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We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so-called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and possesses an up-down symmetry, which makes it similar to other binary opinion models with a single-flip dynamics, such as the majority-vote and the

Models of opinion dynamics are among the most studied models of complex systems [

In this paper, we will focus on a particular subclass of the binary opinion models with a single-flip dynamics, which means that one agent at most can change her/his state in a single update [

We consider

Each individual is described by the dynamical binary variable

Interactions between agents are local; i.e., they take place only if two agents are directly linked.

At each elementary update a single spin is randomly chosen and it can flip to the opposite direction with a probability that depends on the model’s details.

Conformity, i.e., an act of matching opinions, attitudes, beliefs, and/or behaviors to the certain group of influence, is the main type (often the only type) of the social response.

Agents are memoryless, which means that opinion

The differences between models depend mainly on the condition under which the conformity takes place. In some models all neighbors of a given target agent influence her/him (this applies to the majority-vote model [

One may ask what is the motivation to modify the Watts threshold model into the symmetric case? First of all, opinion dynamics concerns not only asymmetrical problems, as the diffusion of innovations, but many symmetric or almost symmetric issues, such as voting to one of two political parties and choosing one of two products on the duopoly market. Although the Watts threshold model was originally introduced to model the diffusion of innovation, the main idea of the threshold is also very natural in the broader context. It has been shown in many social experiments that simple majority (

Because all binary models, mentioned above, have been extensively investigated for years, many modifications and extensions of their original formulations have been proposed; a short review on modifications of the majority-vote model can be found in [

It is clear that conformity increases agreement (ferromagnetic order) in the system, whereas both types of nonconformity act against consensus. As a result of this competition, an order-disorder phase transition emerges. Interestingly, the type of the phase transition (continuous or discontinuous) may depend on the type of nonconformity. For example, it has been shown that within the

The question that naturally arises here is “which factor is responsible for the discontinuous phase transition within the

The paper is organized as follows. In the next section, we describe original versions of binary opinion models with single-flip dynamics. Then we present model’s extensions, which consist of introducing the noise into the models and we describe briefly the results that show how this noise impacts phase transitions. In the following subsection, we modify the original Watts threshold model to make it symmetrical and homogeneous, which makes it comparable to other binary opinion models with single-flip dynamics. Subsequently, we propose two versions of the symmetrical, homogeneous Watts threshold model, one with independence and the second one with anticonformity. Then, we analyze the model on the complete graph, which corresponds to the mean-field approach. We compare results obtained within Monte Carlo simulations results with those obtained via analytical treatment. Moreover, we provide a heuristic explanation of the obtained results. Finally, we discuss results in the context of other binary opinion models with single-flip dynamics.

The general framework of all binary opinion models with a single-flip dynamics has been described above so we will not repeat it here. Instead, we present updating rules that define the dynamics of models within this class. The most extensively studied among all is the linear voter model [

At a given time

Choose randomly

If all

Otherwise, i.e., in lack of unanimity, spin at site

Time is updated

In [

At a given time

Update the opinion

Model

With probability

With probability

Model

With probability

With probability

Time is updated

The generalized versions of the model that consists of a threshold [

The

It has been also shown that in the case of the threshold

Another model with a single-flip dynamics, which has been analyzed in the presence of a noise, is the majority-vote model [

At a given time

With probability

With complementary probability

Time is updated

As seen from the above description, within the original majority-vote model anticonformity takes place with probability

In computational sociology, a particularly popular class of models describing the spread of innovation/idea/behavior are threshold models, based on the idea introduced by Granovetter [

At a given time

An agent at site

Time is updated

There are two characteristic features that make the model different from other models described in the previous subsection. The first visible difference is heterogeneity. In other models it is introduced only by the heterogeneity of a graph; here, agents possess individual thresholds. Originally, each agent is assigned a threshold that is drawn at random from a given probability distribution function (PDF). Of course, as a special case, we can choose a one-point PDF, which takes the value equal to one at

Another difference between Watts threshold model and other models, presented above, is the lack of the up-down symmetry. An agent who ones adopted cannot go back to an unadopted state. However, we can easily modify the model to make it symmetrical, in the following way:

At a given time

An agent at site

if at least a threshold fraction

if at least a threshold fraction

an agent remains in its old state.

Time is updated

It should be noticed that within the above definition, the rules are defined unambiguously only for

Now we are ready to introduce two versions of the model with nonconformity, one with Independence (Model

The algorithm of a single update is as follows:

Model I (with Independence)

With probability

With probability

if at least a threshold fraction

if at least a threshold fraction

an agent remains in its old state.

Model A (with Anticonformity)

With probability

if at least a threshold fraction

if at least a threshold fraction

an agent remains in its old state.

With probability

In this work, analogously as in [

As an aggregated quantity, which fully describes the system in case of the complete graph, we choose an average concentration of agents with positive opinions:

As for all other binary opinion models with a single-flip dynamics, in an elementary time step the number of up spins

Using above probabilities, we obtain a recursive formula for the concentration of up spins:

Probabilities

Within the homogeneous symmetrical threshold model with anticonformity (Model A):

On the other hand,

For the large systems

However, usually, we are more interested in the stationary state than in the time evolution. Especially, the aim of this work is to understand the nature of phase transitions induced by the noise and thus we are interested in the dependence between the stationary value of the concentration of up spins

From (

Phase diagrams for the model with independence for different values of the threshold

Phase diagrams for the model with anticonformity for different values of the threshold

We have investigated the model within the mean-field approach, described in the previous section, as well as via the Monte Carlo simulations. We have conducted simulations for several system sizes varying from

It is seen that, generally, the dependence between

However, there is one crucial difference between the case

Average trajectories for the system of the size

Flow diagrams for the threshold model with

Probably, some of the readers wonder why we use such a formal approach, which eventually led to (

Before we proceed to the heuristic explanation of the observed phenomena,

We start with the

The modification of the Watts threshold model that we have proposed here may be treated as a destruction of the model from the social point of view. However, our aim was not to propose a model describing properly some social phenomena but to understand the nature of the phase transitions observed within models of binary opinions with a single-flip dynamics and up-down symmetry.

It should be noticed that the Model A, proposed here, can be treated as the generalization of the original majority-vote model, which corresponds to

However, results obtained here help not only to understand the difference between the

Threshold

Another possibility of obtaining a discontinuous phase transition has been recently suggested within the generalized threshold

We are aware of the fact that many people may wonder why to care about the type of the phase transition. Is there any reason, other than academic, to distinguish between continuous and discontinuous phase transitions? In the face of the social observations, and more recently also laboratory experiments, it seems that discontinuous phase transitions are particularly important, mainly because of the notion of the social hysteresis and the critical mass [

We realize that more interesting results could be obtained for different homogeneous and heterogeneous networks. Moreover, both noises could be introduced simultaneously as in [

No data were used to support this study (no empirical data were used; only analytical calculations and Monte Carlo simulations were conducted).

The authors declare that they have no conflicts of interest.

This work was supported by funds from the National Science Center (NCN, Poland) through grant no. 2016/21/B/HS6/01256.