Topological bifurcations in a model society of reasonable contrarians

Abstract

People are often divided into conformists and contrarians, the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority. In practice, however, the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion, which denotes a social norm. Such reasonable contrarian behavior is often considered a mark of independent thought and can be a useful strategy in financial markets. We present the opinion dynamics of a society of reasonable contrarian agents. The model is a cellular automaton of Ising type, with antiferromagnetic pair interactions modeling contrarianism and plaquette terms modeling social norms. We introduce the entropy of the collective variable as a way of comparing deterministic (mean-field) and probabilistic (simulations) bifurcation diagrams. In the mean-field approximation the model exhibits bifurcations and a chaotic phase, interpreted as coherent oscillations of the whole society. However, in a one-dimensional spatial arrangement one observes incoherent oscillations and a constant average. In simulations on Watts-Strogatz networks with a small-world effect the mean-field behavior is recovered, with a bifurcation diagram that resembles the mean-field one but where the rewiring probability is used as the control parameter. Similar bifurcation diagrams are found for scale-free networks, and we are able to compute an effective connectivity for such networks.

Topological Phase Transitions in the Nonlinear Parallel Ising Model

“…In order to cover all branches of the pitchfork bifurcations, we used several initial conditions for the magnetization m. These bifurcation diagrams are qualitatively very similar to the ones reported in Ref. [4]. This scenario is reminiscent of that of the logistic map (where, however, the pitchfork bifurcations are not present).…”

“…In previous works [4,5], we studied the "nonlinear" behaviour of contrarians, where the effect of overwhelming majority was present only above a given threshold, denoting them with the term "reasonable contrarian". We investigated the collective behaviour of a society composed either of reasonable contrarian agents, or a mixture of contrarians and conformists.…”

Metastable States in the Parallel Ising Model

“…Thus, the nonlinear voter model can be considered analogous to the Ising model with two-site and multiple-site (plaquette) terms, where the term that couples the given site with all neighbours is infinite (thus producing locally absorbing phases). This model is presented in a companion paper [16], and the absorbing states were used in other opinion models [9], where one can also find a discussion about which model is more representative of a real society.…”

“…It is also to map this attitude onto Ising terms: conformist agents correspond to ferromagnetic coupling and contrarians to antiferromagnetic ones [1]. The presence of contrarian agents in a society have been studied in models related to Ising and the voter model [4][5][6][7][8][9][10].…”

Network inoculation: Heteroclinics and phase transitions in an epidemic model

“…The transcritical bifurcation and its close relatives, the fold and pitchfork bifurcations have been linked to phase transitions in a wide variety of systems including epidemics 4 , collective motion of animals 6,7 , human opinion formation 8,9 , neuronal dynamics 10,11 and others. In a smaller number of models the underlying bifurcation is a Hopf bifurcation, which marks the onset of, at least transient, oscillations [12][13][14] .…”