We show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value pc ≈ 10%, there is a dramatic decrease in the time, Tc, taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < pc, Tc ∼ exp(α(p)N ), while for p > pc, Tc ∼ ln N . We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.

We study the two-dimensional kinetic Ising model below its equilibrium critical temperature, subject to a square-wave oscillating external field. We focus on the multi-droplet regime where the metastable phase decays through nucleation and growth of many droplets of the stable phase. At a critical frequency, the system undergoes a genuine non-equilibrium phase transition, in which the symmetry-broken phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. We investigate the universal aspects of this dynamic phase transition at various temperatures and field amplitudes via large-scale Monte Carlo simulations, employing finite-size scaling techniques adopted from equilibrium critical phenomena. The critical exponents, the fixed-point value of the fourth-order cumulant, and the critical order-parameter distribution all are consistent with the universality class of the two-dimensional equilibrium Ising model. We also study the cross-over from the multi-droplet to the strong-field regime, where the transition disappears.

We study the asymptotic scaling properties of a massively parallel algorithm for discrete-event simulations where the discrete events are Poisson arrivals. The evolution of the simulated time horizon is analogous to a non-equilibrium surface. Monte Carlo simulations and a coarse-grained approximation indicate that the macroscopic landscape in the steady state is governed by the EdwardsWilkinson Hamiltonian. Since the efficiency of the algorithm corresponds to the density of local minima in the associated surface, our results imply that the algorithm is asymptotically scalable.PACS numbers: 89.80.+h, 02.70.Lq, 68.35.Ct To efficiently utilize modern supercomputers requires massively parallel implementations of dynamic algorithms for various physical, chemical, and biological processes. For many of these there are well-known and routinely used serial Monte Carlo (MC) schemes which are based on the realistic assumption that attempts to update the state of the system form a Poisson process. The parallel implementation of these dynamic MC algorithms belongs to the class of parallel discrete-event simulations, which is one of the most challenging areas in parallel computing [1] and has numerous applications not only in the physical sciences, but also in computer science, queueing theory, and economics. For example, in lattice Ising models the discrete events are spin-flip attempts, while in queueing systems they are job arrivals. Since current special-or multi-purpose parallel computers can have 10 4 − 10 5 processing elements (PE) [2], it is essential to understand and estimate the scaling properties of these algorithms.In this Letter we introduce an approach to investigate the asymptotic scaling properties of an extremely robust parallel scheme [3]. This parallel algorithm is applicable to a wide range of stochastic cellular automata with local dynamics, where the discrete events are Poisson arrivals. Although attempts have been made to estimate its efficiency under some restrictive assumptions [4], the mechanism which ensures the scalability of the algorithm in the "steady state" was never identified. Here we accomplish this by noting that the simulated time horizon is analogous to a growing and fluctuating surface. The local random time increments correspond to the deposition of random amounts of "material" at the local minima of the surface. This correspondence provides a natural ground for cross-disciplinary application of wellknown concepts from non-equilibrium surface growth [5] and driven systems [6] to a certain class of massively parallel computational schemes. To estimate the efficiency of this algorithm one must understand the morphology of the surface associated with the simulated time horizon. In particular, the efficiency of this parallel implementation (the fraction of the non-idling processing elements) exactly corresponds to the density of local minima in the surface model. We show that the steady-state behavior of the macroscopic landscape is governed by the EdwardsWilkinson (EW) Hamiltonian [7],...

In a parallel discrete-event simulation (PDES) scheme, tasks are distributed among processing elements (PEs) whose progress is controlled by a synchronization scheme. For lattice systems with short-range interactions, the progress of the conservative PDES scheme is governed by the Kardar-Parisi-Zhang equation from the theory of nonequilibrium surface growth. Although the simulated (virtual) times of the PEs progress at a nonzero rate, their standard deviation (spread) diverges with the number of PEs, hindering efficient data collection. We show that weak random interactions among the PEs can make this spread nondivergent. The PEs then progress at a nonzero, near-uniform rate without requiring global synchronizations.

The NG model is a model for the dissemination of opinions, focusing on the dissemination of mainstream opinions and how individuals move towards consensus in the adoption of a single opinion through discussion. This is a multi-individual model that uses exchange and negotiation between neighbors to share ideas, performing naming games without any central control on spatial networks. we consider the geometric structure in the two dimensions. In a L*L spatial grids, every grid represents an agent which owns his vocabulary, and every agent updates his vocabulary in terms of the word received from his neighbor. Finally all agents through the process of reaching an agreement. The agreement process can be divided into 3 stages. Firstly, the total number of words drop quickly and about 10 words left, and the remain words form several clusters, and the area of the clusters are almost the same. The adjacent agents in the same cluster have the same word. Secondly, the clusters mutually annex with the number of words slow decreasing. Because the word with the biggest area cannot gain momentum in the competition with other words with the similar area, the word with biggest area will be varying all the time. Thirdly, one area of word surpasses 50 percent and this area will be expanding continuously until reaching an agreement.

Abstract. Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to (k i k j ) β with k i and k j being the degrees of the nodes connected by the link. Subject to the constraint that the total network cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at β * =−1. Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of β * . Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.

It has been well established that spatially extended, bistable systems that are driven by an oscillating field exhibit a nonequilibrium dynamic phase transition (DPT). The DPT occurs when the field frequency is on the order of the inverse of an intrinsic lifetime associated with the transitions between the two stable states in a static field of the same magnitude as the amplitude of the oscillating field. The DPT is continuous and belongs to the same universality class as the equilibrium phase transition of the Ising model in zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed that the DPT becomes discontinuous at temperatures below a tricritical point [M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on observations in dynamic Monte Carlo simulations of a multipeaked probability density for the dynamic order parameter and negative values of the fourthorder cumulant ratio. Both phenomena can be characteristic of discontinuous phase transitions. Here we use classical nucleation theory for the decay of metastable phases, together with data from large-scale dynamic Monte Carlo simulations of a two-dimensional kinetic Ising ferromagnet, to show that these observations in this case are merely finite-size effects. For sufficiently small systems and low temperatures, the continuous DPT is replaced, not by a discontinuous phase transition, but by a crossover to stochastic resonance. In the infinite-system limit the stochastic-resonance regime vanishes, and the continuous DPT should persist for all nonzero temperatures.

A classical model for social-influence-driven opinion change is the threshold model. Here we study cascades of opinion change driven by threshold model dynamics in the case where multiple initiators trigger the cascade, and where all nodes possess the same adoption threshold ϕ. Specifically, using empirical and stylized models of social networks, we study cascade size as a function of the initiator fraction p. We find that even for arbitrarily high value of ϕ, there exists a critical initiator fraction pc(ϕ) beyond which the cascade becomes global. Network structure, in particular clustering, plays a significant role in this scenario. Similarly to the case of single-node or single-clique initiators studied previously, we observe that community structure within the network facilitates opinion spread to a larger extent than a homogeneous random network. Finally, we study the efficacy of different initiator selection strategies on the size of the cascade and the cascade window.