Information Spreading on Weighted Multiplex Social Network

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Based on complex network theory [33,34], regarding a user as a node and a connective relation as an edge [35], researchers have mined various underlying factors to investigate the information spreading mechanism and unveil the fundamental laws. Wide-ranging factors have been investigated, such as the initial seed size [36], clustering coe cient [37], community structure [38], temporal network [39], role of synergy [40,41], heterogeneous adoption thresholds [42], and limited imitation [43][44][45]. Additionally, information spreading on a multiplex network [46][47][48] has aroused the interest of researchers who perform numerous investigations about communication channel alternations on multiplex networks [49], heterogeneous behavioral adoption on multiplex networks [16], opinion competition on multiplex networks [50], e ect of multiple social networks on user awareness [51], social support for suppressing epidemics on multiplex networks [52], the interplay of social influence in multiplex networks [53], replicator dynamics on multiplex networks [54], etc.
Moreover, in social networks, because of multiple confirmations of the credibility and legitimacy of information, information spreading exhibits the effect of social reinforcement [55,56]. Repeated transmission of the same information from one neighbor cannot enhance the credibility nor facilitate adoption. erefore, social reinforcement derived from the memory of nonredundant information has been investigated [57], which should be considered in information spreading. Apart from the abovementioned factors, edges represent the connective relations between users and are always modeled in a binary state [58]. In real social networks, different types of individuals, such as colleagues, schoolmates, family members, and strangers may have intimate or distant relations. Even for the same type of individuals, they may exhibit different extents of intimacy.
erefore, similar to epidemic spreading on a weighted network [59], the relations in a social network should be reasonably modeled as weighted edges [60,61]. Noticeably, compared with a one-layered complex network, research on information spreading in multiplex networks is more complicated and challenging. Apart from the intraheterogeneity of weights between edges in the same layer, interheterogeneity of weight distributions between two layers exists as well. Unfortunately, studies performed hitherto lack profound analysis regarding information spreading on weighted multiplex networks with social reinforcement derived from the memory of nonredundant information.
Herein, in terms of social reinforcement effect generated from the memory of nonredundant information and heterogeneous weight distributions in two subnetworks, we propose an information spreading model on a weighted multiplex network. To further understand the intrinsic mechanism of information spreading and quantitatively predict the effects of spreading in time evolution and final steady state, we develop an edge-weight-based compartmental theory. Extensive simulations based on our proposed numerical method suggest that theoretical predictions agree well with numerical simulations. Combining the theoretical predictions and numerical simulations, we discovered that reducing weight distribution heterogeneity does not alter the growth pattern of the final adoption size but can accelerate information spreading and promote the outbreak of information spreading. Furthermore, we discovered that a critical seed size determines the global information outbreak, above which reducing the weight distribution heterogeneity materially affects the facilitation of information spreading. Additionally, reducing the degree distribution heterogeneity can alter the growth pattern of the final adoption size R(∞). e remainder of the paper is organized as follows: In Section 2, we describe the information spreading model on a weighted multiplex network. In Section 3, we propose an edge-weight-based compartmental theory to analyze information spreading on a weighted multiplex network. In Section 4, the numerical simulation method and parameter settings are introduced. In Section 5, we compare the theoretical predictions with the numerical simulation results and unveil the effect mechanism of the weighted multiplex network structure on information spreading. In Section 6, conclusions are presented.

Model Descriptions
For investigating information spreading on a weighted multiplex social network, we constructed an information spreading model based on an N-node two-layer weighted multiplex network, in which layers A and B represent two different independent social networks (illustrated in Figure 1). Correlated nodes in two layers indicate that they are the same individual, and edges between nodes in each subnetwork represent interindividual social relations. To eliminate the unnecessary intradegree-degree correlation, the uncorrelated configuration model [62] was manipulated to construct the multiplex network following two independent degree distributions P A (k A i ) and P B (k B i ), in which k A i and k B i indicate the degrees of node i in layer A and B, respectively. e degree vector of node i is set as In this paper, we assume that there is no correlation between the degree and weight distributions and Selfloop and multiple edges are forbidden. Regarding social reinforcement, we assume that each individual has different adoption willingness in different social subnetworks. Consequently, node i, representing an arbitrary individual, has adoption thresholds T A and T B in layers A and B, respectively, and a smaller adoption threshold suggests more considerable information adoption willingness. Moreover, the intimate relations between individuals typically hold different strengths in the same social subnetwork and in two distinctive subnetworks. erefore, we further construct the weighted multiplex network with two dependent edgeweight distributions, g A (w) and g B (w). Subsequently, if an edge with weight w X ij exists between nodes i and j in the layer X ∈ A, B { }, individual i in the subnetwork X transmits the information to j with probability where β X is the unit transmission probability in the subnetwork X ∈ A, B { }. Given β X , enlarging w X ij can monotonically increase λ X (w X ij ), which coincides with the positive correlation between relationship weight and transmission preference.
In real social networks, individuals always trustfully adopt information after receiving a certain number of it from distinctive neighbors, owing to social reinforcement originating from the memory of nonredundant information transmission [57]. Accordingly, we applied the generalized susceptible-adopted-recovered (SAR) model to describe the state of an individual at each time step in the information spreading process. At the same time step, an individual can only remain in one state: S-state (susceptible), A-state (adopted), or R-state (recovered). e S-state implies that an 2 Complexity individual has not adopted the information.
e A-state indicates that an individual has adopted the information and can transmit the information to neighbors. e R-state means that an individual has lost interest in the information and stops diffusion.
Furthermore, the detailed information spreading process is as follows: Initially, randomly selecting a fraction ρ 0 of nodes as initial seeds (in the A-state). At each time step, each A-state node i attempts to transmit the information to its S-state neighbor j simultaneously through an edge of weight w A ij with probability λ A (w A ij ) in layer A and through an edge of weight w B ij with probability λ B (w B ij ) in layer B according to equation (1). Because of nonredundant information transmission, once an edge between nodes i and j in layer A or B successfully delivers the information from i to j, such an edge is not allowed for repeated information delivery. An adopted node, however, can perform many attempts until it turns into the R-state. If the S-state node j in the layer X ∈ A, B { } successfully receives one piece of information from the A-state neighbor node j, its accumulative number m X j of information pieces in the layer X ∈ A, B { } will increase by one, i.e., m X j ⟶ m X j + 1. en, node j will compare the accumulative number m X j with its adoption threshold T X . In reality, for confirming the legitimacy and credibility of the information, an individual can adopt the information only if he/she corroborates with the information in both subnetworks. erefore, the susceptible node j can enter the A-state when m A j ≥ T A and m B j ≥ T B co-occur in both layers A and B; otherwise, it remains in the S-state. Regarding social reinforcement, because the accumulative number of information pieces determines the information adoption, the information spreading process is characterized by the  Figure 1: Illustration of social information spreading with transformation of three states (susceptible, adopted, and recovered) on a weighted multiplex network. Two different independent social networks A and B, comprising degrees of intimate relations, represent two layers of the weighted multiplex network, in which different degrees of edge-widths denote different edge-weights.
non-Markovian effect. At the same time step, after information transmission, the A-state node j will lose interest in the information and enter the R-state with probability γ, i.e., it will stop diffusing information in both layers A and B. Once the node changes into the R-state, it will retreat from further information spreading. Finally, the disappearance of all A-state nodes signals the termination of the spreading dynamics.

Edge-Weight-Based Compartmental Theory
To include the social reinforcement effect, the mean-filed theory [63] and an advanced approach [64] can be used. Inspired by References [57,65,66], an edge-weight-based compartmental theory is conceived to analyze the spreading process with theoretically strong correlation among the three states of nodes in a multiplex the network featured by strong weight heterogeneity. e theory assumes that the information spreading is on large, sparse, uncorrelated, and local tree-like weighted multiplex networks.
In the edge-weight-based compartmental theory, a node in the cavity state [67] implies that it can only receive information from but cannot transmit information to its neighbors. We denote θ X (t)(X ∈ A, B { }) as the probability that the information has not been passed to the S-state neighbors alongside a randomly selected edge in layer X by time t. Furthermore, in a weight multiplex network, we denote θ X w (t) as the probability that a node has not been informed through a randomly selected edge with weight w in layer X by time t. Following weight distribution g A (w) in layer A and g B (w) in layer B, we can obtain An arbitrary S-state node i with degree vector respectively.
As mentioned in Section 2, an S-state node with degree vector k → � (k A , k B ) can adopt the information only if the cumulative numbers m A ≥ T A and m B ≥ T B occur simultaneously; inversely, the S-state node having received m A and m B pieces of information by time t will remain in the S-state with probability We define the probability that a susceptible node with any degree in the layer X ∈ A, B { } remaining in the S-state if it receives m X pieces of information by time t as Furthermore, the fraction of nodes in the S-state at time t can be expressed as If we wish to obtain S(t), we should solve θ X (t), which depends on θ X w (t) in a weighted network. erefore, we focus on θ X w (t) at first. According to [57], in layer X, the neighbor connected by an edge with weight w to a node in the cavity state can be in the S-state, A-state, or R-state; therefore, θ X w (t)(X ∈ A, B { }) comprises the following three parts: where ξ X S,w (t), ξ X A,w (t), and ξ X R,w (t) indicate the probabilities that a neighbor connected by an edge with weight w to a node in the cavity state remains in the S-, A-, and R-states, respectively, and has not transmitted information by time t.
If neighbor j with degree vector k j → � (k A j , k B j ) is linked by an edge of weight w to the node i in the cavity state in layer A(B), then neighbor j can only receive information from other k A j − 1(k B j − 1) edges in layer A(B). erefore, neighbor j receives n A pieces of information in layer A with probability ϕ n A (k A j − 1, t) and, with no constraint in layer B, receives n B pieces of information in layer B with probability ϕ n V (k B j , t). Subsequently, neighbor j remains in the S-state in layer A with probability Analogously, neighbor j is connected with node i by an edge with weight w in layer B and remains in the S-state with probability Accordingly, given the joint degree distribution p( k → ), in layer X, an edge with weight w connects to an S-state neighbor j with probability 4 Complexity where the expression k X j p(k j → )/〈k X j 〉 denotes the probability an edge connecting to a neighbor with degree k X j in layer X. Next, we proceed with the evolution of ξ X A,w (t) and If, through an edge with weight w, the information is successfully transmitted to the neighbor in layer X with probability λ X w , the change in θ X w (t) is calculated as If neighbor j in the A-state has not transmitted information to node i through an edge of weight w with probability (1 − λ X w ) in layer X but enters into the R-state with probability c at time t, the value of ξ X R,w (t) will increase, and thus the dynamical differential value of ξ X R,w (t) can be obtained by e initial conditions are θ X w (0) � 1 and ξ X R,w (0) � 0; therefore, combining equation (12) with (13), we can obtain After inserting equations (11) and (14) into equation (8), we can obtain In equation (15) When t ⟶ ∞, dθ X w (t)/dt � 0, and we have the steady- Inserting θ X w (∞) into equations (2) and (3), we thus obtain For a convenient expression, and A discontinuous growth pattern will occur [68] when equation (19) is tangent to equation (20) with θ A < 1 and θ B < 1. Moreover, critical conditions can be acquired from the following equation: From the theoretical analysis above, the general solutions for adoption thresholds T A and T B can be derived, but the theoretical analysis itself may not be easily comprehended.
erefore, an intuitive illustration exemplifies the general theory with a particular case, i.e., T A � T B � 1. In this case, equations (9) and (10) are rewritten as Owing to independence, we obtain P(k j → ) � P(k A j )P(k B j ) and substitute (23) and (24) into equation (17) as follows: Complexity where H 0 X (x) � k X P X (k X )x k X represents the generation function of degree distribution P X (k X ), X ∈ A, B { }, and represents the generation function of excess degree distribution P X (k X ). Additionally, the generation functions of degree distribution P( k → ) are expressed as follows: where k → � (k A , k B ). Subsequently, substituting equations (23) and (24) into equation (21), we can deduce the critical conditions.

Numerical Method
Based on Erdös-Rényi (ER) [69] and the scale-free (SF) [62] random network model, extensive numerical simulations on artificial two-layer networks were performed to verify the theoretical analysis above. Without loss of generality, we set the network size N � 10 4 and mean-degree 〈k〉 A � 〈k〉 B � 〈k〉 � 10. For convenience, the information transmission unit probability in the two layers is set as β A � β B � β. Additionally, the weight distribution of the network layer and mean weight 〈ω〉 A � 〈ω〉 B � 〈ω〉 � 8. At the steady state in which no node in the A-state exists, we compute the spreading range by measuring the fraction of nodes in the R-state R(∞). e simulation results are obtained by averaging over at least 10 4 independent dynamical realizations on 30 artificial networks.
To obtain the critical condition from the simulations, the relative variance χ [70] is applied in our numerical verification, using the following method: where 〈. . .〉 is the ensemble average, and 〈R(∞)〉 and 〈R(∞) 2 〉 are the first and secondary moments of R(∞), respectively. e peaks of the curve of relative variance χ versus the information transmission probability correspond to the critical points.

Results and Discussion
In reality, from the perspective of a degree distribution model, layers in some multiplex social networks exhibit the homogeneous creation mechanism, such as MSN versus ICQ, but layers in other multiplex social networks may exhibit heterogeneous creation mechanisms, such as ICQ versus Facebook. erefore, from the aspects of homogeneous and heterogeneous degree distributions of two layers, the effects of weight distribution heterogeneity on information spreading in multiplex social networks is discussed in this section in terms of the parameters in Section 4.

ER-ER Weighted Multiplex
Networks. First, we investigate information spreading on a two-layered weighted ER-ER network with Poisson degree distribution We first explore the time evolutions of node densities in three states (S-state, A-state, and R-state) under different weight distribution exponents α A ω and α B ω , as shown in Figure 2. Under the same information transmission unit probability β but different adoption thresholds and seed sizes, Figure 2 2(h)). e phenomena occurring during the time evolution suggest that increasing the weight distribution exponent (reducing weight distribution heterogeneity) can facilitate information spreading in a weighted multiplex network.
Next, given the adoption thresholds T A , T B , and seed size ρ 0 , we investigate the growth pattern of the final adoption size R(∞) versus information transmission unit probability β. In Figure 3(a), with T A � 1, T B � 1, and ρ 0 � 0.005, increasing the weight distribution exponent such as from α A ω � 3, α B ω � 2.5 to α A ω � 3, α B ω � 3.5 does not change the growth pattern of the final adoption size R(∞) versus β but reduces the critical information transmission unit probability β c , which can be numerically captured by the peaks of relative variance χ in Figure 3(e) with the same settings of Figure 3(a). Figure 3(b), with different T A � 1; T B � 2; ρ 0 � 0.03, exhibits the same pattern as Figure 3(a), with the peaks of relative variance χ in Figure 3(f ). Even enlarging the seed size (ρ 0 � 0.1) in Figure 3(c) with the same adoption threshold of Figure 3 with (e), (f ), and (g), respectively. Particularly, subgraphs (d) and (h) emphasize the time evolution of the R-state node density R(t) under different weight distribution exponents. In both panels, despite the different thresholds and seed sizes, at the same time step, the density of nodes in the R-state under . ese phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading. e basic parameters are N � 10 4 , 〈k〉 � 10, 〈ω〉 � 8, c � 1.0, and β � 0.3.   not alter the growth pattern of R(∞) versus β in any pair of adoption thresholds but can promote the outbreak of information spreading. Noticeably, the theoretical solutions (lines) coincide well with the numerical simulations (symbols).
As shown in Figure 3, the initial seed size affects the information spreading dynamics. We investigate the effect of seed size on information spreading on a weighted multiplex social network, as shown in Figure 4. In Figure 4(a) (T A � 1; T B � 1; α A w � 3, α B w � 2.5), increasing the seed size from ρ 0 � 0.005 to ρ 0 � 0.01 not only changes the growth pattern from discontinuous to continuous, but also decreases the critical probability to promote the outbreak of information spreading. Subsequently, if the weight distribution exponent is enlarged, such as Figure 4(d) and α A w � 3, α B w � 3.5 in Figure 4(g), increasing the seed size ρ 0 still advances information propagation. Comparing Figure 4(a), 4(d), and 4(g) vertically, altering the initial fraction can change the growth pattern of R(∞) versus β but cannot change the fact that increasing the weight distribution exponent (reducing weight distribution heterogeneity) retains the growth pattern while facilitating the outbreak of information spreading. Moreover, in a horizontal comparison, although enlarging the adoption thresholds, such as T A � 1, T B � 2 in Figures 4(b), 4(e), and 4(h), or T A � 1, T B � 3 in Figure 4(c), 4(f ), and 4(i) renders information adoption more difficult, it is clear that increasing ρ 0 still transforms the growth pattern of R(∞) from discontinuous to continuous phase transition and promotes the outbreak of information spreading, as in the case of T A � 1, T B � 1. Additionally, changes in adoption thresholds and seed size do not qualitatively alter phenomena.
For a more comprehensive study, we obtained the results of the final adoption size on the (β, ρ 0 ) parameter plane, as illustrated in Figure 5. Here, we set the subgraphs in column 1 with T A � 1, T B � 1; in column 2 with T A � 1, T B � 2; in column 3 with T A � 1, T B � 3; in row 1 with α A w � 3, α B w � 2.5; in row 2 with α A w � 3, α B w � 3; and in row 3 with α A w � 3, α B w � 3.5. In the following, we discuss the findings based on Figure 5 in detail. In Figure 5 , a critical ρ * 0 � 0.003 exists, below which at any information transmission unit probability β no outbreak of information spreading will occur and above which outbreak and global adoption will occur. In a vertical comparison, we retain the adoption thresholds in Figures 5(a), 5(d), and 5(h) and further increase the weight distribution exponents in Figure 5 e results indicate that the three subgraphs have the same critical seed size ρ * 0 � 0.003, below which increasing the weight distribution exponent (reducing the weight distribution heterogeneity) still cannot induce the outbreak of information spreading, but above which increasing the weight distribution exponent can accelerate the outbreak at a smaller information transmission unit probability β. Moreover, in a horizontal comparison, we retain the weight distribution exponent in Figures 5(a), 5(b), and 5(c) with α A w � 3, α B w � 2.5 and increase the adoption thresholds in Figure  5(b) (T A � 1, T B � 2) and Figure  5(c) (T A � 1, T B � 3). Under these circumstances, we discovered that the critical seed size ρ * 0 is enlarged from 0.003 to 0.06, which implies increasing the difficulty in information adoption. Analogously, other subgraphs exhibit similar findings. e phenomena in Figure 5 suggest that a critical seed size exists, above which outbreak of information spreading and global adoption will occur, and the weight distribution exponent (reducing the weight distribution heterogeneity) is essential for facilitating information spreading.

ER-SF Weighted Multiplex Networks.
For cases of heterogeneous degree distributions, we investigate information spreading on a two-layered weighted ER-SF network with and parameter v B denotes the degree exponent of layer B. Additionally, we set the minimum degree k min B � 4 and maximum degree k max . On the weighted ER-SF networks of distinct degree distribution exponents, the growth pattern is shown in Figure 6, with the common setting T A � 1, T B � 3 in all subgraphs. In Figure 6(a) with degree distribution exponent v B � 2.1, we discovered that enlarging the weight distribution exponent, such as from can increase the final adoption size at the same β and decrease the critical probability to promote the outbreak of information spreading. Moreover, we proceed to increase the degree distribution exponent (reduce the degree distribution heterogeneity) in Figure 6(b) (v B � 3) and Figure 6(c) (v B � 4). Consequently, from Figure 6(a) to 6(c), given α A w and α B w , the growth pattern changes from continuous to the discontinuous growth pattern. However, at any given v B , changing the weight distribution exponent will not alter the growth pattern of R(∞). e phenomena in Figure 6 suggest that changing the degree distribution heterogeneity will not qualitatively alter the fact that reducing weight distribution heterogeneity can promote information spreading.

Conclusions
With the development of the Internet and mobile devices, social networks have prevailed in people's lives. Typically, an individual does not only use one social network, but multiple social networks simultaneously. Moreover, the information does not only diffuse in one social network but also spreads simultaneously on several social networks, e.g., information spreads on Twitter and Facebook simultaneously. Because of different extents of intimacy among friends, colleagues, and family members, connections in social networks should be modeled with different weights. As such, a multiplex social network should be a multilayer weighted network. erefore, information spreading on a multiplex social network should be investigated on a multilayer weighted network. Unfortunately, the mechanism of information spreading on . Given other parameters, increasing the fraction of initial seeds will change the growth pattern of R(∞) from discontinuous to continuous growth. Moreover, vertically, increasing the weight distribution exponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability; horizontally, increasing the adoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability. e phenomena above convey that changing the seed size can alter the growth pattern, and decreasing the adoption threshold can facilitate information spreading. Most importantly, changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distribution heterogeneity can facilitate information spreading. e basic parameters are N � 10 4 , 〈k〉 � 10, 〈ω〉 � 8, and c � 1.0. a weighted multiplex network has not been investigated comprehensively.
Besides, social information spreading possesses the characteristics of social reinforcement derived from the memory of nonredundant information. Herein, in terms of social reinforcement and weight distribution, we first proposed a general information spreading model on a weighted multiplex network to depict the dynamics of information spreading on a two-layered weighted network. Subsequently, we conceived a generalized edge-weight-based compartmental theory to analyze the mechanism of information spreading on a weighted multiplex network with the effect of the memory of nonredundant information. Based on the numerical method, extensive simulations confirmed that the theoretical predictions coincided well with the simulation results.
Combining numerical simulations with theoretical analyses, we discovered that under any adoption threshold of two layers, reducing weight distribution heterogeneity did not alter the growth pattern of the final adoption size but could accelerate information spreading and promote the outbreak of information spreading. Furthermore, we discovered that a critical seed size existed, below which no outbreak of information spreading occurred and above which reducing weight distribution heterogeneity was meaningful to the facilitation of information spreading. Additionally, we discovered that changing degree , and (f ) from row 2 with T A � 1, T B � 2; and (g), (h), and (i) from row 3 with T A � 1, T B � 3. e subgraphs have weight distribution exponents of α A ω � 3, α B ω � 2.5 in column 1; α A ω � 3, α B ω � 3 in column 2; and α A ω � 3, α B ω � 3.5 in column 3. All subgraphs show that a critical seed size ρ * 0 exists, under which no global adoption of information exists and over which a global adoption becomes possible. Additionally, the changes in the subgraphs from the first column to the third confirm that increase the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading. e basic parameters are N � 10 4 , 〈k〉 � 10, 〈ω〉 � 8, ρ 0 � 0.001, and c � 1.0. distribution heterogeneity could transform the growth pattern but did not qualitatively affect the promotion of reducing weight distribution heterogeneity to information spreading. Above all, increasing the weight distribution exponent can decrease critical transmission probability, without the influence of the degree distribution exponent.
In this study, we unveiled the effect of weighted multiplex network structures on information spreading. is study can inspire further studies related to the design of optimized strategies to control information spreading on multiplex social networks. Additionally, more accurate theories to depict the social contagion on real two-layer coupled networks should be further explored.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.
, and (α A ω � 3, α B ω � 3.5), respectively. From subgraphs (a)-(c), increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity (increasing the weight distribution exponent) can facilitate the spread of social behavior.