The Sznajd model (SM) is a one-dimensional voter-like model used to study consensus in systems where information flows outward from like-minded neighboring agents. Here, we introduce long-range interactions to the SM via the parameter p, where p→1 is the mean-field limit (MFL) and p→0 the one-dimensional limit (1DL). Using Monte Carlo simulations and finite size scaling analyses to characterize the exit probability for p > 0, we find a step function reliant on two p-dependent exponents. By examining the exponents' behavior in the 1DL, we comment on the functional form of the exit probability in one dimension—its nature has been an open question. Complimenting this limiting approach, we also simulate the 1D case via two parallelization techniques (task-level and data-level). Finally, we investigate the quantitative nature of consensus time and system magnetization across the p-spectrum.
We find that one of the exponents grows rapidly in the 1DL, its behavior suggesting divergence in this limit; the other stays approximately constant, although more low-p runs are needed to verify both values. Combined, these two exponents give rise to a functional form that well approximates a sigmoidal polynomial that almost exactly fits the original SM simulation results. We also find that consensus time at fixed system size is proportional to p-1.1.
In testing the parallel code, we find that the task-level parallelism approach generates a speedup nearly equal to the number of processors applied; conversely, the data-level parallelism approach results tentatively suggest a superlinear speedup for constant system size.
Garcia, Joseph, "Application of Parallel Computing to Optimize Studies of Critical Exponents in the One-Dimensional Sznajd Model" (2016). Honors College. 386.