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Register a new userAsymmetric games on networks: towards an Ising-model representation
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We here study the Battle of the Sexes game, a textbook case of asymmetric games, on small networks. Due to the conflicting preferences of the players, analytical approaches are scarce and most often update strategies are employed in numerical simulations of repeated games on networks until convergence is reached. As a result, correlations between the choices of the players emerge. Our approach is to study these correlations with a generalized Ising model. Using the response strategy framework, we describe how the actions of the players can bring the network into a steady configuration, starting from an out-of-equilibrium one. We obtain these configurations using game-theoretical tools, and describe the results using Ising parameters. We exhaust the two-player case, giving a detailed account of all the equilibrium possibilities. Going to three players, we generalize the Ising model and compare the equilibrium solutions of three representative types of network. We find that players that are not directly linked retain a degree of correlation that is proportional to their initial correlation. We also find that the local network structure is the most relevant for small values of the magnetic field and the interaction strength of the Ising model. Finally, we conclude that certain parameters of the equilibrium states are network independent, which opens up the possibility of an analytical description of asymmetric games played on networks.
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Many real-world complex systems are best modeled by multiplex networks of interacting network layers. The multiplex network study is one of the newest and hottest themes in the statistical physics of complex networks. Pioneering studies have proven that the multiplexity has broad impact on the systems structure and function. In this Colloquium paper, we present an organized review of the growing body of current literature on multiplex networks by categorizing existing studies broadly according to the type of layer coupling in the problem. Major recent advances in the field are surveyed and some outstanding open challenges and future perspectives will be proposed.
The majority-vote model with noise is one of the simplest nonequilibrium statistical model that has been extensively studied in the context of complex networks. However, the relationship between the critical noise where the order-disorder phase transition takes place and the topology of the underlying networks is still lacking. In the paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the models dynamics that can analytically determine the critical noise $f_c$ in the limit of infinite network size $Nrightarrow infty$. The result shows that $f_c$ depends on the ratio of ${leftlangle k rightrangle }$ to ${leftlangle k^{3/2} rightrangle }$, where ${leftlangle k rightrangle }$ and ${leftlangle k^{3/2} rightrangle }$ are the average degree and the $3/2$ order moment of degree distribution, respectively. Furthermore, we consider the finite size effect where the stochastic fluctuation should be involved. To the end, we derive the Langevin equation and obtain the potential of the corresponding Fokker-Planck equation. This allows us to calculate the effective critical noise $f_c(N)$ at which the susceptibility is maximal in finite size networks. We find that the $f_c-f_c(N)$ decays with $N$ in a power-law way and vanishes for $Nrightarrow infty$. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random $k$-regular networks, Erdos-Renyi random networks and scale-free networks.
We study the problem of identifying macroscopic structures in networks, characterizing the impact of introducing link directions on the detectability phase transition. To this end, building on the stochastic block model, we construct a class of hardly detectable directed networks. We find closed form solutions by using belief propagation method showing how the transition line depends on the assortativity and the asymmetry of the network. Finally, we numerically identify the existence of a hard phase for detection close to the transition point.
We present a detailed investigation of the behavior of the nonlinear q-voter model for opinion dynamics. At the mean-field level we derive analytically, for any value of the number q of agents involved in the elementary update, the phase diagram, the exit probability and the consensus time at the transition point. The mean-field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks. We finally discuss the case of random regular networks and compare analytical results with simulations.
We consider two consensus formation models coupled to Barabasi-Albert networks, namely the Majority Vote model and Biswas-Chatterjee-Sen model. Recent works point to a non-universal behavior of the Majority Vote model, where the critical exponents have a dependence on the connectivity while the effective dimension $D_mathrm{eff} = 2beta/ u + gamma/ u$ of the lattice is unity. We considered a generalization of the scaling relations in order to include logarithmic corrections. We obtained the leading critical exponent ratios $1/ u$, $beta/ u$, and $gamma/ u$ by finite size scaling data collapses, as well as the logarithmic correction pseudo-exponents $widehat{lambda}$, $widehat{beta}+betawidehat{lambda}$, and $widehat{gamma}-gammawidehat{lambda}$. By comparing the scaling behaviors of the Majority Vote and Biswas-Chatterjee-Sen models, we argue that the exponents of Majority Vote model, in fact, are universal. Therefore, they do not depend on network connectivity. In addition, the critical exponents and the universality class are the same of Biswas-Chatterjee-Sen model, as seen for periodic and random graphs. However, the Majority Vote model has logarithmic corrections on its scaling properties, while Biswas-Chatterjee-Sen model follows usual scaling relations without logarithmic corrections.
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