The statistical mechanics of societies: opinion formation dynamics and financial markets
Zubillaga Herrera, Bernardo José
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This work proposes a three-state microscopic opinion formation model based on the stochastic dynamics of the three-state majority-vote model. In order to mimic the heterogeneous compositions of societies, the agent-based model considers two different types of individuals: noise agents and contrarians. We propose an extension of the model for the simulation of the dynamics of financial markets. Agents are represented as nodes in a network of interactions and they can assume any of three distinct possible states (e.g. buy, sell or remain inactive, in a financial context). The time evolution of the state of an agent is dictated by probabilistic dynamics that include both local and global influences. A noise agent is subject to local interactions, tending to assume the majority state of its nearest neighbors with probability 1-q (dissenting from it with a probability given by the noise parameter q). A contrarian is subject to a global interaction with the society as a whole, tending to assume the state of the global minority of said society with probability 1 -q (dissenting from it with probability q). The stochastic dynamics are simulated on complex networks of different topologies, including square lattices, Barabási-Albert networks, Erdös-Rényi random graphs and small-world networks built according to a link rewiring scheme. We perform Monte Carlo simulations to study the second-order phase transition of the system on small-world networks. We perform finite-size scaling analysis and calculate the phase diagram of the system, as well as the standard critical exponents for different values of the rewiring probability. We conclude that the rewiring of the lattice drives the system to different universality classes than that of the three-state majority-vote model on a two dimensional square lattice. The model’s extension for financial markets exhibits the typical qualitative and quantitative features of real financial time series, including heavy-tailed return distributions, volatility clustering and long-term memory for the absolute values of the returns. The histograms of returns are fitted by means of coupled exponential distributions, quantitatively revealing transitions between leptokurtic, mesokurtic and platykurtic regimes in terms of a nonlinear statistical coupling and a shape parameter which describe the complexity of the system.