Abstract:These notes concern the material covered by the authors during 4 classes on the Escola Brasileira de Mecânica Estatística, University of São Paulo at São Carlos, February 2004. They are divided in almost independent sections, each one with a small introduction to the subject and emphasis on the computational strategy adopted.
“…4) as a function of the noise parameter q, for several lattice sizes L. For sufficiently large systems, these curves intercept each other in a sin- gle point U * (q c ). The value of q where occurs the intersection equals the critical noise q c , which is not biased by any assumption about critical exponents since, by construction, the Binder's cumulant presents zero anomalous dimension [16,17]. In Fig.…”
We introduce and study the block voter model with noise on two-dimensional square lattices using Monte Carlo simulations and finite-size scaling techniques. The model is defined by an outflow dynamics where a central set of N(PCS) spins, here denoted by persuasive cluster spins (PCS), tries to influence the opinion of their neighboring counterparts. We consider the collective behavior of the entire system with varying PCS size. When N(PCS)>2, the system exhibits an order-disorder phase transition at a critical noise parameter q(c) which is a monotonically increasing function of the size of the persuasive cluster. We conclude that a larger PCS has more power of persuasion, when compared to a smaller one. It also seems that the resulting critical behavior is Ising-like independent of the range of interaction.
“…4) as a function of the noise parameter q, for several lattice sizes L. For sufficiently large systems, these curves intercept each other in a sin- gle point U * (q c ). The value of q where occurs the intersection equals the critical noise q c , which is not biased by any assumption about critical exponents since, by construction, the Binder's cumulant presents zero anomalous dimension [16,17]. In Fig.…”
We introduce and study the block voter model with noise on two-dimensional square lattices using Monte Carlo simulations and finite-size scaling techniques. The model is defined by an outflow dynamics where a central set of N(PCS) spins, here denoted by persuasive cluster spins (PCS), tries to influence the opinion of their neighboring counterparts. We consider the collective behavior of the entire system with varying PCS size. When N(PCS)>2, the system exhibits an order-disorder phase transition at a critical noise parameter q(c) which is a monotonically increasing function of the size of the persuasive cluster. We conclude that a larger PCS has more power of persuasion, when compared to a smaller one. It also seems that the resulting critical behavior is Ising-like independent of the range of interaction.
“…To determine estimates for the critical point q c , we calculate the Binder fourthorder magnetization cumulant U at different values of the noise q and several network sizes N. Finite size scaling predicts that for sufficiently large systems, these curves should have a unique intersection point U * [27]. The value of q where this crossing occurs is the value of the critical noise q c which is not biased by any assumptions about critical exponents, since by construction, the Binder cumulant presents zero anomalous dimension, therefore it respects the correct critical behavior of the system near q c [27,28].…”
Through Monte Carlo Simulation, the well-known majority-vote model has been studied with noise on directed random graphs. In order to characterize completely the observed order-disorder phase transition, the critical noise parameter q c , as well as the critical exponents β/ν, γ/ν and 1/ν have been calculated as a function of the connectivity z of the random graph.
“…the configurations are reached without the need of the external tuning of any parameter, and the power law suggests the existence of critical behavior. The self organized criticality (SOC) has been discussed in different places [16][17][18][19][20] and has been observed in many different systems [20][21][22][23][24]. In our case, this suggests that, on rare occasions, consensus would spontaneously be reached.…”
Section: Resultsmentioning
confidence: 54%
“…This is called self-organized criticality (SOC) and has been discussed in many different places [12,[22][23][24][25][26]. It has also been observed in a large diversity of systems [12,[22][23][24][25][26][27][28][29][30]. The asymptotic value of the power law exponent of the size distribution shown above lies within the values obtained with the random site percolation (RSP) whose exponent ranges from 2.05 for two-dimensional lattices [13] or 2.186 ± 0.002 for three-dimensional lattices [31] up to 2.5 for the Bethe lattice [13].…”
We propose an opinion model based on agents located at the vertices of a regular lattice. Each agent has an independent opinion (among an arbitrary, but fixed, number of choices) and its own degree of conviction. The latter changes every time two agents which have different opinions interact with each other. The dynamics leads to size distributions of clusters (made up of agents which have the same opinion and are located at contiguous spatial positions) which follow a power law, as long as the range of the interaction between the agents is not too short; i.e., the system self-organizes into a critical state. Short range interactions lead to an exponential cutoff in the size distribution and to spatial correlations which cause agents which have the same opinion to be closely grouped. When the diversity of opinions is restricted to two, a nonconsensus dynamic is observed, with unequal population fractions, whereas consensus is reached if the agents are also allowed to interact with those located far from them. The individual agents' convictions, the preestablished interaction range, and the locality of the interaction between a pair of agents (their neighborhood has no effect on the interaction) are the main characteristics which distinguish our model from previous ones.
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