Anticonformism in the Threshold Model of Collective Behavior
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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 anticonformist. 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.
Keywords
Threshold model Anticonformism Absorbing class Opinion dynamicsJEL Classification
C7 D7 D851 Introduction
Human behavior is governed by many aspects, related to social context, culture, law and other factors. Most of these aspects tend to indicate that our behavior is heavily influenced by the behavior of the other people with whom we are in contact, either directly or indirectly by means of communication devices, information media, etc. Behavior refers here to any kind of action, decision to be taken, or opinion to be held on a given topic. As our environment is constantly changing, behavior and opinion of people, including us, are evolving with time, which makes central the following question: Given a society of agents in a network, 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?
Evidently 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 [14], Abelson [1], French Jr [7], Friedkin and Johnsen [8], Taylor [25], but it has also attracted the attention of many physicists, assimilating agents to particles (this field is usually called “sociophysics”, after the work of Galam [11]; see a survey in Castellano et al. [6]), economists (see, e.g., the monograph of Jackson [16], and the survey by Acemoglu and Ozdaglar [2]), computer scientists and probabilists (by analogy with (probabilistic) cellular automata, see, e.g., Gravner and Griffeath [15] and the survey by Mossel and Tamuz [20]), etc.
One of the simplest model of behavior/opinion dynamics when the opinion or behavior is binary (yes/no, active/inactive, action 1 or 0, etc.) is the threshold model, also called the majority rule model [9], proposed by Granovetter [14], Schelling [24], among others. This model simply says that an agent takes action 1 if sufficiently enough people in his neighborhood takes action 1. The simplicity of the model allows for a deep analysis (see the surveys by Mossel and Tamuz [20] and Castellano et al. [6]), and one remarkable result already observed in the pioneering work of Granovetter [14] 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. Then, after a finite number of steps, all agents become active. Interestingly, the latter study was done in the context of a mob, where the available actions were “to riot” (action 1) or to be inactive (action 0). Then, agents with threshold 0 were called “instigators” as they start to riot alone, and this indeed forms the seed of the cascade effect, ending in a mob rioting. This topic has been very much studied, as demonstrated by a recent monograph on mob control [4], written by researchers in control theory.
So far, most models make the basic assumption that agents tend to follow the trend (they are conformist) and that nobody will have a kind of opposite behavior (anticonformism), choosing action 0 if too many people take action 1. Although the literature on opinion dynamics is vast, very few studies consider that agents may be anticonformist. In game theory, such kind of opposite behavior has been studied however, in what is called anticoordination games, see, e.g., Bramoullé et al. [3], LópezPintado [18], congestion games [23], and fashion games [5]. In sociophysics, the first idea about anticonformist agents seems to have been introduced by Galam [10] under the name of contrarians. Later works include those of SznajdWeron and also Juul and Porter. In Nyczka and SznajdWeron [22], the qvoter 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 SznajdWeron [21]. Close to this model is the recent study of Juul and Porter [17] about the spreading of two competing products, say A and B, where anticonformist agents are called hipsters (see Touboul [26] where this terminology has been introduced). In Juul and Porter [17], starting from a network with all nodes inactive, a single node is uniformly chosen at random to adopt one product, say A, which buries the seed for the spreading process. They assume the threshold of a player (which can be a conformist or a hipster) as the minimum proportion of their active neighbors such that this player becomes active and the transition from active to inactive is a oneway process. Once the player becomes active, they must adopt one product according to the following rules: if he is a conformist, he will adopt the most popular product over his neighborhood; if he is a hipster, he will adopt the less popular product over the whole population. Under this assumption, they found that even a small proportion of hipsters can lead to a reversal of the popularity of two competing products. The model is similar in Nowak and SznajdWeron [21], in the sense that agents are selected at random for updating and their threshold is the same for all agents; however, an agent is not a priori conformist or anticonformist, but is one or the other with some probability.
The present paper also studies a threshold model where both conformist and anticonformist agents coexist, but in a rather different setting compared to Juul and Porter [17] and Nowak and SznajdWeron [21]. Firstly, we assume that updating is done at every period for all agents. Secondly, in our setting the thresholds are drawn from a distribution which means that they are random and different, in general. In addition, the two possible states of an agent are not treated symmetrically. These are the assumptions of the seminal paper of Granovetter [14].
Our paper is in the line of a previous work by the first author [13], whose results will be used at some point in the present paper. Our study bears essentially in answering the main question raised in the first paragraph, that is, on the convergence of the process, analyzing if absorbing states exist (stable state of the society) or if a cycle occurs, or even more chaotic situations. 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 [14] (Sect. 2). We begin by giving a gametheoretic foundation to this model, by means of a mix of coordination and anticoordination games. Then, 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 anticonformist. In a second part (Sect. 3), 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. Most of the proofs can be found in Appendix.
2 The Deterministic Threshold Model with Anticonformists
2.1 The Model
Let \(N=\{1,\ldots , n\}\) be the society of agents. We suppose the existence of an underlying (exogenous) network \(G =(N,E)\) whose nodes are the agents and E is the set of (undirected) edges or links. Each agent i has a set of neighbors \(\varGamma _i=\{j\in N\,:\,\{i,j\}\in E\}\), and \(\varGamma _i=:d_i\) is the degree of agent i. We consider that \(i\in \varGamma _i\) for every agent i.
Two actions (or opinions, states) are available to each agent at every stage: 1 (agree, adopt, join, be active, etc.) and 0 (disagree, refuse, disjoin, be inactive, etc.). The action taken by agent i at stage t is denoted by \(a_i(t)\). For short, we will often use the term “active” for agents taking action 1, and “inactive” for agents taking action 0.
Observe that thresholds 0 and 1 play a particular role. For a conformist agent (respectively, an anticonformist agent), a threshold equal to 0 means that he takes always action 1 (respectively, 0), while a threshold strictly greater than 1 implies to always take action 0 (respectively, 1). We call these agents constant 0player and constant 1player.
Our aim is to study the evolution of the dynamics of actions taken by the agents. To this aim, we define the state of the society at stage t, as the set S (or S(t)) of agents taking action 1 at stage t. Depending which one is more convenient, a state is either denoted as a set \(S\subseteq N\) or as its characteristic vector \(\mathbb {1}_S\) in \(\{0,1\}^N\). The process is deterministic and Markovian, i.e., transitions from S to T (denoted by \(S\rightarrow T\)) are with probability 1 and do not depend on states before S.
We are interested in finding absorbing states, i.e., such that \(S(t)=S(t+1)\) for some value of t, and cycles, i.e., sequences of transitions \(S_1\rightarrow S_2\rightarrow \cdots \rightarrow S_k\) where \(S_k=S_1\).
2.2 A GameTheoretic Foundation of the Threshold Models
It is wellknown that the classical threshold model can be explained by a local coordination game (see, e.g., Morris [19]). We show that the anticonformist threshold model can be explained in a similar way via a local anticoordination game. We recall first the result for the classical model.
Therefore, we can recover the threshold model with the same thresholds as above \(\mu _i=q\) for both conformists and anticonformists. For conformists (respectively, anticonformists), the threshold is the minimum (respectively, maximum) probability that player i assigns to players in his neighborhood for choosing action 1.
2.3 A General Result on Cycles
This general result applies to the case of a network of conformists, taking \(\alpha _{ij}=1\) if \(\{i,j\}\in E\) and 0 otherwise, and \(\theta _i=\mu _i\varGamma _i\), but it also applies to the case of a network where all agents are anticonformist: just put \(\alpha _{ij}=1\) if \(\{i,j\}\in E\) and 0 otherwise, and \(\theta _i=\mu _i\varGamma _i+1\). Hence we have obtained:
Theorem 1
Suppose \(N_c=\emptyset \) or \(N_a=\emptyset \). Then the process converges to either an absorbing state or to a cycle of length 2.
The result is no more true if the network contains both conformists and anticonformists, as the following example shows:
Example 1
2.4 Study of the Complete Network
We suppose in this section that the graph G is complete, i.e., every agent is connected to every other agent, so that the neighborhood \(\varGamma _i\) is N for every agent i.
Theorem 2
The theorem will be illustrated by several examples in the sequel. We begin our study by supposing that the distribution of the threshold is uniform, then the Gaussian case and the general case will be studied.
2.4.1 Uniform Distribution
The case of a uniform distribution permits to get explicit results. It has been studied by Granovetter [14], in order to explain riot phenomena (action 1: take part to a riot, action 0: be inactive). Supposing at the initial state that all agents are inactive, the presence of agents with threshold 0 (called “instigators” as they start rioting alone) initiates the phenomenon of rioting, which by a domino or cascade effect extends to the whole population if the distribution is uniform.
Specifically, we consider the thresholds are uniformly distributed over the set \(\{0,1/n,2/n,\ldots ,n1/n\}\), as in Granovetter [14],^{2} and that w.l.o.g. we may consider that agent 1 has threshold 0, agent 2 has threshold \(1/n\), etc., and agent n has threshold \(n1/n\).^{3} We denote by \(\mu _\ell =\ell /n\) the threshold of agent \(\ell +1\).
The following proposition summarizes the uniform case with one anticonformist.
Proposition 1
Introducing only one anticonformist already considerably changes the dynamic. What is interesting is that the domino effect of the classical case is in a sense “stopped” by the anticonformist: conformist agents become active one by one starting from the agent with threshold 0, till the threshold of the anticonformist is reached, causing it to become inactive, which stops the cascade. Many other absorbing states exist, though, including \(N_c\).
Introducing two anticonformist agents Imagine now that two conformist agents, say \(k_1+1,k_2+1\) with thresholds \(\mu _a^1=k_1/n, \mu _a^2=k_2/n\) become anticonformist with the same thresholds. Assume w.l.o.g. that \(k_1<k_2\).
Suppose for example that \(n=10\) and \(k_1=3\) and \(k_2=5\), which makes agent 4 and agent 6 to be anticonformists (see Fig. 3). According to Theorem 2, the dynamic has no absorbing states since there is no fixed points of G. Instead, there will be a cycle \(S_1 \rightarrow S_2 \rightarrow S_1\) with \(S_1=\{1,2,3,5\}, \quad S_2=\{1,2,3,5,6\}.\) For example, starting from the state vector \((0,\ldots ,0)\), agent 1, 4 and 6 become active, which makes agent 2 and 3 to become active in addition, then agent 4 becomes inactive, which will again activate agent 5 at the next stage, i.e., the state \(S_1=\{1,2,3,5,6\}\) has reached with \(x=5/10\). Thus agent 6 becomes inactive at the next stage with the state \(S_2=\{1,2,3,5\}\) and the cycle \(S_1 \rightarrow S_2 \rightarrow S_1\) has been reached.
Proposition 2
The result with two anticonformists is very different from the case with one anticonformist: several absorbing states vs. a unique cycle. However, the cycle is of length 2 and involves two states which differ only by one agent. Therefore, it looks like an oscillation between these two states, which in addition look similar to the absorbing states which would have been obtained without the first anticonformist.
The general case The previous results tend to indicate that with an odd number of anticonformists, there are absorbing states, while there is no with an even number (only cycles occur). The main result of this section shows that this is indeed the case.
Theorem 3
 (i)
Suppose that there are \(2\ell +1\) (\(\ell \ge 1\)) anticonformist agents with thresholds \(\mu _a^1, \cdots , \mu _a^{2\ell +1}\), respectively, with \(\mu _a^i = k_i/n, (i=1,\ldots ,2\ell +1) \) and \(k_1< k_2< \cdots < k_{2\ell +1}\).^{5} Then G has fixed points \(k_{\ell +1}/n,\cdots ,(k_{\ell +2}1)/n\), whose corresponding absorbing states are given by (5).
 (ii)
Suppose there are \(2\ell \) (\(\ell \ge 1\)) anticonformist agents with thresholds \(\mu _a^1, \cdots , \mu _a^{2\ell }\), respectively, with \(\mu _a^i = k_i/n, (i=1,\ldots ,2\ell ) \) and \(k_1< k_2< \cdots < k_{2\ell }\).^{6} Then there is no absorbing state, but there exist cycles of length 2, corresponding to the pairs of points (x, G(x)), (y, G(y)) such that \(G(y)=x\) and \(y=G(x)\), i.e., x is a fixed point of \(G^{(2)}=G\circ G\). Moreover, there is no cycle of length greater than 2.
The results are qualitatively similar to those obtained with 1 or 2 anticonformists. Basically, no cascade to \(N_c\) or \(N_a\) is possible in general^{7}, and cycles, which are all of length 2 (the major part of the proof is devoted to show this fact), can be seen as oscillations between two states.
When the uniform distribution is on \(\{1/n,\ldots ,1\}\), it is easy to see that the results are the same as in Theorem 3, except that the cases of odd and even numbers of anticonformists are inverted: there are absorbing states when there is an even number of conformists, and no absorbing states but cycles otherwise. This is because in that case, the function G is shifted of \(1/n\) to the right, and \(G(0)=p\), the number of anticonformists.
We illustrate the above result in the case of cycles with the following example.
Example 2
Applying the results above, we get the case where the society is purely anticonformist.
Corollary 1
Consider a group of agents whose thresholds follow a uniform distribution. If all agents are anticonformists, then there exist at most one absorbing state. The existence of absorbing state is decided by the parity of n. If n is even, then there is no absorbing state; if n is odd, then the absorbing state is the action profile associated to the fixed point \((n1)/2\).
Note that this is in accordance with the general result on cycles (Theorem 1).
2.4.2 Gaussian Distribution
Proposition 3

If there are more conformists than anticonformists (\(q>1/2\)), there always exists a stable fixed point \(x_*\) (and possibly two other unstable fixed points) such that \(x_*\ge m\) if \(m\le 1/2\) and \(x_*\le m\) if \(m\ge 1/2\), being solution of (7). When \(m=1/2\), \(1/2\) is a fixed point and two other fixed points exist, also solutions of (7), provided the variance is not greater than \(\frac{1}{\sqrt{2}}\left( q\frac{1}{2}\right) \). The fixed point \(\frac{1}{2}\), when it is not unique, is unstable. No cycle can occur.

If there are more anticonformists than conformists (\(q<1/2\)), there is a unique fixed point \(x_*\) given by solving (7). It is stable if \(G'(x_*)\le 1\), otherwise there exists a limit cycle of length 2.

If there are exactly as many conformists as anticonformists, then there is convergence in one shot from any state S to the absorbing state \(S^*\)corresponding to the fixed point \(1/2\) by (5).
Interestingly, cycles (always of length 2) occur only when there are more anticonformists than conformists, and under the condition that the variance of the threshold distribution is small enough. Otherwise, there is always a stable absorbing state, and there is no cascade effect leading to \(N_c\) or \(N_a\).
The following examples illustrate the above results.
Example 3
\(q=0.9\); \(m=0.5\); \(\sigma =0.1, 0.2, \ldots , 0.9, 1\) (see Fig. 5). This is the case of a majority of conformists and \(m=1/2\). One can observe the fixed point \(1/2\) and the possible existence of 2 others. The limit value of \(\sigma \) for the tangent condition (8) is 0.283. Also, one can observe the asymptotic values q and \(1q\) for G(x).
Example 4
\(q=0.95\); \(m=0.3\); \(\sigma =0.1, 0.2, \ldots , 0.9, 1\) (see Fig. 6). This is the case of a majority of conformists and \(m<1/2\). There is fixed point greater than \(1/2\), whose value is negatively related to the variance, and two other possible ones smaller than \(1/2\).
Example 5
\(q=0.9\); \(m=0.8\); \(\sigma =0.1, 0.2, \ldots , 0.9, 1\) (see Fig. 7): majority of conformists and \(m>1/2\). There is fixed point smaller than \(1/2\), whose value is positively related to the value of the variance.
Example 6
\(q=0.1\); \(m=0.2\); \(\sigma =0.1, 0.2, \ldots , 0.9, 1\) (see Fig. 8). This is the case of a majority of anticonformists and \(m<1/2\). There is fixed point \(x_*\) smaller than \(1/2\). Its value is positively related to the value of the variance. For \(\sigma =0.1\) and \(\sigma =0.2\), the slope at \(x_*\) begin greater than 1, the fixed point is unstable and there is a limit cycle corresponding to the fixed points of \(G\circ G\). The plot of \(G\circ G\) is shown in Fig. 9. One can see that \(G\circ G\) has 1 or 3 fixed points, one of them being the fixed point of G. When there are 3 fixed points (for \(\sigma =0.1\), 0.15 and 0.2), the extreme ones give the coordinates for the limit cycle. For \(\sigma =0.1\) the coordinates are (0.1, 0.7731) and (0.7731, 0.1), while for \(\sigma =0.2\) they are (0.1116, 0.6366) and (0.6366, 0.1116). The cycles are materialized in Fig. 8.
2.4.3 General Distribution
Recall that if \(N_a \ne \emptyset \), the function G may not have fixed points because the presence of anticonformists makes it nonmonotonic. It is therefore difficult to get precise results in the general case. The next proposition elucidates the situation when there is only one type of anticonformist agent.
Proposition 4
 (i)If there is no agent i such that \(\mu _i=\mu _a\), there exist absorbing states if and only if the thresholds and corresponding fractions violate one of the following inequalities:^{8}$$\begin{aligned} \left\{ \begin{aligned}&\delta _a \ge \mu _1 \\&\delta _a + \displaystyle \sum _{i=1}^{i_0} q_i \ge \mu _{i_0+1}&(i_0=1,2,\ldots ,k1)\\&\delta _a + \displaystyle \sum _{i=1}^{k} q_i \ge \mu _a\\&\displaystyle \sum _{i=1}^{k} q_i< \mu _a\\&\displaystyle \sum _{i=1}^{i_0} q_i < \mu _{i_0}&(i_o=k+1,k+2,\ldots ,p) \\ \end{aligned} \right. \end{aligned}$$(9)
 (ii)Otherwise, there exist absorbing states if and only if the thresholds and corresponding fractions violate one of the following inequalities:^{9}$$\begin{aligned} \left\{ \begin{aligned}&\delta _a \ge \mu _1 \\&\delta _a + \displaystyle \sum _{i=1}^{i_0} q_i \ge \mu _{i_0+1}&(i_0=1,2,\ldots ,k1)\\&\displaystyle \sum _{i=1}^{i_0} q_i < \mu _{i_0}&(i_o=k,k+1,\ldots ,p) \\ \end{aligned} \right. \end{aligned}$$(10)
Distribution of agents’ thresholds with conformists and one type of anticonformists
Proportion  Threshold  Behavior characteristics 

\(q_1\)  \(\mu _1\)  Conformism 
\(\cdots \)  \(\cdots \)  \(\cdots \) 
\(q_k\)  \(\mu _k\)  Conformism 
\(\delta _a\)  \(\mu _a\)  Anticonformism 
\(q_{k+1}\)  \(\mu _{k+1}\)  Conformism 
\(\cdots \)  \(\cdots \)  \(\cdots \) 
\(q_{p}\)  \(\mu _p\)  Conformism 
The following example illustrates the case where there is no absorbing state.
Example 7
We consider \(n=10\), with \(N_c=\{1,2,3,4,5,6\}\) and 4 anticonformists. The parameters are \(\mu _a=\delta _a=4/10\), \(\mu _1=q_1=q_2=q_3=q_4=q_5=q_6=1/10\), \(\mu _2=2/10\), \(\mu _3=3/10\), \(\mu _4=5/10\), \(\mu _5=6/10\), \(\mu _6=7/10\) (see Fig. 11). There is no absorbing state but a cycle: \(\{1,2,3,4,5,6\} \rightarrow \{1,2,3,4,5\} \rightarrow \{1,2,3,4\} \rightarrow \{1,2,3\} \rightarrow \{1,2,3\}\cup N_a \rightarrow \{1,2,3,4,5,6\}\) with group opinion x: \(6/10 \rightarrow 5/10 \rightarrow 4/10 \rightarrow 3/10 \rightarrow 7/10 \rightarrow 6/10\).
The previous example has shown the existence of cycles. The next theorem establishes that there could be at most one cycle, whose length has an upper bound.
Theorem 4
Consider the same assumptions and notation as in Proposition 4. Then the process has either absorbing states or a unique cycle of length at most \(m+2\), where m is the number of values \(\mu _i\) in the interval \(\big ]\sum _{i=1}^k q_i,\sum _{i=1}^k q_i+\delta _a\big ]\). The upper bound of the length of the cycle, considering any possible values for the thresholds and fractions, is \(n_a+1\), where \(n_a=n\delta _a\) is the number of anticonformists.
Compared to the case of the uniform and the Gaussian distributions, we see that it is possible to obtain cycles of length greater than 2. However, as in the uniform case with one type of anticonformist, there could be several absorbing states, but only one cycle.
The next example illustrates the theorem and shows that the cycle can be shorter than \(m+2\) and that its length can be far below the upper bound \(n_a+1\). Note that in Example 7, this bound is attained.
Example 8
We consider \(n=100\), with the following parameters: \(\delta _a=0.4\) (40 anticonformists), \(\mu _a=0.5\), \(\mu _1=0.2\), \(q_1=0.1\), \(\mu _2=0.3\), \(q_2=0.15\), \(\mu _3=0.7\), \(q_3=0.3\), \(\mu _4=0.8\), \(q_4=0.05\). Then \(k=2\), and by the theorem, the cycle should be of length at most 3, while the upper bound \(n_a+1\) yields 41. One sees in Fig. 12 that the cycle (in green) has in fact length 2 and is formed of the two points (0.25, 0.5), (0.5, 0.25).
3 Random Sampling Models
The previous section considered a mechanism of diffusion with a complete and undirected network, where each agent was permanently in contact with all other agents. As this assumption may be unrealistic in some situations, we consider here a different mechanism where agents meet other agents at random (random neighborhood), with a certain size of the neighborhood to be either fixed or drawn from a distribution. When the size is fixed and identical for all agents, we speak of a homogeneous neighborhood. We make the following assumptions: (1) the agents in a given neighborhood are picked uniformly at random in N (2) There is no symmetry in the sense that if i selects j in its neighborhood, it does not necessarily imply that j has i in its neighborhood. We denote by P(k) the distribution of the degree of each agent, who are supposed to have the same distribution.
This can be interpreted in two ways. In the first one, at each time step, a random graph on N realizes, where directed links are picked at random so as to follow the specified degree distribution. In the second one, we consider the network to be the complete graph on N, meaning that any agent may potentially meet any other agent. Then, at each time step, for each agent a subset of agents is chosen randomly (i.e., links are drawn from the uniform distribution), so as to obey the specified degree distribution. We follow in the sequel the latter interpretation.
This model is a good approximation of many reallife situations, especially communication via online social networks like Twitter, Sina weibo, etc. Taking Twitter as an example, an individual receives at some time several tweets not only from his/her friends but also from strangers by checking the latest or hottest tweets. Moreover, he/she also can access information proactively on a certain topic by searching keywords or hashtags. At a different time, different users post some new tweets that attract this individual’s attention, which can be seen as a random sample (neighborhood) whose size obeys some distribution. Another example is that we meet different people everyday and obtain information either by communicating directly with each other or observing their behavior.
To avoid intricacies, it is convenient to consider that the random neighborhood of agent i, knowing that its degree is d, is taken as a random subset of N, of size d. This means that sometimes i is in its neighborhood, sometimes it is not. Still, the agents are considered to have a threshold, which can be drawn from a distribution or is fixed.
An important consequence of the model is that the process of updating of the opinion is no more deterministic, but still obeys a Markov chain. Its analysis is therefore much more complex, as not only absorbing states and cycles can exist but also aperiodic and periodic absorbing classes, where a class is a set of states such that a chain of transitions exists from any state to any other state in the set, and which is maximal for this property. It is easy to see that a state T different from \(\emptyset \) and N cannot be absorbing anymore: this is because the neighborhood being random and smaller than N, it is not guaranteed that it will contain T at each period. However, \(\emptyset \) and N can still be absorbing. The next lemma clarifies this point.
Lemma 1
\(\emptyset \) is absorbing (resp., N is absorbing) iff all anticonformists are constant 0players (resp., constant 1players), while there is no constant conformist player (i.e., \(0<\mu _i\le 1\) for all \(i\in N_c\)).
Proof
Suppose \(T=\emptyset \). By assumption, every conformist will take action 0 with certainty. Now, \(i\in N_a\) takes action 0 iff \(\mu _i=0\). Hence, any anticonformist must be a constant 0player.
The argument for \(T=N\) is much the same. \(\square \)
The existence of nontrivial absorbing classes will be shown in Sect. 3.1.1, where a complete analysis is done in a simple case (only two different thresholds, one for conformists and one for anticonformists). The complexity of the results shows that it seems out of reach to get a complete study in more general cases. Nevertheless, general results, although not exhaustive, can be obtained (see Sect. 3.1.2).
We start by focusing on the case of fixed degree (homogeneous neighborhood).
3.1 Homogeneous Neighborhood
We suppose in this section that the neighborhood of every agent has a fixed size d. A complete study of this case is possible when all conformist agents have the same threshold \(\mu _c\), and all the anticonformist agents have threshold \(\mu _a\). Then we give a result in the general case. We begin by some general considerations.
3.1.1 Case with Two Thresholds \(\mu _a,\mu _c\)
We assume here that there are two types of agents: anticonformist agents with threshold \(\mu _a\) and conformist agents with threshold \(\mu _c\), where \(0<\mu _a, \mu _c \le 1\).
Observe that \(p_i^1(s)\) depends only on whether i belongs to \(N_a\) or \(N_c\). Specifically, for a conformist agent i, \(p_i^1(S)= \mathbb {P}(\overline{a}\ge \mu _c\mid S)\) is a nondecreasing function of \(s=S\in \{0,1,\ldots ,n\}\) to [0, 1], depending only on \(\mu _c\), n and d. In addition, we have \(p_i^1(0)=0\) and \(p_i^1(n)=1\). Similarly, if i is anticonformist, \(p_i^1(s)\) is a nonincreasing function of s, starting at 1 with \(s=0\) and finishing at 0 with \(s=n\). Thus, we fall into the framework studied in Grabisch et al. [13] on an anonymous model of anticonformism where each agent i has the probability \(p_i(s)\) to take action 1 at next step knowing that the current state is S, with \(s=S\), and \(p_i(s)\) is a nondecreasing (respectively, nonincreasing) function reaching values 0 and 1 when i is conformist (respectively, anticonformist).
 (1)
\(N_a\) if and only if \(n_c \ge \max \{nl^c, nl^a\}\);
 (2)
\(N_c\) if and only if \(n_c \ge \max \{nr^c, nr^a\}\);
Cycles: sequence of states made of the infinite repetition of a pattern.
 (3)
\(N_a \xrightarrow {1} \emptyset \xrightarrow {1} N_a\) if and only if \(nl^c \le n_c \le r^a\);
 (4)
\(N_c \xrightarrow {1} N \xrightarrow {1} N_c\) if and only if \(nr^c \le n_c \le l^a\);
 (5)
\(N_a \xrightarrow {1} N_c \xrightarrow {1} N_a\) if and only if \(n_c \le \min \{l^c,l^a, r^c, r^a\}\);
 (6)
\(\emptyset \xrightarrow {1} N_a \xrightarrow {1} N_c \xrightarrow {1} \emptyset \) if and only if \(n_c \le \min \{r^c,r^a, l^c\}\) and \(n_c \ge nr^a\);
 (7)
\(N_a \xrightarrow {1} N \xrightarrow {1} N_c \xrightarrow {1} N_a\) if and only if \(n_c \le \min \{l^c, l^a, r^c\}\) and \(n_c \ge nl^a\);
Fuzzy cycles: the pattern contains states but also intervals of states. This means that there is no exact repetition of the same pattern, but at each repetition a state is picked at random in the interval.
 (8)
\(N_a \xrightarrow {1} [\emptyset ,N_c] \xrightarrow {1} N_a \) if and only if \(n_c \le \min \{l^c,l^a,r^a\}\) and \(r^c< n_c < nl^c\);
 (9)
\(N_c \xrightarrow {1} [N_a,N] \xrightarrow {1} N_c \) if and only if \(n_c \le \min \{r^c,r^a,l^a\}\) and \(l^c< n_c < nr^c\);
 (10)
\([\emptyset , N_c] \xrightarrow {1} [N_a,N] \xrightarrow {1} [\emptyset , N_c] \) if and only if \(\max \{r^c,l^c\} < n_c \le \min \{r^a, l^a, nl^c1, nr^c1\}\);
Fuzzy polarization: the polarization is defined by an interval, which means that at each time step, a state is picked at random in the interval, representing the set of active agents.
 (11)
\([\emptyset , N_a]\) if and only if \(\max \{nl^c, r^a+1\} \le n_c < nl^a\);
 (12)
\([N_c, N]\) if and only if \(\max \{nr^c, l^a+1\} \le n_c < nr^a\);
Chaotic polarization: similar to the previous case but more complex as several intervals are involved.
 (13)
\([\emptyset , N_a] \cup [\emptyset , N_c]\) if and only if \(l^c \ge nr^a\) and \(n_c \in (]r^c, nl^c[ \cap ]l^a, nr^c[) \cup ((]l^a, nr^a[ \cup ]l^c, nr^c[) \cap ]0,r^c[)\);
 (14)
\([N_a, N] \cup [N_c, N]\) if and only if \(l^a \ge nr^c\) and \(n_c \in (]l^c, nr^c[ \cap ]r^a, nl^c[) \cup ((]r^a, nl^a[ \cup ]r^c, nl^c[) \cap ]0,l^c[)\);
Chaos: at each time step a state is picked at random among all possible states.
 (15)
\(2^N\) otherwise.
Whenntends to infinity Assume that the number of agents tends to infinity. For simplicity, divide the previous parameters \(n^a, n^c, l^a, l^c, r^a, r^c, d\) by n, keeping the same notation so that these parameters now take value in [0, 1]. Thus \(l^c=d\mu _c\) and \(r^c=d(1\mu _c)\), and similarly for \(l^a,r^a\).
 Situation 1: \(l^a=l^c=:l\) and \( r^a=r^c=:r\). This implies \(\mu _c=\mu _a=:\mu \), i.e., all agents have the same threshold. Only the following four absorbing classes remain possible in this situation:The general tendency is that as the proportion of anticonformist agents increases, the society goes from consensus, to polarization or cascade, then to a chaos, finally to a cycle. A cascade effect (i.e., a convergence with probability 1 to \(N_c\) or with probability 1 to \(N_a\), whatever the initial state) is likely to occur, all the more \(l+r\) is close to 1 (i.e., the functions \(p_i^1\) are close to threshold function). When l is smaller than \(1/2\) and \(n^a\) is greater than l but sufficiently below \(1l\), it will lead to a cascade with all conformist agents saying yes. When l is greater than \(1/2\) and \(n_a\) between \(1l\) and l, it will lead to a cascade with all anticonformist agents saying yes.

\(N^a\) iff \(n^a \le l\)

\(N^c \) iff \(n^a \le dl=d(1\mu )\)

cycle \(N^a \xrightarrow {1} N^c \xrightarrow {1} N^a\) iff \(n^a \ge 1d\mu \) and \(n^a \ge 1d(1\mu )\)

\(2^N\) otherwise

 Situation 2: \(l^a=l^c= r^a=r^c=\frac{d}{2}\). This implies \(\mu _c=\mu _a=1/2\). The three possible absorbing classes in this situation are:The possible absorbing classes of “fuzzy cycle” and “fuzzy polarization” mentioned in Grabisch et al. [13] become impossible since there is a constraint \(l^a+r^a=l^c+r^c\) in this special context. Note that polarizations \(N^c\) and \(N^a\) always appear together, implying that there is no cascade effect.

\(N^a\), \(N^c\) iff \(n^a \le d/2\) (“polarization”)

cycle \(N^a \xrightarrow {1} N^c \xrightarrow {1} N^a\) iff \(n^a \ge 1d/2\) (“cycle”)

\(2^N\) otherwise (“chaos”)

 Situation 3: \(n^a\) tends to 0. Assume that \(n^a=\epsilon >0 \) arbitrarily small, therefore \(n^c = 1 \epsilon \). Among the initial 15 possible absorbing classes, only 7 of them remain possible:Again there is no cascade effect in this situation since two possible polarizations always appear together. When \(l_c,r_c<\epsilon \), which means that d is very small, only chaos (\(2^N\)) appears.

(1) \(N^a\) iff \(\min (l^a, l^c) \ge \epsilon \);

(2) \(N^c \) iff \(\min (r^c,r^a) \ge \epsilon \)’

(3) \(N^a \xrightarrow {1} \emptyset \xrightarrow {1} N^a\) iff \(l^c \ge \epsilon \) and \(r^a \ge 1 \epsilon \);

(4) \(N^c \xrightarrow {1} N \xrightarrow {1} N^c\) iff \(r^c \ge \epsilon \) and \(l^a \ge 1 \epsilon \);

(11) \([\emptyset , N^a]\) iff \(l^a < \epsilon \), \(l^c\ge \epsilon \) and \(r^a<1\epsilon \);

(12) \([N^c, N]\) iff \(r^a < \epsilon \), \(r^c \ge \epsilon \) and \(l^a < 1 \epsilon \);

(15) \(2^N\) otherwise.

3.1.2 General Case
With more than two thresholds, the complexity of the previous study indicates that it seems to be hopeless to get exact and complete results. This negative conclusion is tempered by our next result, established with an arbitrary distribution of thresholds. It shows that in most cases, only chaos can occur, i.e., the only absorbing class is \(2^N\).
Theorem 5
Suppose \(n_a\ge d\), \(n_c\ge d\) and suppose that there is no constant player (i.e., \(0<\mu _i\le 1\) for every player i). Then \(2^N\) is the only absorbing class, i.e., the transition matrix is irreducible.
For example, if the distribution of thresholds has support \(\{1/d,\ldots , 1\}\) for the conformists and the anticonformists. Then \(2^N\) is the only absorbing class. Indeed, the assumption implies that there are at least d members in \(N_a, N_c\).
This result is in accordance with those found in the previous section with two thresholds \(\mu _a,\mu _c\). Indeed, one can check that under the condition \(n_a,n_c\ge d\), none of the absorbing classes from (1) to (14) is possible. This is because we always have all four quantities \(l^c,l^a,r^c,r^a\) strictly smaller than d. Therefore, \(n_a\ge d\) implies that \(nn_a=n_c\ge nl^c\) and \(n_c\ge nr^c\) are impossible (and similarly with \(n_c\ge d\)).
3.2 Arbitrary Degree Distribution
We suppose now that the degree of the neighborhood is not fixed but follows a distribution P(d). We try to generalize the results of the homogeneous case.
3.2.1 Case with Two Thresholds \(\mu _a,\mu _c\)
The introduction of a distribution over the degree does not change the behavior of \(p^1_i(S)\): there are still nonincreasing or nondecreasing functions of s taking boundary values 0 and 1. The identification of the absorbing classes depends only on the width of the domain where these functions take values 0 and 1, hence their exact form is unimportant for this purpose.
An important consequence is the following: suppose that the distribution of d gives a positive probability to \(d=1\). Then we find \(l^c=r^c=l^a=r^a=0\). By inspection of the conditions of existence of the 15 absorbing classes, it follows that only the case of the chaos (\(2^N\)) remains possible. Note that this assumption is often satisfied (e.g., for the Poisson distribution, which arises when any pair of vertices is connected with a fixed probability).
3.2.2 General Case
We suppose now that each agent has a fixed threshold but possibly different among agents. A generalization of Theorem 5 is possible: under mild assumptions, only chaos can occur. Let us denote by \({\underline{d}},\overline{d}\) the lowest and greatest values of d with a positive probability, and by \({\underline{\mu }},{\overline{\mu }}\) the lowest and highest threshold values among the agents.
Theorem 6
Note that the conditions on \(n_a,n_c\) can be written equivalently as \(n_a\ge {\overline{\mu }}{\underline{d}}\) and \(n_a> {\underline{d}}(1{\underline{\mu }})\) (same for \(n_c\)). Again, observe that if \({\underline{d}}=1\), these conditions are always satisfied.
4 Concluding Remarks
We have performed in this paper a detailed study of convergence of the threshold model incorporating anticonformist agents. Two models were considered: a deterministic model supposing a complete graph, and a random neighborhood model, both corresponding to useful real situations. The first one represents a connected society where every agent is informed about the number of agents being in state 1 or 0 (active or inactive) at the present time, through media, etc. It is to be noted that no other information about the society is possessed by an agent, e.g., if there are anticonformists and how many. The second model represents a society communicating via social networks like Twitter, receiving randomly messages from other agents indicating their state. Here also, a given agent has no information on the type of his neighbor (conformist or anticonformist). We have given a gametheoretic foundation of the threshold mechanism with anticonformists, using coordination and anticoordination games, which permits to interpret the threshold value of an agent as a minimum or maximum probability that this agent assigns to agents in his/her neighborhood for choosing action 1.
For the deterministic model, we have found that, generally speaking, the presence of anticonformists causes the appearance of much more absorbing states, and cycles of length possibly greater than 2 (when only (anti)conformist agents are present cycles can only be of length 2). We have performed a complete and exact study when the distribution of threshold is uniform, generalizing the results of Granovetter [14]. We have also studied the case of a Gaussian distribution, where we showed the existence of unstable fixed points and limit cycles of length 2, and the case of an arbitrary distribution, where it is possible to find cycles of length greater than 2.
Based on a previous study, we have performed a complete and exact analysis of the random model when there are only two thresholds, one for the conformists, and another for the anticonformists. The introduction of randomness causes a variety of absorbing classes to appear: polarization, periodic classes of more or less complex structure, and chaos, i.e., any state of the society can be reached. When thresholds are randomly distributed, such an analysis is no more possible, however, we have shown that in most cases, only chaos occurs.

The presence of anticonformists introduces instability in the process, causing a multiplicity of absorbing states and a variety of cycles, periodic classes and chaos. Also, small variations in the parameters defining the society may induce important changes in the convergence: the model is highly sensitive, e.g., in the number of anticonformists, the threshold values, etc. For example, it has been seen in the case of a uniform threshold distribution that introducing or deleting only one anticonformist agent changes the convergence from a stable state to a cyclic behavior or vice versa.

In the case of a random neighborhood, the process converges to chaos (every state is possible) for most values of the parameters defining the society (it suffices that there are more conformists and more anticonformists than the size of a smallest neighborhood). Otherwise, cascades may occur: we have proved their existence in the case of fixed thresholds for conformists and anticonformists. This shows that introducing a small proportion of anticonformists in a society may lead, not only to chaotic situations, but also to permanent opinion reversal.
Footnotes
 1.
When adoption of action 1 stays for ever, one speaks of “switch”.
 2.
We may adopt another definition where the thresholds value from \(1/n\) to 1 (note that there is no constant player then). As we will see below, there is no fundamental change in the results, except for the case \(N_a=\emptyset \), where the domino effect would not start and \(\emptyset \) would be the only absorbing state.
 3.
We consider for ease of notation that only one agent has a given value of threshold. We may consider a more general situation where several agents have the same threshold. This will be considered in Sect. 2.4.3 with arbitrary distribution, however, in the case of a uniform distribution, this has no interest as uniformity obliges to have for each value of the threshold the same number of agents, so that everything goes exactly the same as the case of one agent per threshold value.
 4.
\(k_1,k_2 \in \{0,1,\ldots , n1\}\).
 5.
\(k_1, k_2, \ldots , k_{2l+1} \in \{0,1,\ldots , n1\}\).
 6.
\(k_1, k_2, \ldots , k_{2l} \in \{0,1,\ldots , n1\}\).
 7.
It is not difficult to check that \(N_c,N_a\) can be reached only if there is one anticonformist.
 8.Note that if \(\mu _a < \mu _1\), we can think it as \(k=0\) and delete all the terms related to nonpositive indices. Thus inequalities (9) become$$\begin{aligned} \left\{ \begin{aligned}&\delta _a \ge \mu _a \\&\displaystyle \sum _{i=1}^{i_0} q_i < \mu _{i_0}&(i_0=1,2,\ldots ,p) \\ \end{aligned} \right. \end{aligned}$$
 9.Note that if \(\mu _a = \mu _1\), we can think it as \(k=1\) and delete all the terms related to nonpositive indices. Thus inequalities (10) will be$$\begin{aligned} \left\{ \begin{aligned}&\delta _a \ge \mu _1 \\&\displaystyle \sum _{i=1}^{i_0} q_i < \mu _{i_0}&(i_0=1,2,\ldots ,p) \\ \end{aligned} \right. \end{aligned}$$
Notes
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