Anti-conformism in the Threshold Model of Collective Behavior

Abstract

We provide a detailed study of the threshold model, where both conformist and anticonformist agents coexist. Our study bears essentially on the convergence of the opinion dynamics in the society of agents, i.e., finding absorbing classes, cycles, etc. Also, we are interested in the existence of cascade effects, as this may constitute an undesirable phenomenon in collective behavior. We divide our study into two parts. In the first one, we basically study the threshold model supposing a fixed complete network, where every one is connected to every one, like in the seminal work of Granovetter. We study the case of a uniform distribution of the threshold, of a Gaussian distribution, and finally give a result for arbitrary distributions, supposing there is one type of anti-conformist. In a second part, we suppose that the neighborhood of an agent is random, drawn at each time step from a distribution. We distinguish the case where the degree (number of links) of an agent is fixed, and where there is an arbitrary degree distribution. We show the existence of cascades and that for most societies, the opinion converges to a chaotic situation.

On games with coordinating and anti-coordinating agents

“…Granovetter (1978) studied the synchronous best-response dynamics of coordinating agents as a linear threshold model. Similar results have been generalized to configuration models by Rossi et al (2017) and to games with a mixture of coordinating and anti-coordinating agents by Grabisch and Li (2019). Other results concern time of convergence of asynchronous bestresponse dynamics and its dependence on the network structure (Ellison (1993), Kandori et al (1993), Montanari and Saberi (2010)).…”

Equilibria and learning dynamics in mixed network coordination/anti-coordination games

“…This is a dynamical system where, synchronously, all players update their action choosing the one adopted by the majority in their neighborhood. This model has been extended [14]- [16] considering also the possible presence of anticonformist individuals that choose to play the action played by the minority in their neighborhood. Various versions of the linear threshold model have been introduced depending on how matches are treated and depending if certain transitions are considered or not to be irreversible.…”

A Survey on Nonstrategic Models of Opinion Dynamics

“…The classical threshold model is clearly conformist. There exist recent studies on an anticonformist version of the threshold model, see, e.g., Nowak and Sznajd-Weron [43], Vanelli et al [81], Grabisch and Li [82]. We give some excerpts of the latter reference.…”

“…Theorem 2. [82] Assume an arbitrary distribution of the thresholds with lowest value µ > 0 and highest value µ, and an arbitrary degree (size of the neighborhood) distribution, with lowest value d.…”