Ising model simulation in directed lattices and networks

On directed lattices, with half as many neighbours as in the usual undirected lattices, the Ising model does not seem to show a spontaneous magnetisation, at least for lower dimensions. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law on the square and simple cubic lattice. On directed Barabasi-Albert networks with two and seven neighbours selected by each added site, Metropolis and Glauber algorithms give similar results, while for Wolff cluster flipping the magnetisation decays exponentially with time.Comment: Expanded to 8 pages: additional author, additional result

Ising model with spins S=1/2 and 1 on directed and undirected Erd\"os-R\'enyi random graphs

Using Monte Carlo simulations we study the Ising model with spin S=1/2 and 1 on {\it directed} and {\it undirected} Erd\"os-R\'enyi (ER) random graphs, with $z$ neighbors for each spin. In the case with spin S=1/2, the {\it undirected} and {\it directed} ER graphs present a spontaneous magnetization in the universality class of mean field theory, where in both {\it directed} and {\it undirected} ER graphs the model presents a spontaneous magnetization at $p = z/N$ ($z=2, 3, ...,N$), but no spontaneous magnetization at $p = 1/N$ which is the percolation threshold. For both {\it directed} and {\it undirected} ER graphs with spin S=1 we find a first-order phase transition for z=4 and 9 neighbors.Comment: 11 pages, 8 figure

Evolution of ethnocentrism on undirected and directed Barabási-Albert networks

Using Monte Carlo simulations, we study the evolution of contigent cooperation and ethnocentrism in the one-move game. Interactions and reproduction among computational agents are simulated on undirected and directed Barabási-\ud Albert (BA) networks. We first replicate the Hammond-Axelrod model of in-group favoritism on a square lattice and then generalize this model on undirected and directed BA networks for both asexual and sexual reproduction cases. Our simulations demonstrate that irrespective of the mode of reproduction, ethnocentric strategy becomes common even though cooperation is individually costly and mechanisms such as reciprocity or conformity are absent. Moreover, our results indicate that the spread of favoritism toward similar others highly depends on the network topology and the associated heterogeneity of the studied population

Potts model with q=3 and 4 states on directed Small-World network

Monte Carlo simulations are performed to study the two-dimensional Potts models with q=3 and 4 states on directed Small-World network. The disordered system is simulated applying the Heat bath Monte Carlo update algorithm. A first-order and second-order phase transition is found for q=3 depending on the rewiring probability $p$, but for q=4 the system presents only a first-order phase transition for any value $p$ . This critical behavior is different from the Potts model on a square lattice, where the second-order phase transition is present for $q\le4$ and a first-order phase transition is present for q>4.Comment: 5 pages, 8 figures. arXiv admin note: text overlap with arXiv:1001.184

Three-state majority-vote model on square lattice

Here, the model of non-equilibrium model with two states ($-1,+1$) and a noise $q$ on simple square lattices proposed for M.J. Oliveira (1992) following the conjecture of up-down symmetry of Grinstein and colleagues (1985) is studied and generalized. This model is well-known, today, as Majority-Vote Model. They showed, through Monte Carlo simulations, that their obtained results fall into the universality class of the equilibrium Ising model on a square lattice. In this work, we generalize the Majority-Vote Model for a version with three states, now including the zero state, ($-1,0,+1$) in two dimensions. Using Monte Carlo simulations, we showed that our model falls into the universality class of the spin-1 ($-1,0,+1$) and spin-1/2 Ising model and also agree with Majority-Vote Model proposed for M.J. Oliveira (1992) . The exponents ratio obtained for our model was $\gamma/\nu =1.77(3)$, $\beta/\nu=0.121(5)$, and $1/\nu =1.03(5)$. The critical noise obtained and the fourth-order cumulant were $q_{c}=0.106(5)$ and $U^{*}=0.62(3)$.Comment: 13 pages, 6 figure

A Biased Review of Sociophysics

Various aspects of recent sociophysics research are shortly reviewed: Schelling model as an example for lack of interdisciplinary cooperation, opinion dynamics, combat, and citation statistics as an example for strong interdisciplinarity.Comment: 16 pages for J. Stat. Phys. including 2 figures and numerous reference

The Harris-Luck criterion for random lattices

The Harris-Luck criterion judges the relevance of (potentially) spatially correlated, quenched disorder induced by, e.g., random bonds, randomly diluted sites or a quasi-periodicity of the lattice, for altering the critical behavior of a coupled matter system. We investigate the applicability of this type of criterion to the case of spin variables coupled to random lattices. Their aptitude to alter critical behavior depends on the degree of spatial correlations present, which is quantified by a wandering exponent. We consider the cases of Poissonian random graphs resulting from the Voronoi-Delaunay construction and of planar, ``fat'' $\phi^3$ Feynman diagrams and precisely determine their wandering exponents. The resulting predictions are compared to various exact and numerical results for the Potts model coupled to these quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one figure added for clarification, minor re-wordings and typo cleanu

Opinion dynamics: models, extensions and external effects

Recently, social phenomena have received a lot of attention not only from social scientists, but also from physicists, mathematicians and computer scientists, in the emerging interdisciplinary field of complex system science. Opinion dynamics is one of the processes studied, since opinions are the drivers of human behaviour, and play a crucial role in many global challenges that our complex world and societies are facing: global financial crises, global pandemics, growth of cities, urbanisation and migration patterns, and last but not least important, climate change and environmental sustainability and protection. Opinion formation is a complex process affected by the interplay of different elements, including the individual predisposition, the influence of positive and negative peer interaction (social networks playing a crucial role in this respect), the information each individual is exposed to, and many others. Several models inspired from those in use in physics have been developed to encompass many of these elements, and to allow for the identification of the mechanisms involved in the opinion formation process and the understanding of their role, with the practical aim of simulating opinion formation and spreading under various conditions. These modelling schemes range from binary simple models such as the voter model, to multi-dimensional continuous approaches. Here, we provide a review of recent methods, focusing on models employing both peer interaction and external information, and emphasising the role that less studied mechanisms, such as disagreement, has in driving the opinion dynamics. [...]Comment: 42 pages, 6 figure

Critical behavior of the Ising and Blume-Capel models on directed two-dimensional small-world networks

The critical properties of the two-dimensional Ising and Blume-Capel model on directed small-world lattices with quenched connectivity disorder are investigated. The disordered system is simulated by applying the Monte Carlo method with heat bath update algorithm and histogram re-weighting techniques. The critical temperature, as well as the critical exponents are obtained. For both models the critical parameters have been obtained for several values of the rewiring probability p. It is found that these disorder systems do not belong to the same universality class as two-dimensional ferromagnetic model on regular lattices. In particular, the Blume-Capel model, with zero crystal field interaction, on a directed small-world lattice presents a second-order phase transition for p  pc, where pc ≈ 0.25. The critical exponents for p < pc are different from those of the same model on a regular lattice, but are identical to the exponents of the Ising model on directed small-world lattice

Is Kaniadakis κ-generalized statistical mechanics general?

In this Letter we introduce some field-theoretic approach for computing the critical properties of systems undergoing continuous phase transitions governed by the κ-generalized statistics, namely κ-generalized statistical field theory. In particular, we show, by computations through analytic and simulation results, that the κ-generalized Ising-like systems are not capable of describing the nonconventional critical properties of real imperfect crystals, e.g. of manganites, as some alternative generalized theory is, namely nonextensive statistical field theory, as shown recently in literature. Although κ-Ising-like systems do not depend on κ, we show that a few distinct systems do. Thus the κ-generalized statistical field theory is not general, i.e. it fails to generalize Ising-like systems for describing the critical behavior of imperfect crystals, and must be discarded as one generalizing statistical mechanics. For the latter systems we present the physical interpretation of the theory by furnishing the general physical interpretation of the deformation κ-parameter