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Traditional empirical models of propagation consider individual contagion as an independent process, thus spreading in isolation manner. In this paper, we study how different contagions interact with each other as they spread through the network in order to propose an alternative dynamics model for information propagation. The proposed model is a novel combination of Lotka-Volterra cooperative model and competitive model. It is assumed that the interaction of one message on another is flexible instead of always negative. We prove that the impact of competition depends on the critical speed of the messages. By analyzing the differential equations, one or two stable equilibrium points can be found under certain conditions. Simulation results not only show the correctness of our theoretical analyses but also provide a more attractive conclusion. Different types of messages could coexist in the condition of high critical speed and intense competitive environment, or vice versa. The messages will benefit from the high critical speed when they are both competitive, and adopting a Tit-for-Tat strategy is necessary during the process of information propagation.

With the rapid development of information technology and the widespread use of intelligent communication tools, we progress further into the age of information explosion and have been bombarded with massive amounts of data. For instance, there are more than 368 million users in Sina Weibo and 100 million messages are distributed every day [

Many researches have been devoted to the propagation dynamics. The significant model for information or rumor spreading was introduced by Daley and Kendall many years ago [

As the continuous development of research on complex networks, a number of recent studies confirm that the network topology has a marked influence on the whole diffusion process. In the implementation of the MK model, Zanette [

Besides the epidemic dynamics models mentioned above, other widely used models describing information propagation are diffusion models in social networks. For instance, the threshold models developed by Granovetter [

Although extensive progress has been made in understanding the dynamics of the information spreading, most studies focused on one type of information or topic through a complex network in the context of dynamics of information spreading. Only a few of unsystematic studies concern the case that more than one type of information coexists and spreads among the population, such as two rumors with different probabilities of acceptance spreading among nodes [

Nevertheless, all of these studies explored different contagions competing with each other as they spread over the network but neglected the phenomena of the cooperation as they mutually help each other in spreading. In fact, there are multiple pieces of information about an event spreading through the OSN simultaneously. Moreover, these pieces of information do not spread in isolation manner from all other information currently diffusing in the network. The built-in mechanism underlying these complex interactions between different types of contagions makes a noticeable impact on dynamic spreading. For instance, competing contagions decrease each other’s probability of spreading, while cooperating contagions help each other in being adopted throughout the network [

As the main contribution of this paper, we propose a cooperative and competitive model for the different contagions of information diffusion in OSN. First of all, the proposed model is based on the population dynamics models to describe the development tendency and interaction between two different types of information. It provides a more realistic description of this process comparing with previous models by Lotka-Volterra and others. Secondly, we use differential equations of stability theory to analyze the proposed model. By means of the approximate analytical and exact numerical solutions of these equations, we examine the steady-state of information propagation in OSN. Finally, the analytic results with the real data collected from Sina Weibo indicate that the proposed model is effective for understanding and explaining some social phenomena presented in the process of online information spreading, especially the vital role of the cooperation and competition in the information propagation.

The rest of this paper is organized as follows. In Section

In this section, we describe the basic information spreading system and then establish a general co-competition model based on Lotka-Volterra model.

In OSNs websites, once a user publishes a message on Twitter or other microblog websites, the message will be transmitted to his/her followers. If the message is so interesting that some followers decide to repost it [

Considering the special time attributes of the information spreading in OSNs, we choose the user growth rate

Assume that there is a message published at time

We discuss the main features of (

(i) When

(ii) When

(iii) When the speed remains unchanged,

The spreading of abundance of information to which we are exposed through online social networks is a complex sociopsychological process. In the real world, these contagions not only propagate at the same time but also interplay with each other as they spread over the networks. This phenomenon is similar to the mutualism-competition interaction among multiple species. During the spreading of pieces of information, the cooperation happens when the different types of information are at low speed, and the competition happens when they are at high speed. Hence, according to [

We study the common case that two different types of information interact with each other and then make a natural extension of the classical two-species Lotka-Volterra competitive model and Lotka-Volterra cooperative model. We assume that the spread of each message is affected by the internal attractiveness itself and the external pressure among messages simultaneously. Considering two different types of information on the same event spreading around the same time in Twitter or other OSNs, they will compete with each other for the reason that users only choose to believe one of them. In the meantime, if all the messages are so important or interesting, users would be more likely to adopt and share them. It is considered that they mutually help each other in spreading through the network. We term the two types of information as message 1 and message 2, and they will interact with each other when they spread at the same time. The propagation process and the interactions between the two types of information can be governed by the following set of rules.

Each message has a relatively stable natural growth coefficient

Since a signal message could not spread indefinitely, there exists a maximum speed

Two messages will compete or cooperate with one another and take negative or positive effect, represented by

Supported by the rules (i) and (ii), we have the following model:

Here

According to the rule (iii), when the messages 1 and 2 coexist in the network, the interactive influence among the two messages produces positive effect or negative effect as they cooperate or compete with each other. Based on the analysis above, (

In order to facilitate the analysis, set

Equation (

The smoothed parts of the isoclines

As shown in (

Since the system (

Conversely, when the messages are at high speed, for instance,

By systematic comparison of the system (

In this section, we discuss the intersection of two isoclines

In order to show the existence of intersection of two isoclines

According to [

There are two different roots

There is double root

There is root

The parabolic zero growth isoclines of message 1 (a) and message 2 (b).

Based on the above analysis, we can conclude that trajectories in phase space always intersect isoclines either in the horizontal or in the vertical direction. In the next section, we make further analysis about the stability of intersection.

In order to study the final results of interactive effect between two different types of messages, we need to conduct a comprehensive analysis for the stability of the equilibrium points of simultaneous differential equations.

System (

Let

According to the Bendixson-Dulac theorem [

We study the evolution of

(i) For the equilibrium point

Since our aim is to show the coexistence of competition and cooperation by the introduction of

(ii) For the equilibrium point

Condition

(iii) For the equilibrium point

Condition

Assume that the conditions in (

While the line segments

Suppose that

Dynamics of system (

Three positive equilibria

Condition

Similarly, The Jacobian matrix of system (

The condition

The Jacobian matrix of system (

Two positive equilibria

Since

The unique positive equilibrium

Equilibrium points on the line segment

The unique positive equilibrium

To better understand the proposed model, we carry out computer simulations with real data from Sina Weibo, a Chinese microblog. Our data collected by a crowd sourcing Weibo visual analytic system [

We first consider the case that both messages have less competitive

(a)

The change trends of message 1 and message 2 can be observed from Figure

In the case of

(a)

The change trends of message 1 and message 2 in Figure

As shown in Figure

Positive or negative effects for different critical speed with

Generally, more competitive messages disseminate more rapidly than less competitive messages over network. Unfortunately, the situation is often not the case. It can be observed that under the conditions with the same

The majority of studies on information propagation have focused on a single contagion spreading through the social network without considering the influence of other contagions. In this paper, we have explored the information spreading mechanism impacted by cooperative and competitive effectiveness during the process of the propagation of two types of messages over networks. We developed a new OSN information propagation model based on Lotka-Volterra models to demonstrate the dynamics of a specific cooperation-competition system of these two messages. The stability of the system was proved by the differential equations and the phase trajectory of the stability theory. The results of stability analysis further indicated that the differential equations admit no periodic orbit, and the system can reach steady state in certain condition.

We proved that two types of messages usually cannot spread together peacefully, and only one type message survives till final state. Nevertheless, it produced stable points in the condition of high (or low) critical speed, which shows that it is possible for the coexistence of both messages in competitive propagation of different contagions. Using the real data collected from Sina Weibo, the results validated the theoretical analysis and presented another interesting conclusion that the messages will benefit from the high critical speed when they are both competitive. If there already is a competitive message in OSN, it is wise to adopt a Tit-for-Tat strategy.

As the great massive data are generated every moment in OSN, the rapid progress of information technology provides us with an easy way to collect data and observe the phenomenon that various types of information flow in the form of cascades on Twitter or other OSNs. Naturally, the proposed dynamic information propagation model should be tested and applied in real OSN with large topology network and vast numbers of the nodes and connections. These issues will be addressed in future research.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by the National Nature Science Foundation of China (no. 71271186), Education Ministry Fund of Humanities and Social Science (no. 12YJA630191), Natural Science Foundation of Hebei (no. G2013203237), and Brazilian National Council for Scientific and Technological Development-CNPq (no. 304903/2013-2).