# Competing Complex Information Spreading in Multiplex Social Network

Authors: Xiang Li; Bocheng Hou
Journal: Complexity
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Social networks, e.g., WeChat, Twitter, and Facebook, are the convenient way to express and share information with friends [1–5]. Students may retransmit an article to support or oppose the opinions of the authors. To describe the interaction of two types of information spreading dynamics on social networks, researchers used the framework of coevolution spreading dynamics on complex networks [6]. Generally speaking, scholars used complex networks to describe the social network, in which the nodes represent the individuals and edges stand for the relationships among individuals. Extensive empirical analyses revealed that social network exhibits degree heterogeneity, high clustering, multilayer, and temporality [7–18]. Including some of those characteristics, many research studies have been designed (see some important reviews in this field, e.g., reference [19–21]).Based on the shared information’s content, we can classify the information spreading dynamics into simple and complex information [22–24]. The simple information means that a single contact between the informed and susceptible individuals can trigger the infection transmission and widely used to describe epidemic spreading and simple news diffusion. In contrast, single contact cannot eliminate some complex information’s legality and reliability (e.g., political networks). Therefore, social reinforcement is included in the dynamics of complex information spreading. In the following content, we present the progress for simple and complex information spreading detailedly.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 [25–27]. Besides, the high clustering promotes the breakout of the information while suppresses the spreading size for large transmission probability [28]. Once several information are transmitted simultaneously, scholars revealed more novel phenomena. When two pieces of information are spreading on networks consecutively, the second information’s outbreak threshold is larger than the threshold of the first information [29]. Karrer and Newman [30] further considered two competing spreading dynamics on the single network simultaneously. By using the competing percolation theory, they found that two epidemic cannot coexist in the thermodynamic limit (i.e., when the network size is extremely large). Sahneh and Scoglio [31] investigated two competing information on multiplex networks and computed the absolute-dominance and coexistence regions. When two information collaborate, i.e., if the acceptance probability for another information is enlarged when an individual accepts one information, the system exhibits a discontinuous phase transition [32–34]. Scholars considered another situation, asymmetric interacting, i.e., one spreading dynamic promotes the other. In contrast, the other one suppresses the spreading and reveals that the global epidemic will be greatly suppressed once the information spreading is included [35–37].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 [22, 38–40]. Setting the final spreading size as the order parameter, researchers revealed the discontinuous phase transition in the system. More detailedly, the final spreading size versus the transmission probability exhibits a discontinuous growth. Liu et al. [41] investigated two complex information spreading consecutively on single complex network and revealed a transition for continuous and discontinuous phase transitions. Furthermore, Liu et al. [42] studied two complex synergetic information spreading dynamics on multiplex networks and revealed that the synergetic induced the discontinuous phase transition. To the best of our knowledge, there is still lacking a mathematical model to investigate two competing complex information spreading dynamics on multiplex networks. We will propose a mathematical model in section 2 and then develop a heterogeneous mean-field theory in section 3. We further study the complex competing information spreading model by using Monte Carlo simulation in section 4. Finally, we conclude in section 5.


## 2. Competing Complex Information Spreading Model

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


### 2.1. Network Description

Students usually discuss the events happened and diffuse the information or opinions through social platforms. We here use a multiplex networkG=NA,NB,EA,EB to describe the multiplex social network with two layers A and B, where NA and NB, respectively, represent the sizes of networks A and B and EA and EB, respectively, stand for the number of edges in of networks A and B. Nodes stand for the student users, and edges mean the relationships among those users. If two students i and j communicate with each other in network AB, an edge exists between them.We assume there aren=NA=NB students in the social network and eA=EA and eB=EB connections among them. Therefore, the average degrees of the social network are kA˜=2eA/n and kB˜=2eB/n, respectively. The two networks’ degree distribution is denoted as PAkA and PBkB, respectively. Nodes with large degrees mean that they have more communication with friends and always act as hubs. A node in two networks represents that it communicates with others through two platforms. To build the multiplex network, we first generate the intralayer edges using the uncorrelated configuration model proposed in reference [43] according the degree distributions PAkA and PBkB. Then, we build the interlayer edges randomly, that is to say, we randomly match two nodes in the two subnetworks.


### 2.2. Competing Spreading Dynamics

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 asℓA and ℓB. We assume information ℓA and ℓB, respectively, spread on networks A and B simultaneously. The competing between two dynamics means that a node cannot adopt two information simultaneously. The two competing spreading dynamics evolve as follows. Initially, we randomly select a node infected by ℓA and ℓB in two subnetworks. At every time step, every node informed by information ℓh (h∈A,B) will transmit the information to its neighboring nodes in network h with probability λ. To eliminate the information’s credibility, we include the social reinforcement and assume a node adopting information ℓh must receive T>1 pieces. A susceptible node i, respectively, obtains mA≥T and mB≥T pieces of ℓA and ℓB information at time step t, and it adopts information ℓA with probability mA/mA+mB and adopts information ℓB with probability mB/mA+mB. The informed nodes are recovered with probability γ. The two competing spreading dynamics evolve until there are no nodes in the informed state. The pseudocode of the competing spreading dynamics as shown in Algorithm 1. At each time step, the time complexity of the algorithm is OE, where E is the number of edges in the network. The space complexity is ON.Algorithm 1: Competing spreading dynamics.
(1)
Input: networkG, and dynamical parameters λ, T, γ, w;(2)
Output: spreading sizes of informationℓA and ℓB and denoted as RA and RB;(3)
t←0;(4)
Randomly a seed for informationℓA and ℓB, and put them into queue QA;(5)
whileQA is not empty do(6)
mA←0;(7)
mB←0;(8)
InitializeQB to be empty;(9)
δt←1/lengthQA;(10)
fori=1 to lengthQAdo(11)
if Node i adopt the information ℓAthen Node i transmits the information to every susceptible neighbor j in subnetwork A with probability λ;(12)
if Node j received the information ℓA successfully then(13)
mA←mA+1;(14)
end if(15)
end if(16)
if Node i adopt the information ℓBthen(17)
Nodei transmits the information to every susceptible neighbor in subnetwork B with probability λ;(18)
if Node j received the information ℓA successfully then(19)
mA←mA+1;(20)
end if(21)
end if(22)
ifmA≥T and mB≥Tthen(23)
Nodej adopts information ℓA with probability mA/mA+mB;(24)
if Node j adopts information ℓAthen(25)
Adding nodej into queue QB;(26)
end if(27)
Nodej adopts information ℓB with probability mB/mA+mB;(28)
if Node j adopts information ℓBthen(29)
Adding nodej into queue QB;(30)
end if(31)
end if(32)
ifmA≥T and mB<Tthen(33)
Nodej adopts information ℓA with probability 1;(34)
Adding nodej into queue QB;(35)
end if(36)
ifmA<T and mB≥Tthen(37)
Nodej adopts information ℓB with probability 1;(38)
Adding nodej into queue QB;(39)
end if(40)
end for(41)
fori=1 to lengthQAdo(42)
Recovering nodei with probability γ;(43)
if Node i recovers then(44)
Delete nodei from queue QA;(45)
end if(46)
end for(47)
fori=1 to lengthQBdo(48)
Adding nodei to queue QA;(49)
Deleting nodei from queue QB;(50)
end for(51)
t←t+δt(52)
end while


## 2.1. Network Description

Students usually discuss the events happened and diffuse the information or opinions through social platforms. We here use a multiplex networkG=NA,NB,EA,EB to describe the multiplex social network with two layers A and B, where NA and NB, respectively, represent the sizes of networks A and B and EA and EB, respectively, stand for the number of edges in of networks A and B. Nodes stand for the student users, and edges mean the relationships among those users. If two students i and j communicate with each other in network AB, an edge exists between them.We assume there aren=NA=NB students in the social network and eA=EA and eB=EB connections among them. Therefore, the average degrees of the social network are kA˜=2eA/n and kB˜=2eB/n, respectively. The two networks’ degree distribution is denoted as PAkA and PBkB, respectively. Nodes with large degrees mean that they have more communication with friends and always act as hubs. A node in two networks represents that it communicates with others through two platforms. To build the multiplex network, we first generate the intralayer edges using the uncorrelated configuration model proposed in reference [43] according the degree distributions PAkA and PBkB. Then, we build the interlayer edges randomly, that is to say, we randomly match two nodes in the two subnetworks.


## 2.2. Competing Spreading Dynamics

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 asℓA and ℓB. We assume information ℓA and ℓB, respectively, spread on networks A and B simultaneously. The competing between two dynamics means that a node cannot adopt two information simultaneously. The two competing spreading dynamics evolve as follows. Initially, we randomly select a node infected by ℓA and ℓB in two subnetworks. At every time step, every node informed by information ℓh (h∈A,B) will transmit the information to its neighboring nodes in network h with probability λ. To eliminate the information’s credibility, we include the social reinforcement and assume a node adopting information ℓh must receive T>1 pieces. A susceptible node i, respectively, obtains mA≥T and mB≥T pieces of ℓA and ℓB information at time step t, and it adopts information ℓA with probability mA/mA+mB and adopts information ℓB with probability mB/mA+mB. The informed nodes are recovered with probability γ. The two competing spreading dynamics evolve until there are no nodes in the informed state. The pseudocode of the competing spreading dynamics as shown in Algorithm 1. At each time step, the time complexity of the algorithm is OE, where E is the number of edges in the network. The space complexity is ON.Algorithm 1: Competing spreading dynamics.
(1)
Input: networkG, and dynamical parameters λ, T, γ, w;(2)
Output: spreading sizes of informationℓA and ℓB and denoted as RA and RB;(3)
t←0;(4)
Randomly a seed for informationℓA and ℓB, and put them into queue QA;(5)
whileQA is not empty do(6)
mA←0;(7)
mB←0;(8)
InitializeQB to be empty;(9)
δt←1/lengthQA;(10)
fori=1 to lengthQAdo(11)
if Node i adopt the information ℓAthen Node i transmits the information to every susceptible neighbor j in subnetwork A with probability λ;(12)
if Node j received the information ℓA successfully then(13)
mA←mA+1;(14)
end if(15)
end if(16)
if Node i adopt the information ℓBthen(17)
Nodei transmits the information to every susceptible neighbor in subnetwork B with probability λ;(18)
if Node j received the information ℓA successfully then(19)
mA←mA+1;(20)
end if(21)
end if(22)
ifmA≥T and mB≥Tthen(23)
Nodej adopts information ℓA with probability mA/mA+mB;(24)
if Node j adopts information ℓAthen(25)
Adding nodej into queue QB;(26)
end if(27)
Nodej adopts information ℓB with probability mB/mA+mB;(28)
if Node j adopts information ℓBthen(29)
Adding nodej into queue QB;(30)
end if(31)
end if(32)
ifmA≥T and mB<Tthen(33)
Nodej adopts information ℓA with probability 1;(34)
Adding nodej into queue QB;(35)
end if(36)
ifmA<T and mB≥Tthen(37)
Nodej adopts information ℓB with probability 1;(38)
Adding nodej into queue QB;(39)
end if(40)
end for(41)
fori=1 to lengthQAdo(42)
Recovering nodei with probability γ;(43)
if Node i recovers then(44)
Delete nodei from queue QA;(45)
end if(46)
end for(47)
fori=1 to lengthQBdo(48)
Adding nodei to queue QA;(49)
Deleting nodei from queue QB;(50)
end for(51)
t←t+δt(52)
end while


## 3. Theoretical Analysis

Mathematically, the competing information spreading dynamics on social networks can be described by using the heterogeneous mean-field approach [25, 26, 44], which is widely used in investigating the dynamics on networks. There are some basic assumptions behind the heterogeneous mean-field approach as follows:(i)
The network size is large enough.(ii)
There are no degree-degree correlations in the network.(iii)
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 notationsskAℓAt, ρkAℓAt, and rkAℓAt to represent the probability that a node with degree kA in subnetwork A is in the susceptible, informed, and recovered state for information ℓA at time t. Similarly, we use skBℓBt, ρkBℓBt, and rkBℓBt to represent the probability that a node with degree kB in subnetwork B is in the susceptible, informed, and recovered state for information ℓB at time t. Since the symmetry of the two spreading dynamics, we only describe the evolution equations for one of them. At time step t, the fraction of nodes in susceptible, informed, and recovered for information ℓhh∈A,B is Sℓht=∑Phkhskhℓht, ρℓht=∑Phkhρkhℓht, and Rℓht=∑Phkhrkℓht, respectively. In the following, we derive the evolutions of skhℓht, ρkhℓht, and rkhℓht.The fraction ofskhℓht decreases once node i with degree kkh in subnetwork h becomes informed. On the one hand, node i must receive mℓh≥T pieces of information ℓh in subnetwork h at time t. In uncorrelated network, the probability of node i connecting to an informed node j with degree kh′ is kh′Phkh/kh. Therefore, the probability of node j transmits the information to node i successfully which is λkh′Phkh/khρkhℓht. The pieces of information ℓh that node i received at time t are(1)mℓh=khλ1kh∑kh′kh′Phkhρkhℓht.According to descriptions of the model, we know nodei adopts information ℓh with probability(2)ξℓh=mℓhδmℓh≥Tmℓhδmℓh≥T+mℓgδmℓg≥T,where δmℓh≥T=1 if mℓh≥T; otherwise, mℓh≥T=0. Note that ℓg is the other type of information. Based the above discussions, we know the evolution of skℓht as(3)dskhℓhtdt=−skhℓhtskhℓgtξℓh.With a similar discussion, we know the evolutions ofρkhℓht and rkhℓht as(4)dρkhℓhtdt=−γρkhℓht+skhℓhtskhℓgtξℓh,dρkhℓhtdt=γρkhℓht,respectively.


## 4. Results Analysis

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., networksA and B are Erdös–Rényi (ER), in which the degree distribution follows a Poisson distribution. That is to say, the degree distribution of subnetwork h∈A,B is Phkh=khkh/kh!e−kh, where kh is the average degree of subnetwork h. We assume there is no interlayer degree-degree correlations, and thus the joint degree distribution is PkA,kB=kAkA/kA!e−kAkBkB/kB!e−kB. For ER-ER multiplex networks, the degree distribution is homogeneous. In reality, many real-data revealed that the degree distribution is heterogeneous. We therefore use the SF-SF multiplex networks, in which the two subnetworks are both scale-free networks. The degree distribution of subnetwork h∈A,B is Phkh=1/εhkh−γh, where εh=∑khkh−γh. Similar with ER-ER multiplex networks, we assume there is no interlayer degree-degree correlations, and the joint degree distribution is PkA,kB=1/εAεBkA−γAkB−γB. In numerical simulations, we set the network size n=NA=NB=2×104 and kA=kB=10. The results presented in the paper are averaged over at least 1000 times.


### 4.1. ER-ER Multiplex Networks

Because the social reinforcement is included in the information spreading dynamics, we first study the effects of initial seed size in Figure1. To study the fraction of new informed nodes by each information, we investigate RA−ρ0 and RB−ρ0 versus λA for different values of λB and ρ0. For small values of λB, e.g., λB=0.2, the fraction of new informed nodes by the seeds depends on the fraction of seeds shown in Figure 1(a). When the seed is small (e.g., ρ0=0.01), any values of information transmission probability λA cannot trigger the global outbreak for information ℓA. Since the social reinforcement is included, a small fraction of seeds cannot further adopt the information. With the increase in ρ0, the global outbreak of ℓA becomes possible when λA is above the threshold point λAc. Note that λAc can be numerically determined by studying the variability χ of RA−ρ0, and at the threshold point, χ exhibits a peak. We do not show χ in this paper. Note that the global information of ℓB cannot outbreak when λB=0.2, as shown in Figure 1(b). For larger values of λB, such as λB=0.5 and 0.8, we find that a small fraction of seeds cannot trigger the global outbreak for the two types of information (see Figures 1(c)–1(f)). For larger values of ρ0=0.05 and 0.1, RA−ρ0 increases with λA, meanwhile RB−ρ0 decreases with λA. We note that two types of information may coexist, and the coexistence conditions are discussed detailedly in Figure 2.Figure 1
Competing information spreading on ER-ER multiplex networks: (a) The new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB as a function of λA with λB=0.2; (c)RA−ρ0; (d)RB−ρ0λA with λB=0.5; (e)RA−ρ0; (f)RB−ρ0λA with λB=0.8. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, and we set the recovery probability γ=0.5.
(a)(b)(c)(d)(e)(f)Figure 2
Competing information spreading on ER-ER multiplex networks versus information transmission probability: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB; (c)RA−RA as a function of λA,λB. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, and we set the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)(c)In Figure2, we investigate the fractions of new informed nodes by the two types of information triggered by the initial seeds on a plane λA,λB. As shown in Figure 2(a), the global information can not trigger small values λA. Previous studies indicated that the global information outbreak threshold of information ℓA is λA0=1/kA=0.1 when there is only information ℓA spreading on network A. However, when the two competing information spreads on multiplex networks simultaneously, the global information cannot break when λA slightly is more extensive than λA0. We know that the competing information suppresses the spreading dynamics, and the global information outbreak threshold λAc is much larger than λA0. With the two information spreading dynamics’ symmetry mechanisms, we find the similar phenomena of RB−ρ0 in Figure 2(b).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 studyRA−RB in Figure 2(c). For the case of RA−RB>0, we know information ℓA dominates. When RA<RB, the information ℓB dominates. When RA≈RB, the two types of information compete for the nodes. Combing Figures 2(a)–2(c), we speculate information ℓA dominates when λA>λA0 and λA≫λB. Similarly, ℓB dominates when λB>λB0=1/k and λB≫λA. The two types of information are competing with the nodes and can coexist when λA≈λB, λA>λA0, and λB>λB0.In Figure3, we further study the effects of the average degrees on the competing information spreading dynamics when λA=λB. When the two types of information spreading on the two networks with the same average degree, i.e., kA=kB=10, we find that RA−ρ0 and RB−ρ0 exhibit the similar behavior, that is to say, RA−ρ0 and RB−ρ0 increase monotonously with the increase in λA=λB. However, RA−ρ0 first increases with λA=λB and then decreases when kA>kB, e.g., kA=15 and kA=10. That is to say, there exists a peak at which RA−ρ0 reaches its maximum. Meanwhile, RB−ρ0 always increases with λA=λB regardless the values of kA and kB.Figure 3
Competing information spreading on ER-ER multiplex networks: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB function of λA=λB. We set the threshold T=2 and the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)


### 4.2. SF-SF Multiplex Networks

Finally, we study the two competing information spreading on SF-SF multiplex networks in Figure4 and investigate the effects of degree heterogeneity of the two networks on the spreading dynamics. Compared with the ER-ER networks, the SF-SF networks promote the spreading dynamics since the existence of some nodes with large degrees. In addition, we find that the degree heterogeneity does not qualitatively affect the phenomena presented in Figure 2. That is to say, the domination and coexistence regions are the same. The domination region of information ℓA is λA>λA0 and λA≫λB. Similarly, information ℓB domination region is λB>λB0=1/k and λB≫λA. The coexistence region is λA≈λB, λA>λA0, and λB>λB0.Figure 4
Competing information spreading on SF-SF multiplex networks versus information transmission probability: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB; (c)RA−RA as a function of λA,λB. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, the degree exponents are γA=γB=3.0, and we set the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)(c)


## 4.1. ER-ER Multiplex Networks

Because the social reinforcement is included in the information spreading dynamics, we first study the effects of initial seed size in Figure1. To study the fraction of new informed nodes by each information, we investigate RA−ρ0 and RB−ρ0 versus λA for different values of λB and ρ0. For small values of λB, e.g., λB=0.2, the fraction of new informed nodes by the seeds depends on the fraction of seeds shown in Figure 1(a). When the seed is small (e.g., ρ0=0.01), any values of information transmission probability λA cannot trigger the global outbreak for information ℓA. Since the social reinforcement is included, a small fraction of seeds cannot further adopt the information. With the increase in ρ0, the global outbreak of ℓA becomes possible when λA is above the threshold point λAc. Note that λAc can be numerically determined by studying the variability χ of RA−ρ0, and at the threshold point, χ exhibits a peak. We do not show χ in this paper. Note that the global information of ℓB cannot outbreak when λB=0.2, as shown in Figure 1(b). For larger values of λB, such as λB=0.5 and 0.8, we find that a small fraction of seeds cannot trigger the global outbreak for the two types of information (see Figures 1(c)–1(f)). For larger values of ρ0=0.05 and 0.1, RA−ρ0 increases with λA, meanwhile RB−ρ0 decreases with λA. We note that two types of information may coexist, and the coexistence conditions are discussed detailedly in Figure 2.Figure 1
Competing information spreading on ER-ER multiplex networks: (a) The new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB as a function of λA with λB=0.2; (c)RA−ρ0; (d)RB−ρ0λA with λB=0.5; (e)RA−ρ0; (f)RB−ρ0λA with λB=0.8. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, and we set the recovery probability γ=0.5.
(a)(b)(c)(d)(e)(f)Figure 2
Competing information spreading on ER-ER multiplex networks versus information transmission probability: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB; (c)RA−RA as a function of λA,λB. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, and we set the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)(c)In Figure2, we investigate the fractions of new informed nodes by the two types of information triggered by the initial seeds on a plane λA,λB. As shown in Figure 2(a), the global information can not trigger small values λA. Previous studies indicated that the global information outbreak threshold of information ℓA is λA0=1/kA=0.1 when there is only information ℓA spreading on network A. However, when the two competing information spreads on multiplex networks simultaneously, the global information cannot break when λA slightly is more extensive than λA0. We know that the competing information suppresses the spreading dynamics, and the global information outbreak threshold λAc is much larger than λA0. With the two information spreading dynamics’ symmetry mechanisms, we find the similar phenomena of RB−ρ0 in Figure 2(b).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 studyRA−RB in Figure 2(c). For the case of RA−RB>0, we know information ℓA dominates. When RA<RB, the information ℓB dominates. When RA≈RB, the two types of information compete for the nodes. Combing Figures 2(a)–2(c), we speculate information ℓA dominates when λA>λA0 and λA≫λB. Similarly, ℓB dominates when λB>λB0=1/k and λB≫λA. The two types of information are competing with the nodes and can coexist when λA≈λB, λA>λA0, and λB>λB0.In Figure3, we further study the effects of the average degrees on the competing information spreading dynamics when λA=λB. When the two types of information spreading on the two networks with the same average degree, i.e., kA=kB=10, we find that RA−ρ0 and RB−ρ0 exhibit the similar behavior, that is to say, RA−ρ0 and RB−ρ0 increase monotonously with the increase in λA=λB. However, RA−ρ0 first increases with λA=λB and then decreases when kA>kB, e.g., kA=15 and kA=10. That is to say, there exists a peak at which RA−ρ0 reaches its maximum. Meanwhile, RB−ρ0 always increases with λA=λB regardless the values of kA and kB.Figure 3
Competing information spreading on ER-ER multiplex networks: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB function of λA=λB. We set the threshold T=2 and the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)


## 4.2. SF-SF Multiplex Networks

Finally, we study the two competing information spreading on SF-SF multiplex networks in Figure4 and investigate the effects of degree heterogeneity of the two networks on the spreading dynamics. Compared with the ER-ER networks, the SF-SF networks promote the spreading dynamics since the existence of some nodes with large degrees. In addition, we find that the degree heterogeneity does not qualitatively affect the phenomena presented in Figure 2. That is to say, the domination and coexistence regions are the same. The domination region of information ℓA is λA>λA0 and λA≫λB. Similarly, information ℓB domination region is λB>λB0=1/k and λB≫λA. The coexistence region is λA≈λB, λA>λA0, and λB>λB0.Figure 4
Competing information spreading on SF-SF multiplex networks versus information transmission probability: (a) the new informed information spreading sizeRA−ρ0 of information ℓA; (b) new informed information spreading size RB−ρ0 of information ℓB; (c)RA−RA as a function of λA,λB. We set the threshold T=2, the average degrees of the two networks are kA=kB=10, the degree exponents are γA=γB=3.0, and we set the initial fraction seed of the two types of information as ρ0=0.1 and γ=0.5.
(a)(b)(c)


## 5. Discussion

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 isλA≤λA0 and λB≤λB0; information ℓA dominant is λA>λA0 and λA≫λB; information ℓB dominant is λB>λB0 and λB≫λA; and two information coexistence region is λA≈λB, λA>λA0, and λB>λB0. When the two competing complex information spread on a multiplex with heterogeneous degree distribution, the phenomena do not qualitatively be affected. The results presented in this paper may shed some light into further investigating the competing spreading, such as design of effective strategy to make one spreading always win regardless of the other spreading dynamics and network structures.



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*Source: 9923837-2021-05-11.xml*