Competing Complex Information Spreading in Multiplex Social Network

Coevolution spreading dynamics on complex networks is a hot topic, which attracts much attention in network science.(is 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.


Introduction
Social networks, e.g., WeChat, Twitter, and Facebook, are the convenient way to express and share information with friends [1][2][3][4][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][8][9][10][11][12][13][14][15][16][17][18]. Including some of those characteristics, many research studies have been designed (see some important reviews in this field, e.g., reference [19][20][21]).
Based on the shared information's content, we can classify the information spreading dynamics into simple and complex information [22][23][24]. e 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). erefore, 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.
e 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][26][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][33][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][36][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][39][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.

Competing Complex Information Spreading Model
is section proposes the competing information spreading model on social network to describe different opinions evolving in the campus.

Network Description.
Students usually discuss the events happened and diffuse the information or opinions through social platforms. We here use a multiplex network G � (N A , N B , E A , E B ) to describe the multiplex social network with two layers A and B, where N A and N B , respectively, represent the sizes of networks A and B and E A and E B , 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 A (B), an edge exists between them.
We assume there are n � |N A | � |N B | students in the social network and e A � |E A | and e B � |E B | connections among them. erefore, the average degrees of the social network are k A � (2e A /n) and k B � (2e B /n), respectively. e two networks' degree distribution is denoted as P A (k A ) and P B (k B ), 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 P A (k A ) and P B (k B ). en, we build the interlayer edges randomly, that is to say, we randomly match two nodes in the two subnetworks.

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. erefore, 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. e 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.
e two competing information spreading dynamics are denoted as ℓ A and ℓ B . We assume information ℓ A and ℓ B , respectively, spread on networks A and B simultaneously. e competing between two dynamics means that a node cannot adopt two information simultaneously. e two competing spreading dynamics evolve as follows. Initially, we randomly select a node infected by ℓ A and ℓ B in two subnetworks. At every time step, every node informed by information ℓ h (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 ℓ h must receive T > 1 pieces. A susceptible node i, respectively, obtains m A ≥ T and m B ≥ T pieces of ℓ A and ℓ B information at time step t, and it adopts information ℓ A with probability m A /(m A + m B ) and adopts information ℓ B with probability e informed nodes are recovered with probability c. e two competing spreading dynamics evolve until there are no nodes in the informed state. e pseudocode of the competing spreading dynamics as shown in Algorithm 1. At each time step, the time complexity of the algorithm is O(E), where E is the number of edges in the network. e space complexity is O(N).

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.
ere are some basic assumptions behind the heterogeneous mean-field approach as follows: (1) Input: network G, and dynamical parameters λ, T, c, w; (2) Output: spreading sizes of information ℓ A and ℓ B and denoted as R A and R B ; (3) t←0; (4) Randomly a seed for information ℓ A and ℓ B , and put them into queue Q A ; (11) if Node i adopt the information ℓ A then Node i transmits the information to every susceptible neighbor j in subnetwork A with probability λ; (12) if Node j received the information ℓ A successfully then (13) m A ←m A + 1; (14) end if (15) end if (16) if Node i adopt the information ℓ B then (17) Node i transmits the information to every susceptible neighbor in subnetwork B with probability λ; (18) if Node j received the information ℓ A successfully then (19) m A ←m A + 1; (20) end if (21) end if (22) if m A ≥ T and m B ≥ T then (23) Node j adopts information ℓ A with probability m A /(m A + m B ); (24) if Node j adopts information ℓ A then (25) Adding node j into queue Q B ; (26) end if (27) Node j adopts information ℓ B with probability m B /(m A + m B ); (28) if Node j adopts information ℓ B then (29) Adding node j into queue Q B ; (30) end if (31) end if (32) if m A ≥ T and m B < T then (33) Node j adopts information ℓ A with probability 1; (34) Adding node j into queue Q B ; (35) end if (36) if m A < T and m B ≥ T then (37) Node j adopts information ℓ B with probability 1; (38) Adding node j into queue Q B ; (39) end if (40) end for (41) for i � 1 to length(Q A ) do (42) Recovering node i with probability c; (43) if Node i recovers then (44) Delete node i from queue Q A ; To describe the evolution equations, we use the following mathematical notations s to represent the probability that a node with degree k A in subnetwork A is in the susceptible, informed, and recovered state for information ℓ A at time t. Similarly, we use s to represent the probability that a node with degree k B in subnetwork B is in the susceptible, informed, and recovered state for information ℓ B 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 respectively. In the following, we derive the evolutions of s On the one hand, node i must receive m ℓ h ≥ T pieces of information ℓ h in subnetwork h at time t. In uncorrelated network, the probability of node i connecting to an informed node j with degree k h ′ is k h ′ P h (k h )/〈k h 〉.
erefore, the probability of node j transmits the information to node i successfully which is λk h ′ P h (k h )/〈k h 〉ρ (1) According to descriptions of the model, we know node i adopts information ℓ h with probability where Note that ℓ g is the other type of information. Based the above discussions, we know the evolution of s With a similar discussion, we know the evolutions of ρ respectively.

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. e first one is ER-ER multiplex network, i.e., networks A and B are Erdös-Rényi (ER), in which the degree distribution follows a Poisson distribution. at is to say, the degree distribution of subnetwork where 〈k h 〉 is the average degree of subnetwork h. We assume there is no interlayer degree-degree correlations, and thus the joint degree dis- 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. e degree dis- h . Similar with ER-ER multiplex networks, we assume there is no interlayer degree-degree correlations, and the joint degree distribution is P( In numerical simulations, we set the network size n � N A � N B � 2 × 10 4 and 〈k A 〉 � 〈k B 〉 � 10.
e results presented in the paper are averaged over at least 1000 times.

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 Figure 1. To study the fraction of new informed nodes by each information, we investigate R A − ρ 0 and R B − ρ 0 versus λ A for different values of λ B and ρ 0 . For small values of λ B , e.g., λ B � 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., ρ 0 � 0.01), any values of information transmission probability λ A cannot trigger the global outbreak for information ℓ A . Since the social reinforcement is included, a small fraction of seeds cannot further adopt the information. With the increase in ρ 0 , the global outbreak of ℓ A becomes possible when λ A is above the threshold point λ c A . Note that λ c A can be numerically determined by studying the variability χ of R A − ρ 0 , and at the threshold point, χ exhibits a peak. We do not show χ in this paper. Note that the global information of ℓ B cannot outbreak when λ B � 0.2, as shown in Figure 1(b). For larger values of λ B , such as λ B � 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 ρ 0 � 0.05 and 0.1, R A − ρ 0 increases with λ A , meanwhile R B − ρ 0 decreases with λ A . We note that two types of information may coexist, and the coexistence conditions are discussed detailedly in Figure 2.
In Figure 2, we investigate the fractions of new informed nodes by the two types of information triggered by the initial seeds on a plane (λ A , λ B ). As shown in Figure 2  indicated that the global information outbreak threshold of information ℓ A is λ 0 A � 1/〈k A 〉 � 0.1 when there is only information ℓ A spreading on network A. However, when the two competing information spreads on multiplex networks simultaneously, the global information cannot break when λ A slightly is more extensive than λ 0 A . We know that the competing information suppresses the spreading dynamics, and the global information outbreak threshold λ c A is much larger than λ 0 A . With the two information spreading dynamics' symmetry mechanisms, we find the similar phenomena of R B − ρ 0 in Figure 2 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 R A − R B in Figure 2 In Figure 3, we further study the effects of the average degrees on the competing information spreading dynamics when λ A � λ B . When the two types of information spreading on the two networks with the same average degree, i.e., 〈k A 〉 � 〈k B 〉 � 10, we find that R A − ρ 0 and R B − ρ 0 exhibit the similar behavior, that is to say, R A − ρ 0 and R B − ρ 0 increase monotonously with the increase in λ A � λ B . However, R A − ρ 0 first increases with λ A � λ B and then decreases when 〈k A 〉 > 〈k B 〉, e.g., 〈k A 〉 � 15 and 〈k A 〉 � 10.
at is to say, there exists a peak at which R A − ρ 0 reaches its maximum. Meanwhile, R B − ρ 0 always increases with λ A � λ B regardless the values of 〈k A 〉 and 〈k B 〉.

SF-SF Multiplex
Networks. Finally, we study the two competing information spreading on SF-SF multiplex networks in Figure 4 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.
at is to say, the domination and coexistence regions are the same. e domination region of (c)R A − R A as a function of (λ A , λ B ). We set the threshold T � 2, the average degrees of the two networks are 〈k A 〉 � 〈k B 〉 � 10, and we set the initial fraction seed of the two types of information as ρ 0 � 0.1 and c � 0.5. 6 Complexity

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. rough extensive numerical simulations, we revealed the conditions of different regions. Specifically, no global information outbreak region is λ A ≤ λ 0 A and λ B ≤ λ 0 B ;  (c)R A − R A as a function of (λ A , λ B ). We set the threshold T � 2, the average degrees of the two networks are 〈k A 〉 � 〈k B 〉 � 10, the degree exponents are c A � c B � 3.0, and we set the initial fraction seed of the two types of information as ρ 0 � 0.1 and c � 0.5.
information ℓ A dominant is λ A > λ 0 A and λ A ≫ λ B ; information ℓ B dominant is λ B > λ 0 B and λ B ≫ λ A ; and two information coexistence region is λ A ≈ λ B , λ A > λ 0 A , and λ B > λ 0 B . When the two competing complex information spread on a multiplex with heterogeneous degree distribution, the phenomena do not qualitatively be affected. e 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.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.