Gyorgy Korniss

Social consensus through the influence of committed minorities.

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 p(c) ≈ 10%, there is a dramatic decrease in the time T(c) taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < pc, T(c) ~ exp [α(p)N], whereas for p>p(c), T(c) ~ ln N. We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.

Suppressing Roughness of Virtual Times in Parallel Discrete-Event Simulations

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.

Dynamic phase transition, universality, and finite-size scaling in the two-dimensional kinetic Ising model in an oscillating field

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 multidroplet 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 nonequilibrium 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 multidroplet regime to the strong-field regime, where the transition disappears.

Absence of first-order transition and tricritical point in the dynamic phase diagram of a spatially extended bistable system in an oscillating field.

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 of 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 fourth-order 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.

Evolution of opinions on social networks in the presence of competing committed groups.

Public opinion is often affected by the presence of committed groups of individuals dedicated to competing points of view. Using a model of pairwise social influence, we study how the presence of such groups within social networks affects the outcome and the speed of evolution of the overall opinion on the network. Earlier work indicated that a single committed group within a dense social network can cause the entire network to quickly adopt the groups opinion (in times scaling logarithmically with the network size), so long as the committed group constitutes more than about of the population (with the findings being qualitatively similar for sparse networks as well). Here we study the more general case of opinion evolution when two groups committed to distinct, competing opinions and , and constituting fractions and of the total population respectively, are present in the network. We show for stylized social networks (including Erdös-Rényi random graphs and Barabási-Albert scale-free networks) that the phase diagram of this system in parameter space consists of two regions, one where two stable steady-states coexist, and the remaining where only a single stable steady-state exists. These two regions are separated by two fold-bifurcation (spinodal) lines which meet tangentially and terminate at a cusp (critical point). We provide further insights to the phase diagram and to the nature of the underlying phase transitions by investigating the model on infinite (mean-field limit), finite complete graphs and finite sparse networks. For the latter case, we also derive the scaling exponent associated with the exponential growth of switching times as a function of the distance from the critical point.

Threshold-limited spreading in social networks with multiple initiators

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.

Network synchronization in a noisy environment with time delays: fundamental limits and trade-offs.

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrary network. Using the known exact threshold value from the theory of differential equations with delays, we provide the synchronizability threshold for an arbitrary network. Further, by constructing the scaling theory of the underlying fluctuations, we establish the absolute limit of synchronization efficiency in a noisy environment with uniform time delays, i.e., the minimum attainable value of the width of the synchronization landscape. Our results also have strong implications for optimization and trade-offs in network synchronization with delays.

From massively parallel algorithms and fluctuating time horizons to nonequilibrium surface growth

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 nonequilibrium surface. Monte Carlo simulations and a coarse-grained approximation indicate that the macroscopic landscape in the steady state is governed by the Edwards-Wilkinson 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.

Synchronization in weighted uncorrelated complex networks in a noisy environment: optimization and connections with transport efficiency.

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 (kikj)beta with ki and kj being the degrees of the nodes connected by the link. Subject to the constraint that the total edge cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at beta*= -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 beta*. 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.

Parallelization of a Dynamic Monte Carlo Algorithm

We experiment with a massively parallel implementation of an algorithm for simulating the dynamics of metastable decay in kinetic Ising models. The parallel scheme is directly applicable to a wide range of stochastic cellular automata where the discrete events (updates) are Poisson arrivals. For high performance, we utilize a continuous-time, asynchronous parallel version of the n-fold way rejection-free algorithm. Each processing element carries an l×l block of spins, and we employ fast one-sided communication routines on a distributed-memory parallel architecture. Different processing elements have different local simulated times. To ensure causality, the algorithm handles the asynchrony in a conservative fashion. Despite relatively low utilization and an intricate relationship between the average time increment and the size of the spin blocks, we find that the algorithm is scalable and for sufficiently large l it outperforms its corresponding parallel Metropolis (non-rejection-free) counterpart. As a sample application, we present results for metastable decay in a model ferromagnetic or ferroelectric film, observed with a probe of area smaller than the total system.