder…
The massive amount of text data on the web has facilitated research on the quantitative analysis of public opinion, which could not be visualized earlier. In this paper, we propose a new opinion dynamics theory. This theory that is intended to explain agreement formation and opinion breakup division in opinion exchanges on social media such as Twitter. With the popularization of the public network, we have become able to communicate with instantaneity and interactivity beyond the temporal and spatial constraints.Research on quantitatively analyzing the distribution of opinion on public opinion…
We introduce a class of stochastic models for the dynamics of two linguistic variants that are competing to become the single, shared convention within an unstructured community of speakers. Different instances of the model are distinguished by the way agents handle variability in the language (i.e., multiple forms for the same meaning). The class of models includes as special cases two previously-studied models of language dynamics, the Naming Game, in which agents tend to standardise on variants they have encountered most frequently, and the Utterance Selection Model, in which agents tend to…
We consider a general class of three--state models where individuals hold one of two opposite opinions, or are neutral, and exchange opinions in generic pairwise interactions. We show that when opinions spread in a population where a fraction of individuals (zealots) maintain unshakably their opinion, one of four qualitatively distinct kinds of collective dynamics arise, depending on the specific rules governing the social interactions. Unsurprisingly, when their density is high, zealots drive the whole population to consensus on their opinion. For low densities a rich phase diagram emerges: a…
We study the opinion dynamics in a generalized voter model in which voters are additionally influenced by two antagonistic news sources, whose effect is to promote political polarization. We show that, as the influence of the news sources is increased, the mean time to reach consensus is anomalously long, the time to reach a politically polarized state is quite short, and the steady-state opinion distribution exhibits a transition from a near consensus state to a politically polarized state.
We study a two states opinion formation model driven by PageRank node influence and report an extensive numerical study on how PageRank affects collective opinion formations in large-scale empirical directed networks. In our model the opinion of a node can be updated by the sum of its neighbor nodes' opinions weighted by the node influence of the neighbor nodes at each step. We consider PageRank probability and its sublinear power as node influence measures and investigate evolution of opinion under various conditions. First, we observe that all networks reach steady state opinion after a cert…
We consider a stochastic, continuous state and time opinion model where each agent's opinion locally interacts with other agents' opinions in the system, and there is also exogenous randomness. The interaction tends to create clusters of common opinion. By using linear stability analysis of the associated nonlinear Fokker-Planck equation that governs the empirical density of opinions in the limit of infinitely many agents, we can estimate the number of clusters, the time to cluster formation and the critical strength of randomness so as to have cluster formation. We also discuss the cluster dy…
We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the ot…
Within the framework of a simple model for social influence, the Taylor model, we analytically investigate the role of stubborn agents in the overall opinion dynamics of networked systems. Similar to zealots, stubborn agents are biased towards a certain opinion and have a major effect on the collective opinion formation process. Based on a modified version of the network Laplacian we derive quantities capturing the transient dynamics of the system and the emerging stationary opinion states. In the case of a single stubborn agent we characterize his/her ability to coherently change a prevailing…
The q-voter model, a variant of the classic voter model, has been analyzed by several authors: while allowing 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 non-conformists, embedded on heterogeneous network topologies (as small-world and scale-free). In fact, although it is already known that conformity enhances the rea…
Inspired by the analysis of several empirical online social networks, we propose a simple reaction-diffusion-like coevolving model, in which individuals are activated to create links based on their states, influenced by local dynamics and their own intention. It is shown that the model can reproduce the remarkable properties observed in empirical online social networks; in particular, the assortative coefficients are neutral or negative, and the power law exponents are smaller than 2. Moreover, we demonstrate that, under appropriate conditions, the model network naturally makes transition(s) f…
Majority-vote model is a much-studied model for social opinion dynamics of two competing opinions. With the recent appreciation that our social network comprises a variety of different layers forming a multiplex network, a natural question arises on how such multiplex interactions affect the social opinion dynamics and consensus formation. Here, the majority-vote model will be studied on multiplex networks to understand the effect of multiplexity on opinion dynamics. We will discuss how global consensus is reached by different types of voters: AND- and OR-rule voters on multiplex-network and v…
The adaptive voter model is a paradigmatic model in the study of opinion formation. Here we propose an extension for this model, in which conflicts are resolved by obtaining another opinion, and analytically study the time required for consensus to emerge. Our results shed light on the rich phenomenology of both the original and extended adaptive voter models, including a dynamical phase transition in the scaling behavior of the mean time to consensus.
To reach consensus among interacting agents is a problem of interest for social, economical, and political systems. A computational and mathematical framework to investigate consensus dynamics on complex networks is naming games. In general, naming is not an independent process but relies on perception and categorization. Existing works focus on consensus process of vocabulary evolution in a population of agents. However, in order to name an object, agents must first be able to distinguish objects according to their features. We articulate a likelihood category game model (LCGM) to integrate f…
