Absorbing states of zero-temperature Glauber dynamics in random networks

Order-disorder transition in the zero-temperature Ising model on random graphs

The zero-temperature Ising model is known to reach a fully ordered ground state in sufficiently dense random graphs. In sparse random graphs, the dynamics gets absorbed in disordered local minima at magnetization close to zero. Here, we find that the nonequilibrium transition between the ordered and the disordered regime occurs at an average degree that slowly grows with the graph size. The system shows bistability: The distribution of the absolute magnetization in the reached absorbing state is bimodal, with peaks only at zero and unity. For a fixed system size, the average time to absorption behaves nonmonotonically as a function of average degree. The peak value of the average absorption time grows as a power law of the system size. These findings have relevance for community detection, opinion dynamics, and games on networks.

Nonequilibrium dynamics in a three-state opinion-formation model with stochastic extreme switches

We investigate the nonequilibrium dynamics of a three-state kinetic exchange model of opinion formation, where switches between extreme states are possible, depending on the value of a parameter q. The mean field dynamical equations are derived and analyzed for any q. The fate of the system under the evolutionary rules used in S. Biswas et al. [Physica A 391, 3257 (2012)0378-437110.1016/j.physa.2012.01.046] shows that it is dependent on the value of q and the initial state in general. For q=1, which allows the extreme switches maximally, a quasiconservation in the dynamics is obtained which renders it equivalent to the voter model. For general q values, a "frozen" disordered fixed point is obtained which acts as an attractor for all initially disordered states. For other initial states, the order parameter grows with time t as exp[α(q)t] where α=1-q/3-q for q≠1 and follows a power law behavior for q=1. Numerical simulations using a fully connected agent-based model provide additional results like the system size dependence of the exit probability and consensus times that further accentuate the different behavior of the model for q=1 and q≠1. The results are compared with the nonequilibrium phenomena in other well-known dynamical systems.

Long route to consensus: Two-stage coarsening in a binary choice voting model

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 of average behavior of a large number of individuals in a statistical sense. However, in absence of external noise, they tend to reach a frozen state with fragmented and polarized states, i.e., two or more groups of similarly opinionated groups with frozen dynamics. Here we show in a kinetic exchange opinion model considered on L×L square lattices, that while such frozen states could be avoided, an exponentially slow approach to consensus is manifested. Specifically, the system could either reach consensus in a time that scales as L^{2} or a long-lived metastable state (termed a "domain-wall state") for which formation of consensus takes a time scaling as L^{3.6}. The latter behavior is comparable to some voterlike models with intermediate states studied previously. The late-time anomaly in the timescale is reflected in the persistence probability of the model. Finally, the interval of zero crossing of the average opinion, i.e., the time interval over which the average opinion does not change sign, is shown to follow a scale-free distribution, which is compared with that seen in the opinion surveys regarding Brexit and associated issues since the late 1970s. The issue of minority spreading is also addressed by calculating the exit probability.

Fragmentation transition in a coevolving network with link-state dynamics

We study a network model that couples the dynamics of link states with the evolution of the network topology. The state of each link, either A or B, is updated according to the majority rule or zero-temperature Glauber dynamics, in which links adopt the state of the majority of their neighboring links in the network. Additionally, a link that is in a local minority is rewired to a randomly chosen node. While large systems evolving under the majority rule alone always fall into disordered topological traps composed by frustrated links, any amount of rewiring is able to drive the network to complete order, by relinking frustrated links and so releasing the system from traps. However, depending on the relative rate of the majority rule and the rewiring processes, the system evolves towards different ordered absorbing configurations: either a one-component network with all links in the same state or a network fragmented in two components with opposite states. For low rewiring rates and finite-size networks there is a domain of bistability between fragmented and nonfragmented final states. Finite-size scaling indicates that fragmentation is the only possible scenario for large systems and any nonzero rate of rewiring.