Consensus time and conformity in the adaptive voter model

Analysis of a continuous-time adaptive voter model

In this paper, we study a variant of the voter model on adaptive networks in which nodes can flip their spin, create new connections, or break existing connections. We first perform an analysis based on the mean-field approximation to compute asymptotic values for macroscopic estimates of the system, namely, the total mass of present edges in the system and the average spin. However, numerical results show that this approximation is not very suitable for such a system, for which it does not capture key features such as the network breaking into two disjoint and opposing (in spin) communities. Therefore, we propose another approximation based on an alternate coordinate system to improve accuracy and validate this model through simulations. Finally, we state a conjecture dealing with the qualitative properties of the system, corroborated by numerous numerical simulations.

Macroscopic approximation methods for the analysis of adaptive networked agent-based models: Example of a two-sector investment model

In this paper, we propose a statistical aggregation method for agent-based models with heterogeneous agents that interact both locally on a complex adaptive network and globally on a market. The method combines three approaches from statistical physics: (a) moment closure, (b) pair approximation of adaptive network processes, and (c) thermodynamic limit of the resulting stochastic process. As an example of use, we develop a stochastic agent-based model with heterogeneous households that invest in either a fossil-fuel- or renewables-based sector while allocating labor on a competitive market. Using the adaptive voter model, the model describes agents as social learners that interact on a dynamic network. We apply the approximation methods to derive a set of ordinary differential equations that approximate the macrodynamics of the model. A comparison of the reduced analytical model with numerical simulations shows that the approximation fits well for a wide range of parameters. The method makes it possible to use analytical tools to better understand the dynamical properties of models with heterogeneous agents on adaptive networks. We showcase this with a bifurcation analysis that identifies parameter ranges with multistabilities. The method can thus help to explain emergent phenomena from network interactions and make them mathematically traceable.

Adaptive voter model on simplicial complexes

Collective decision making processes lie at the heart of many social, political, and economic challenges. The classical voter model is a well-established conceptual model to study such processes. In this work, we define a form of adaptive (or coevolutionary) voter model posed on a simplicial complex, i.e., on a certain class of hypernetworks or hypergraphs. We use the persuasion rule along edges of the classical voter model and the recently studied rewiring rule of edges towards like-minded nodes, and introduce a peer-pressure rule applied to three nodes connected via a 2-simplex. This simplicial adaptive voter model is studied via numerical simulation. We show that adding the effect of peer pressure to an adaptive voter model leaves its fragmentation transition, i.e., the transition upon varying the rewiring rate from a single majority state into a fragmented state of two different opinion subgraphs, intact. Yet, above and below the fragmentation transition, we observe that the peer pressure has substantial quantitative effects. It accelerates the transition to a single-opinion state below the transition and also speeds up the system dynamics towards fragmentation above the transition. Furthermore, we quantify that there is a multiscale hierarchy in the model leading to the depletion of 2-simplices, before the depletion of active edges. This leads to the conjecture that many other dynamic network models on simplicial complexes may show a similar behavior with respect to the sequential evolution of simplices of different dimensions.

Nonuniversal opinion dynamics driven by opposing external influences

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.

Group-based rewiring rules of binary opinion competition dynamics

The dynamics of competing opinions on networks has attracted multi-disciplinary research. Most modelling approaches assume uniform or heterogeneous behaviour among all individuals, while the role of distinctive group behaviour is rarely addressed. Here, we consider competition occurring between two opinion groups with bound rewiring rules, i.e., opinion-preferred rewiring, degree-preferred rewiring and random rewiring. When two opinions share a balanced initial proportion, opinion-preferred rewiring is superior to the other rules under low rewiring rates, and coexistence occurs under high rewiring rates. For unbalanced proportions, the best response rule for the minority/majority is unfixed, and this depends on the initial proportion and rewiring frequency. Furthermore, we find evolution processes for all competing cases belong to two categories. Evolution Category I shows an obvious correlation between opinion proportions and the density of discordant edges (connecting nodes with different opinions), and these trends can be effectively described by numerical approximations. However, for Evolution Category II, no such correlation exists for individuals or linking pairs, and an analysis of local structures reveals the emergence of large numbers of open triads with the same opinions, denoting group prevalence. This work broadens the understanding of opinion competition and inspires exploring group strategies employed in social dynamic systems.

Strategic tradeoffs in competitor dynamics on adaptive networks

Recent empirical work highlights the heterogeneity of social competitions such as political campaigns: proponents of some ideologies seek debate and conversation, others create echo chambers. While symmetric and static network structure is typically used as a substrate to study such competitor dynamics, network structure can instead be interpreted as a signature of the competitor strategies, yielding competition dynamics on adaptive networks. Here we demonstrate that tradeoffs between aggressiveness and defensiveness (i.e., targeting adversaries vs. targeting like-minded individuals) creates paradoxical behaviour such as non-transitive dynamics. And while there is an optimal strategy in a two competitor system, three competitor systems have no such solution; the introduction of extreme strategies can easily affect the outcome of a competition, even if the extreme strategies have no chance of winning. Not only are these results reminiscent of classic paradoxical results from evolutionary game theory, but the structure of social networks created by our model can be mapped to particular forms of payoff matrices. Consequently, social structure can act as a measurable metric for social games which in turn allows us to provide a game theoretical perspective on online political debates.

Bias, belief, and consensus: Collective opinion formation on fluctuating networks

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.

Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one-dimensional rivers

Upstream migration of fish schools in 1-D rivers as an optimal control problem is formulated where their swimming velocity and the horizontal oblateness are taken as control variables. The objective function to be maximized through a migration process consists of the biological and ecological profit to be gained at the upstream-end of a river, energetic cost of swimming against the flow, and conceptual cost of forming a school. Under simplified conditions where the flow is uniform in both space and time and the profit to be gained at the goal of migration is sufficiently large, the optimal control variables are determined from a system of algebraic equations that can be solved in a cascading manner. Mathematical analysis of the system reveals that the optimal controls are uniquely found and the model is exactly solvable under certain conditions on the functions and parameters, which turn out to be realistic and actually satisfied in experimental fish migration. Identification results of the functional shapes of the functions and the parameters with experimentally observed data of swimming schools of Plecoglossus altivelis (Ayu) validate the present mathematical model from both qualitative and quantitative viewpoints. The present model thus turns out to be consistent with the reality, showing its potential applicability to assessing fish migration in applications.

Stationary solutions for metapopulation Moran models with mutation and selection

We construct an individual-based metapopulation model of population genetics featuring migration, mutation, selection, and genetic drift. In the case of a single "island," the model reduces to the Moran model. Using the diffusion approximation and time-scale separation arguments, an effective one-variable description of the model is developed. The effective description bears similarities to the well-mixed Moran model with effective parameters that depend on the network structure and island sizes, and it is amenable to analysis. Predictions from the reduced theory match the results from stochastic simulations across a range of parameters. The nature of the fast-variable elimination technique we adopt is further studied by applying it to a linear system, where it provides a precise description of the slow dynamics in the limit of large time-scale separation.

Solution of the voter model by spectral analysis

An exact spectral analysis of the Markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the voter model, this method is applied to compute analytically several key quantities such as exact expressions for the m time-step propagator of the voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte Carlo simulations are presented to validate the spectral method.

Fast-mode elimination in stochastic metapopulation models

We investigate the stochastic dynamics of entities which are confined to a set of islands, between which they migrate. They are assumed to be one of two types, and in addition to migration, they also reproduce and die. Birth and death events are later moderated by weak selection. Systems which fall into this class are common in biology and social science, occurring in ecology, population genetics, epidemiology, biochemistry, linguistics, opinion dynamics, and other areas. In all these cases the governing equations are intractable, consisting as they do of multidimensional Fokker-Planck equations or, equivalently, coupled nonlinear stochastic differential equations with multiplicative noise. We develop a methodology which exploits a separation in time scales between fast and slow variables to reduce these equations so that they resemble those for a single island, which are amenable to analysis. The technique is generally applicable, but we choose to discuss it in the context of population genetics, in part because of the extra features that appear due to selection. The idea behind the method is simple, its application is systematic, and the results are in very good agreement with simulations of the full model for a range of parameter values.

Asymmetric coevolutionary voter dynamics

We consider a modification of the adaptive contact process that, interpreted in the context of opinion dynamics, breaks the symmetry of the coevolutionary voter model by assigning to each node type a different strategy to promote consensus: Orthodox opinion holders spread their opinion via social pressure and rewire their connections following a segregationist strategy; heterodox opinion holders adopt a proselytic strategy, converting their neighbors through personal interactions, and relax to the orthodox opinion according to its representation in the population. We give a full description of the phase diagram of this asymmetric model, using the standard pair approximation equations and assessing their performance by comparison with stochastic simulations. We find that although global consensus is favored with regard to the symmetric case, the asymmetric model also features an active phase. We study the stochastic properties of the corresponding metastable state in finite-size networks, discussing the applicability of the analytic approximations developed for the coevolutionary voter model. We find that, in contrast to the symmetric case, the final consensus state is predetermined by the system's parameters and independent of initial conditions for sufficiently large system sizes. We also find that rewiring always favors consensus, both by significantly reducing convergence times and by changing their scaling with system size.