Coevolution spreading dynamics on complex networks is a hot topic, which attracts much attention in network science. This paper proposes a mathematical model to describe the two competing complex information spreading dynamics on multiplex networks. An individual can only accept one of the two pieces of information. A heterogeneous mean-field theory is developed to describe the spreading dynamics. We reveal different regions through Monte Carlo simulations of the competing complex information spreading dynamics: no global information, one information dominant, and two information coexistence. We finally find that the heterogeneity of the multiplex networks’ degree distributions does not qualitatively affect the results.

Social networks, e.g., WeChat, Twitter, and Facebook, are the convenient way to express and share information with friends [

Based on the shared information’s content, we can classify the information spreading dynamics into simple and complex information [

The study about the simple information spreading on complex networks independently is the most widely investigated and revealed some critical results. For instance, simple information can always spread in the social network regardless of the values of information transmission probability when the heterogeneity of degree distribution is extremely strong [

To include the social reinforcement in the dynamics of complex information spreading, the linear threshold model is the most widely used, in which an individual accepts the information only when the received information is larger than a threshold [

This section proposes the competing information spreading model on social network to describe different opinions evolving in the campus.

Students usually discuss the events happened and diffuse the information or opinions through social platforms. We here use a multiplex network

We assume there are

In social networks, students always express different opinions on the same event, and as a result, additional information is competing with each other. In reality, adopting an idea is risky. Therefore, social reinforcement is included. Social reinforcement means that the adoption of the information requires multiple verifications from friends. We here adopt a generalized susceptible-informed-recovered model to describe the two information spreading dynamics. The susceptible node means that it does not know the information and can adopt the information. An informed node stands for that adopted the information and willing to transmit it to friends. A node in the recovered state means that it has lost interest in the information.

The two competing information spreading dynamics are denoted as

Input: network

Output: spreading sizes of information

Randomly a seed for information

Initialize

Node

Node

Adding node

Node

Adding node

Node

Adding node

Node

Adding node

Recovering node

Delete node

Adding node

Deleting node

Mathematically, the competing information spreading dynamics on social networks can be described by using the heterogeneous mean-field approach [

The network size is large enough.

There are no degree-degree correlations in the network.

The infection probability of informed nodes is independent, i.e., there are no dynamical correlations among nodes. Based on the above assumptions, we derive the evolution equations for competing information spreading dynamics.

To describe the evolution equations, we use the following mathematical notations

The fraction of

According to descriptions of the model, we know node

With a similar discussion, we know the evolutions of

In this section, we numerically study the competing information spreading on multiplex networks detailedly. We study the competing information spreading on two types of multiplex networks. The first one is ER-ER multiplex network, i.e., networks

Because the social reinforcement is included in the information spreading dynamics, we first study the effects of initial seed size in Figure

Competing information spreading on ER-ER multiplex networks: (a) The new informed information spreading size

Competing information spreading on ER-ER multiplex networks versus information transmission probability: (a) the new informed information spreading size

In Figure

One important question is the domination and coexistence of regions. Domination means that one information spreads to most nodes, while the other only transmits to a few nodes. Coexistence stands for the two types of information spread to a large fraction of nodes. To this end, we study

In Figure

Competing information spreading on ER-ER multiplex networks: (a) the new informed information spreading size

Finally, we study the two competing information spreading on SF-SF multiplex networks in Figure

Competing information spreading on SF-SF multiplex networks versus information transmission probability: (a) the new informed information spreading size

Coevolution spreading dynamics aims at describing interacting dynamics and revealing the phenomena induced by distinct interacting mechanisms. In this paper, we proposed a competing complex information model to describe two types of information diffusion on multiplex networks. A heterogeneous mean-field theory is developed to describe the model. Through extensive numerical simulations, we revealed the conditions of different regions. Specifically, no global information outbreak region is

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was partially supported by the Ministry of Education Science and Technology Development Center University Research Innovation Fund: “Qingtai Digital Intelligence Integration” Collaborative Innovation, no. 2020QT20.