Carmen P. C. Prado

ROBUSTNESS OF SCALE INVARIANCE IN MODELS WITH SELF-ORGANIZED CRITICALITY

A random neighbor extremal stick-slip model is introduced. In the thermodynamic limit, the distribution of states has a simple analytical form and the mean avalanche size, as a function of the coupling parameter, is exactly calculable. The system is critical only at a special point Jc in the coupling parameter space. However, the critical region around this point, where approximate scale invariance holds, is very large, suggesting a mechanism for explaining the ubiquity of scale invariance in Nature.

A direct calculation of the spectrum of singularities f(α) of multifractals

Abstract We present a numerical method to perform a direct evaluation of the spectrum of singularities f(α) of a multifractal set. Using this direct method, we avoid the distortions of f(α) for q

Network of epicenters of the Olami-Feder-Christensen model of earthquakes

We study the dynamics of the Olami-Feder-Christensen (OFC) model of earthquakes, focusing on the behavior of sequences of epicenters regarded as a growing complex network. Besides making a detailed and quantitative study of the effects of the borders (the occurrence of epicenters is dominated by a strong border effect which does not scale with system size), we examine the degree distribution and the degree correlation of the graph. We detect sharp differences between the conservative and nonconservative regimes of the model. Removing border effects, the conservative regime exhibits a Poisson-like degree statistics and is uncorrelated, while the nonconservative has a broad power-law-like distribution of degrees (if the smallest events are ignored), which reproduces the observed behavior of real earthquakes. In this regime the graph has also an unusually strong degree correlation among the vertices with higher degree, which is the result of the existence of temporary attractors for the dynamics: as the system evolves, the epicenters concentrate increasingly on fewer sites, exhibiting strong synchronization, but eventually spread again over the lattice after a series of sufficiently large earthquakes. We propose an analytical description of the dynamics of this growing network, considering a Markov process network with hidden variables, which is able to account for the mentioned properties.

Exit probability of the one-dimensional q-voter model: analytical results and simulations for large networks.

We discuss the exit probability of the one-dimensional q-voter model and present tools to obtain estimates about this probability, both through simulations in large networks (around 10(7) sites) and analytically in the limit where the network is infinitely large. We argue that the result E(ρ) = ρ(q)/ρ(q) + (1-ρ)(q), that was found in three previous works [F. Slanina, K. Sznajd-Weron, and P. Przybyła, Europhys. Lett. 82, 18006 (2008); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008), for the case q = 2; and P. Przybyła, K. Sznajd-Weron, and M. Tabiszewski, Phys. Rev. E 84, 031117 (2011), for q > 2] using small networks (around 10(3) sites), is a good approximation, but there are noticeable deviations that appear even for small systems and that do not disappear when the system size is increased (with the notable exception of the case q = 2). We also show that, under some simple and intuitive hypotheses, the exit probability must obey the inequality ρ(q)/ρ(q) + (1-ρ) ≤ E(ρ) ≤ ρ/ρ + (1-ρ)(q) in the infinite size limit. We believe this settles in the negative the suggestion made [S. Galam and A. C. R. Martins, Europhys. Lett. 95, 48005 (2001)] that this result would be a finite size effect, with the exit probability actually being a step function. We also show how the result that the exit probability cannot be a step function can be reconciled with the Galam unified frame, which was also a source of controversy.

Generalized Sznajd model for opinion propagation.

Instituto de Física, Universidade de São Paulo Caixa Postal 66318, 05314-970 São Paulo São Paulo Brazil (Dated: May 4, 2009) Abstract In the last decade the Sznajd Model has been successfully emp loyed in modeling some properties and scale features of both proportional and majority elections . We propose a new version of the Sznajd model with a generalizedbounded confidence rule a rule that limits the convincing capability of agents and that is essential to allow coexistence of opinions in the stationar y state. With an appropriate choice of parameters it can be reduced to previous models. We solved this new model both in a mean-field approach (for an arbitrary number of opinions) and numerically in a Barabási -Albert network (for three and four opinions), studying the transient and the possible stationary states. W built the phase portrait for the special cases of three and four opinions, defining the attractors and their ba sins of attraction. Through this analysis, we were able to understand and explain discrepancies between meanfield and simulation results obtained in previous works for the usual Sznajd Model with bounded confidence and t hree opinions. Both the dynamical system approach and our generalized bounded confidence rule are qui te general and we think it can be useful to the understanding of other similar models.

Coexistence of interacting opinions in a generalized Sznajd model.

The Sznajd model is a sociophysics model that mimics the propagation of opinions in a closed society, where the interactions favor groups of agreeing people. It is based in the Ising and Potts ferromagnetic models and, although the original model used only linear chains, it has since been adapted to general networks. This model has a very rich transient, which has been used to model several aspects of elections, but its stationary states are always consensus states. In order to model more complex behaviors, we have, in a recent work, introduced the idea of biases and prejudices to the Sznajd model by generalizing the bounded confidence rule, which is common to many continuous opinion models, to what we called confidence rules. In that work we have found that the mean field version of this model (corresponding to a complete network) allows for stationary states where noninteracting opinions survive, but never for the coexistence of interacting opinions. In the present work, we provide networks that allow for the coexistence of interacting opinions for certain confidence rules. Moreover, we show that the model does not become inactive; that is, the opinions keep changing, even in the stationary regime. This is an important result in the context of understanding how a rule that breeds local conformity is still able to sustain global diversity while avoiding a frozen stationary state. We also provide results that give some insights on how this behavior approaches the mean field behavior as the networks are changed.

SZNAJD MODEL AND PROPORTIONAL ELECTIONS: THE ROLE OF THE TOPOLOGY OF THE NETWORK

The Sznajd model (SM) has been employed with success in the last years to describe opinion propagation in a community. In particular, it has been claimed that its transient is able to reproduce some scale properties observed in data of proportional elections, in different countries, if the community structure (the network) is scale-free. In this work, we investigate the properties of the transient of a particular version of the SM, introduced by Bernardes and co-authors in 2002. We studied the behavior of the model in networks of different topologies through the time evolution of an order parameter known as interface density, and concluded that regular lattices with high dimensionality also leads to a power-law distribution of the number of candidates with v votes. Also, we show that the particular absorbing state achieved in the stationary state (or else, the winner candidate), is related to a particular feature of the model, that may not be realistic in all situations.

Connections between the Sznajd model with general confidence rules and graph theory.

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabási-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q>2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the q-voter model).

Dealing with transients in models with self-organized criticality

The problems of identifying and eliminating long transients are common to various numerical models in statistical mechanics. These problems are particularly relevant for models of self-organized criticality, as the Olami–Feder–Christensen (OFC) model, for which most of the results were, and still are, obtained through numerical simulations. In order to obtain reliable numerical results, it is usually necessary to simulate models on lattices as large as possible. However, in general, this is not an easy task, because transients increase fast with lattice size. So it is often necessary to wait long computer runs to obtain good statistics. In this paper we present an efficient algorithm to reduce transient times and to identify with a certain degree of precision if the statistical stationary state is reached, avoiding long runs to obtain good statistics. The efficiency of the algorithm is exemplified in the OFC model for the dynamics of earthquakes, but it can be useful as well in many other situations. Our analysis also shows that the OFC model approaches stationarity in qualitatively different ways in the conservative and non-conservative cases.

Evolution of Chaos in the Matsumoto-Chua Circuit: a Symbolic Dynamics Approach

We use symbolic dynamics to follow the evolution of the Matsumoto-Chua circuit in the chaotic regime. We consider the evolution of the whole population of unstable periodic orbits and of the associated trajectories, in four chaotic attractors generated by the circuit. Symbolic planes and first return maps are built for different values of the control parameter. The bifurcation mechanism suggests the possibility of the existence of a homoclinic orbit.