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Rumors have been widely spread in online social networks and they become a major concern in modern society. This paper is devoted to the design of a cost-effective rumor-containing scheme in online social networks through an optimal control approach. First, a new individual-based rumor spreading model is proposed, and the model considers the influence of the external environment on rumor spreading for the first time. Second, the cost-effectiveness is recommended to balance the loss caused by rumors against the cost of a rumor-containing scheme. On this basis, we reduce the original problem to an optimal control model. Next, we prove that this model is solvable, and we present the optimality system for the model. Finally, we show that the resulting rumor-containing scheme is cost-effective through extensive computer experiments.

Nowadays, with the rapid development of Internet technology, online social network (OSN), ranging from Facebook and Twitter to YouTube and LinkedIn, has become a popular platform for people to communicate [

In order to protect the security of cyberspace, it is urgent to propose some effective measures to control the spread of rumors. In recent years, researchers have suggested many measures to mitigate the impact of rumors, such as blocking or isolating some OSN users to prevent them from further spreading rumors to OSN and releasing convincing rumor-containing messages to OSN users. These measures have been proven to be effective in controlling the spread of rumors [

From the above discussion, it is noticeable that the rumor-containing agency faces the following challenging problem:

Rumor-containing problem: supposing that a rumor is spreading over an OSN, design a cost-effective rumor-containing scheme.

Designing a cost-effective rumor-containing scheme is a valuable research problem. There are new rumors appearing in OSNs all the time. However, due to the fact that there is no one approach that can completely control the spread of rumors, the rumor-containing problem is essentially a management problem that requires continuous investment of resources. In practice, the budget of the rumor-containing agency is limited. If the economic costs incurred in the process of controlling rumors cannot be managed well, it will be difficult to continuously implement measures to control the spread of rumors.

In this paper, we propose a novel individual-based rumor spreading model, where the effect of the external environment on the spread of the rumor is accounted for. Thereby, we estimate the cost-effectiveness of a rumor-containing scheme. On this basis, we reduce the rumor-containing problem to an optimal control model, where each control stands for a rumor-containing scheme, and the objective functional stands for the cost-effectiveness of a rumor-containing scheme. We prove that this model admits an optimal control, guaranteeing the solvability of the model. We derive the optimality system for the model, which can be used to solve the model. Through extensive comparative experiments, we show that the rumor-containing scheme so obtained is cost-effective.

This paper makes a theoretical study on rumor-containing problem, and the research results can provide some theoretical guidance for taking measures to suppress the spread of rumors. In addition, the research method proposed in this paper can be applied to analyze the cost management of the rumor containment process, and the new model proposed in this paper can be used to study the influence of some parameters on the spread of rumors. Finally, new research ideas can also be extended to other cyberspace security problems, such as malware propagation [

The subsequent materials are organized in this fashion: Section

This section is devoted to reviewing the previous work that is related to the present paper. First, rumor-containing problem is introduced. Second, some rumor spreading models are discussed. Third, the optimal control approach used to deal with rumor-containing problem is introduced.

Rumor-containing problem is devoted to finding effective strategies to control or limit the propagation of rumor in a network, so that the losses caused by rumor can be reduced. In recent years, rumor-containing problem has received significant attention of researchers. Toward this direction, there exist two main types of rumor-containing strategies [

Modeling the spreading process of rumor lays a theoretical basis for studying rumor-containing problem. Existing rumor spreading models can be classified into three categories:

In recent years, optimal control theory has been applied to deal with rumor-containing problem. Optimal control theory is devoted to finding a control scheme for a dynamical system so that a certain optimality criterion is met [

In the present paper, we deal with the rumor-containing problem through optimal control approach but from a more practical perspective. First, we consider the influence of external environment on rumor spreading and propose a new rumor spreading model. Second, we regard the growth rate function of the rumor-containing cost as a rumor-containing scheme, and we define the cost-effectiveness of a rumor-containing scheme. Finally, we modeled the rumor-containing problem as the problem of finding the most cost-effective rumor-containing scheme, and we solve the problem by applying the optimal control theory.

This section is devoted to the modeling of the rumor-containing problem. Based on a novel individual-based rumor spreading model, we measure the cost-effectiveness of a rumor-containing scheme. On this basis, we reduce the rumor-containing problem to an optimal control model.

Consider an OSN of

Suppose a rumor is spreading over the network

For ease in realization, we assume that all feasible rumor-containing schemes are Lebesgue integrable [

Combined with sociological evidence and rational analysis, we can know that at any time

In practice, the first seven parameters can be estimated through online questionnaire survey, and the last two functions can be approximated through regression based on historical data. In particular, we introduce the parameter

The diagram of state transition of the user

Let

Since

The evolutionary process of the expected network state is described by the following system of ordinary differential equations:

Let

System (

For

Then, model (

Next, we show that model (

Let

Obviously,

Hence,

In what follows, let

For our modeling purpose, we need to estimate the total loss caused by rumor, and then we will define cost-effectiveness. To this end, let

The expected total loss of all network users in the time range

Let

The rumor-containing cost in the time range

Let

Based on Theorems

In this paper, we use the quantity

Based on the above discussions, we model the rumor-containing problem as the following optimal control problem:

Here,

We refer to the optimal control problem as the rumor-containing model. Each instance of the model is given by the 14-tuple as follows:

This section aims to deal with the rumor-containing model (

The following lemma is a direct corollary of a theorem in [

The rumor-containing model (

There exists

There exist

We are ready to show the solvability of the rumor-containing model.

The rumor-containing model (

First, let

Next, the convexity of

The Hamiltonian of the rumor-containing model (

We are ready to present a necessary condition for the optimal control of model (

Let

Moreover,

It follows from Pontryagin’s Minimum Principle [

System (

System (

Based on the previous discussions, we get that the optimality system for model (

At the end of the previous section, we proposed the notion of promising rumor-containing scheme. In this section, we assess the cost-effectiveness of this scheme through comparative experiments.

In each of the following experiments, we conduct the following operations: (1) generate an instance of the rumor-containing model (

Studies show that some social platforms such as Facebook [

Three real-world OSNs: (a)

The cost-effectiveness is the focus in this section. Common OSN platforms generally have information security agencies. When rumors break out on OSN, the agency is responsible for collecting and disseminating truths to dispel rumors, for example, when President Trump declared on Twitter that mail voting would lead to a “rigged election.” In order to control the spread of the rumor, Twitter tagged Trump’s tweets with the label “Getting the facts about mailing votes” and redirected users to a fact-checking page to provide comprehensive investigation information for the misleading article. In practice, the agency’s budget is limited. In order to control the spread of rumors, the agency needs to find a cost-effective rumor control scheme; that is, the scheme can minimize the total loss caused by rumors.

In all the following experiments, let

In particular, the static rumor-containing scheme

For the rumor-containing model

In practice, the closed-form formula for the functions

The effect of

Consider three instances of the rumor-containing model:

Furthermore, we compare the cost-effectiveness between the promising control strategy

Similarly, supposing that there is a rumor spreading on the network

The promising rumor-containing scheme

A comparison between the three promising controls

Consider three instances of the rumor-containing model:

Furthermore, we compare the cost-effectiveness between the promising control strategy

Again, supposing that there is a rumor spreading on the network

The promising rumor-containing scheme

A comparison between the three promising controls

Consider three instances of the rumor-containing model:

Furthermore, we compare the cost-effectiveness between the promising control strategy

Based on the results of Experiments

The promising rumor-containing scheme

A comparison between the three promising controls

In this study, we have studied the problem of developing a cost-effective rumor-containing scheme. Based on a node-level rumor spreading model that takes account of the effect of external environment, we have measured the impact of rumors. On this basis, we have modeled the rumor-containing problem as an optimal control problem. The optimization goal of the problem is to find a rumor-containing scheme that minimizes the total loss, and simulation results show that the proposed scheme is cost-effective. This work has studied the propagation of rumor from theoretical modeling and cost management perspectives. The research results can provide some theoretical guidance for taking measures to suppress the spread of rumors. The new model proposed in this paper can be used to study the influence of some parameters on the spread of rumors, and the research ideas can also be applied to study other cyberspace security problems.

The spread of rumors in the real world may be more complicated, and there are some open problems. First, since there is more than one OSN in the real world [

The data used to support the findings of this study are available, and the sources of the datasets have been given in the paper.

The authors declare that there are no conflicts of interest.

Da-Wen Huang contributed to formal analysis, software, and roles/writing of the original draft. Lu-Xing Yang contributed to formal analysis, investigation, and validation. Xiaofan Yang contributed to supervision, methodology, and writing, reviewing, and editing of the paper. Yuan Yan Tang contributed to supervision and writing, reviewing, and editing of the paper. Jichao Bi contributed to writing, reviewing, and editing of the paper.