Abstract
We provide a first study of the threshold model, where both conformist and anti-conformist agents coexist. The paper is in the line of a previous work by the first author (Grabisch et al., 2018), whose results will be used at some point in the present paper. Our study bears essentially in answering the following question: Given a society of agents with a certain topology of the network linking these agents, given a mechanism of influence for each agent, how the behavior/opinion of the agents will evolve with time, and in particular can it be expected that it converges to some stable situation, and in this case, which one? Also, we are interested by the existence of cascade effects, as this may constitute a 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 work of Granovetter (1978). 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, the graph is no more complete and 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.
Highlights
Human behavior is governed by many aspects, related to social context, culture, law and other factors
The question has been studied by sociologists and psychologists, and a number of pioneering models of “opinion dynamics” have been proposed by them, e.g., Granovetter (1978), Abelson (1964), French Jr (1956), Friedkin and Johnsen (1990), Taylor (1968), but it has attracted the attention of many physicists, assimilating agents to particles (this field is usually called “sociophysics”, after the work of Galam (2012); see a survey in Castellano et al (2009)), economists (see, e.g., the monograph of Jackson (2008), and the survey by Acemoglu and Ozdaglar (2011)), computer scientists and probabilists
The simplicity of the model allows for a deep analysis (see the surveys by Mossel and Tamuz (2017) and Castellano et al (2009)), and one remarkable result already observed in the pioneering work of Granovetter (1978) was that a cascade effect occurs, supposing that the population of agents starts from an initial state where nobody is active, and that the distribution of the threshold value is uniform over the population
Summary
Human behavior is governed by many aspects, related to social context, culture, law and other factors. The simplicity of the model allows for a deep analysis (see the surveys by Mossel and Tamuz (2017) and Castellano et al (2009)), and one remarkable result already observed in the pioneering work of Granovetter (1978) was that a cascade effect occurs, supposing that the population of agents starts from an initial state where nobody is active, and that the distribution of the threshold value is uniform over the population. In Nyczka and Sznajd-Weron (2013), the q-voter model is studied, where it is supposed that agents may adopt with some probability an anticonformist attitude, while the threshold model is considered under this assumption in Nowak and Sznajd-Weron (2019) Close to this model is the recent study of Juul and Porter (2019) about the spreading of two competing products, say A and B, where anticonformist agents are called hipsters (see Touboul (2014) where this terminology has been introduced).
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