1. Introduction
Rumors are unconfirmed statements that may cause adverse effects on our society’s stability [
1,
2]. With the rapid development of social networks, the spread of rumors has become faster, and its impact is more widespread. In order to reduce the negative impact of rumor propagation, the government has invested a lot of resources in refuting rumors to control the spread of rumors. For example, the recent spread of the COVID-19 has caused a lot of rumors about the epidemic, and some of them have seriously affected people’s life and social stability [
3,
4]. Fortunately, the government implements the rumor refutation strategies and promptly announces authoritative information to the public, which can greatly reduce the negative impact of rumors on society globally. Moreover, the spontaneous rumor refutation strategies of non-governmental organizations (NGOs) have a certain inhibitory effect on the spread of rumors locally. With the joint efforts of the government and some NGOs, the spread of rumors will be effectively controlled. Therefore, it is necessary for us to study the global and local influence of the rumor refutation mechanism to better guide public opinion.
As early as the 1960s, researchers began to study the mechanism of rumor propagation. Daley and Kendal comparatively studied the similarity between the spread of epidemics and rumors and proposed the DK model, which revealed the differences between the two spread mechanisms [
5]. The MK model proposed by Maki and Thompson suggested that rumors were spread through direct contact between spreaders and other individuals [
6]. With further research on complex network models, Newman et al. believed that the classic model could only describe the propagation process on small-scale social networks because they did not consider the characteristics of social network topology [
7]. In addition, scholars further studied the influence of the complex topology of social networks on the dynamics of rumor propagation [
8]. Zanette et al. first applied the theory of complex networks to the study of rumor propagation and proposed the rumor propagation model on static and dynamic small-world networks [
9]. Moreno et al. used the Monte Carlo simulation method to study the random MK model on scale-free networks [
10]. Based on the traditional rumor propagation model, many scholars used different methods to study the rumor propagation mechanism [
11], thus developing many new rumor propagation models, such as SIS (Susceptible-Infected-Susceptible) model [
12], SIR (Susceptible-Infected-Removed) model [
13], SEIR (Susceptible-Exposed-Infected-Removed) model [
14], SIHR (Susceptible-Infected-Hibernator-Removed) model [
15], ILSR (Ignorant-Lurker-Spreader-Removal) model [
16], IWSR (Ignorant-Wiseman-Spreader-Stifler) model [
17], and SCIR (Susceptible-Infective-Counterattack-Refractory) model [
18], etc.
Some scholars have found that the activity rate of individuals will also affect the process of rumor propagation. An individual’s activity rate represents the individual’s willingness to spread rumors. Individuals with a higher rate of activity are usually more active in spreading rumors. Liu et al. considered the influence of individual activity rates on the spread of rumors and obtained a new critical curve consisting of pairs of critical activity rate and infection rate [
19]. Huo et al. believed that the activity rate of the individuals in complex networks would affect the process of rumor propagation [
20].
In reality, the negative impact of the large-scale spread of rumors has attracted official attention. The government began to formulate some measures to refute the rumors, called the rumor refutation mechanism. Some scholars discovered that the rumor refutation mechanism could effectively control rumor propagation. Zhao et al. described the dynamic process of rumor propagation by accounting for the refutation mechanism in homogeneous social networks [
21]. Zhang et al. developed a dynamic eight-state ICSAR rumor propagation model considering the official rumor refutation mechanism [
22]. Zhang et al. introduced a new two-stage rumor propagation and refutation model with time effect for online social networks [
23]. Einwiller and Kamins considered the moderating effect of rumors-targeted identity on rumor impact and refutation effectiveness [
24]. Jia et al. studied a rumor propagation model with regime switching considering colored noise and white noise and calculated a threshold Rs that guarantees the existence of rumor demise and stationary distribution [
25].
In addition, the propagation of rumors is influenced not only by the global social environment but also the local environment. For instance, in terms of the global impact of the spread of rumors, Zhao et al. studied the SIHR rumor propagation model considering the forgetting and remembering mechanisms in a homogeneous network [
26]. Dong et al. established an SEIR rumor propagation model to describe online social networks with different total user numbers and user deactivation rates [
27]. Zhu studied the comprehensive influence of psychological factors, propagation delay, network topology, and control strategies on the spread of rumors on online social networks [
28]. Li et al. considered the influence of government punishment on SIS and SIR rumor propagation models [
29]. Li et al. proposed a model considering the symmetry and complexity of the social strengthening effect [
30]. In terms of the local influence of the spread of rumors, Wang et al. studied the influence of differences in network individual identification capabilities on rumor propagation [
31]. Xu et al. introduced an enhanced mechanism to describe the individual’s willingness to spread rumors [
32]. Cheng et al. consider the influence of psychological factors such as the weakening of personal interest and the cumulative effect of memory on rumor propagation [
33]. Hu et al. considered and discussed different attitudes of individuals to the spread of rumors and established a sensitive-hesitant-affected-resistance (SHAR) model [
34]. Chen et al. divided the population into two categories: radical and stable and proposed the SEIsIrR model [
35].
From the above literature research, it can be seen that the rumor refutation mechanism and individual activity rates have a certain impact on rumor propagation. In the previous literature, the impact of the rumor refutation mechanism was not analyzed into two aspects: global and local. Therefore, on the basis of the classic ISR rumor propagation model, this paper considers the global and local rumor refutation mechanism and individual activity rates, establishes the mean-field equation, deduces the rumor propagation model, calculates the threshold of rumor propagation, and conducts simulation analysis on a heterogeneous network. The arrangement of this paper is as follows.
Section 2 introduces the improved rumor propagation model in detail. In
Section 3, we use the mean-field equation to derive the corresponding equations of the model. In
Section 4, the numerical study is carried out to verify the theoretical analysis.
Section 5 presents the conclusions.
2. The Dynamic Rumor Propagation Model
In this paper, we considered two aspects that may affect the spread of rumors: the global and the local influences. On the one hand, in the process of rumor propagation, the government will launch corresponding refutation strategies based on the scale and severity of the spread of rumors. In other words, the government will determine the strength of rumor refutation according to the severity of the rumor influence to restrain the spread of rumors. Under the global influence of the rumor refutation mechanism, the public opinion atmosphere of the whole society began to change, thus changing people’s attitude towards the rumor and lowering the threshold of rumor propagation. On the other hand, NGOs and some people will also spontaneously refute rumors and have an impact on individuals around them. The rumor refutation mechanism of NGOs also affects people’s attitude towards rumors and reduces people’s willingness to spread rumors. In the heterogeneous networks, we regard individuals as nodes.
In the traditional rumor propagation ISR model, each node can be in one of the following three states: ignorant (, those who have never heard the rumor), spreader (, those who spread the rumor), stifler (, those who have heard the rumor but not to spread), and each link represent the contact between nodes. In this paper, we also used the ISR model to describe the dynamic process of rumor propagation. Moreover, we noticed that different individuals have different activity rates. In the process of rumor propagation, there were roughly two types of individuals: active and inactive. Individuals with different activity rates will react differently when faced with rumors. These two types of people have different attitudes towards rumor propagation. Active individuals are very proactive in the rumor propagation and willing to share information or gossip with people around them. This is to say, active individuals can interact or exchange information with all individuals. Inactive individuals have a negative attitude towards rumor propagation and will not take the initiative to contact other people and spread rumors, which means that two inactive individuals may not exchange information with each other. However, as their opinions change, inactive individuals may become active, and active individuals may become inactive. We assume that active individuals and inactive individuals can transform into each other with a certain probability and introduce the activity rate to express this difference. We set represents the probability of a node is in active state and is in inactive state. Therefore, all nodes will be in one of the following six states: active ignorant , inactive ignorant , active spreader , inactive spreader and active stifler , inactive stifler , in which the superscript denotes an active state and inactive state. Since stifler has no effect on the process of rumor propagation, we use instead of and .
We divided all nodes into three groups according to their degree and gave the mean-field analysis to study the spread dynamics of rumor propagation. The number of nodes in each group is denoted by . Then, we use , , to represent the expected number of nodes with degree at time in ignorant, spreader and stifler state, and , , as their density respectively. Apparently, , , and .
In our model, the global rumor refutation mechanism is dominated by the government and has an impact on all people in the whole society. Time delay is not considered in this model. We assume that once the rumors begin to spread, the government will take measures to refute rumors according to the situation. Moreover, the more people infected, the stronger the government’s efforts to refute rumors. After activating the rumor refutation mechanism, the social environment of rumor propagation has changed, which is called global influence, denoted by . The local rumor refutation mechanism is dominated by the NGOs. It affects a small range of people through the contact among individuals. Under the influence of the local rumor refutation mechanism, the individual’s willingness to spread rumors will be reduced accordingly. We call it local influence, which is represented by .
When the rumor spreads with probability
, affected by the global and local influence of rumor refutation mechanism, the spread probability when an active ignorant node
contacts the surrounding
spreader nodes at time
who has heard the rumor in the time interval
is
. Therefore, the probability of an active ignorant
becoming an active spreader when there are
spreaders around is
In the process of rumor propagation, active nodes play a relatively major role. Since active and inactive nodes can be transformed into each other with probability
and
. For inactive nodes, the probability of transformation from ignorant to spreaders will be affected by
. To simplify the calculation, we consider the influence of
as the influence of the whole network. Moreover, the probability of an inactive ignorant node
becoming an inactive spreader when there are
active spreaders around is
where
is global influence of the rumor refutation mechanism, which means that the government decides the strength of rumor refutation based on the proportion of infected nodes in the whole network. Moreover, we assume the global influence of the rumor refutation mechanism as a set of linear relationships, that is, the function of the government’s rumor refutation intensity and the density of the rumor spreaders. In addition, considering that inactive nodes will only be affected by active spreaders, we use
instead of
and
to represent the number of spreaders around the ignorant node
. Therefore,
, where
is the strength of the government to refute rumors and
is taken to be representative of the density of spreaders in whole network. Similarity,
is taken to be representative of the local influence of rumor refutation mechanism, where
is the coefficient of NGOs rumor refutation mechanism and
is the total number of its neighbors.
The contacts rules between the active and inactive individuals are as follows:
In the process of rumor propagation, affected by the global and local influence of rumor refutation mechanism, when an active spreader contacts an active or inactive ignorant ( or ), the active ignorant becomes an active spreader with probability or , respectively. Moreover, we have , where . To simplify the calculation, we let , thus . We use instead of and . The inactive ignorant becomes an inactive spreader with probability ;
For some reasons, the active and inactive individuals can transform into each other with a certain probability. The probability that the active individual becomes the inactive state is , otherwise . The active ignorant can be infected by active and inactive spreader, but the inactive ignorant only can be infected by active spreader;
When the spreader contacts another spreader or stifler, the spreader loses the motivation to spread the rumors and becomes a stifler with probability ;
Due to the forgetting mechanism, active or inactive spreaders change their states and become stiflers spontaneously with probability .
The reaction process can be schematically represented by
Spontaneously stop spreading rumors
According to the above description, we can obtain the rumor propagation process as shown in
Figure 1.
Based on the above assumptions, we establish an improved ISR model that considers the rumor refutation mechanism and activity rates. In
Section 3, we will derive the mean-field equation for the rumor propagation model and calculate the critical threshold.
3. Mean-Field Equations of Spreading Dynamics
In this section, we consider the mean-field equations of the above spreading dynamic model on the heterogeneous networks. According to the improved model proposed in
Section 2, let
be the probability that a node with degree
randomly selected edge points to a spreader, thus we have
where
is the degree distribution of the network,
is the average degree and
is the density of spreaders with degree
at time
.
Generally, active ignorant individuals can be infected by any infected individual, but inactive ignorant individuals can only be infected by active infected individuals. Next, we will divide the discussion into two cases.
The probability that a node with a degree
has
spreading neighbors is given by
For further derivation, we let
and
. Combined with Equation (1), the probability that an active ignorant keeps its state unchanged for arbitrary
at time
is
By employing L’Hôpital’s rule, we obtain
Similarity, the probability that an inactive ignorant keeps its state unchanged for arbitrary
at time
is
Based on above analysis, the probability of an ignorant node with degree
to stay in ignorant state at time
is
The discrete-time rumor propagation process with some simplification can be described as the following equation according to Equation (3)
Dividing both sides of Equation (4) by
and letting
, Equation (4) can be written as a continuous-time equation
Then, we calculate the value of
where
,
. Similarity, we can obtain
where
,
and
In order to further calculation, we set an intermediate variable
as follows:
In the following formula we use the following simplified symbols
Since
and
, according to Equation (5), we have
Due to
, by removing the high order term of
, we can obtain
Combining Equations (5) and (8), we can derive
In order not to lose generality, we assume the initial distribution of ignorant is
. In the limit
we have
,
, then we can then obtain
There is a zero solution
of Equation (9), and the necessary condition for the existence of non-zero solution of Equation (9) is
which means the spreading threshold of ISR dynamic is
But we must notice that
is not always make sense.
will be greater than 1 when
. This means that when the activity rate of a node is below a certain threshold, the propagation scale will be relatively small regardless of the spread rate. We can also draw the threshold of activity rate and the strength of the global rumor refutation
From Equation (11), it can be seen that when the strength of government refutation is small, that is, when , the rumors cannot be controlled. Only when the strength of rumor refutation is greater than , can it affect the threshold of rumor propagation.
Similarity the threshold of local influence is
From Equation (12), we can conclude that the rumor propagation threshold
is related to the individual activity rate
, the global influence of the rumor refutation mechanism
and the local influence of the rumor refutation mechanism
. Since
, we obtain the final scale of rumor as follows:
According to equation Equation (10), we can infer that the critical rumors propagation threshold is affected by the strength of rumor refutation
, the strength of spontaneous refutation
and the individual activity rate
. When
,
and
, that is, regardless of the global and local influence of rumor refutation and the nodes in the whole network are active, the spreading threshold is
, the model is a classical rumor propagation ISR model [
11]. In particular, when
and
, it means that the government and NGOs do not refute rumors. At this time, only the influence of activity rate on the spread of rumors is considered [
19].