# Information Spreading on Weighted Multiplex Social Network

Authors: Xuzhen Zhu; Jinming Ma; Xin Su; Hui Tian; Wei Wang; Shimin Cai
Journal: Complexity
 (2019)

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## Abstract

Information spreading on multiplex networks has been investigated widely. For multiplex networks, the relations of each layer possess different extents of intimacy, which can be described as weighted multiplex networks. Nevertheless, the effect of weighted multiplex network structures on information spreading has not been analyzed comprehensively. We herein propose an information spreading model on a weighted multiplex network. Then, we develop an edge-weight-based compartmental theory to describe the spreading dynamics. We discover that under any adoption threshold of two subnetworks, reducing weight distribution heterogeneity does not alter the growth pattern of the final adoption size versus information transmission probability while accelerating information spreading. For fixed weight distribution, the growth pattern changes with the heterogeneous of degree distribution. There is a critical initial seed size, below which no global information outbreak can occur. Extensive numerical simulations affirm that the theoretical predictions agree well with the numerical results.


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## Body

## 1. Introduction

The Internet [1, 2], web 2.0 [3], and intelligent mobile terminals [4] have completely transformed people’s daily lives; these infrastructures have assisted people in constructing virtual social networks [5–8], which map people’s connections in real life into virtual cyberspaces and extend people’s social scope significantly, not only with schoolmates, colleagues, and family members, but also with strangers in different countries. Previously, various types of social networks such as Facebook [9], Twitter [10], Weibo [11], and Wechat [12] have been used. Social networks are important in information transmission between users [13], such as commodity recommendation [14], news spreading [15, 16], advertisement diffusion [17, 18], health behavior adoption [19], discovery of new friends [20], and social innovation propagation [21]. In addition, understanding the mechanism of information spreading in social networks can facilitate in the design of antivirus strategies [22–24], control rumors [25], avoid economic risks [26, 27], restrain social unrest [28], etc. Because of important values in economic activities and engineering applications, social information spreading has attracted significant attention [28–32].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 coefficient [37], community structure [38], temporal network [39], role of synergy [40, 41], heterogeneous adoption thresholds [42], and limited imitation [43–45]. Additionally, information spreading on a multiplex network [46–48] 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], effect 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. Therefore, 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. Therefore, 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 sizeR∞.The remainder of the paper is organized as follows: In Section2, 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.


## 2. Model Descriptions

For investigating information spreading on a weighted multiplex social network, we constructed an information spreading model based on anN-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 PAkiA and PBkiB, in which kiA and kiB indicate the degrees of node i in layer A and B, respectively. The degree vector of node i is set as ki→=kiA,kiB. In this paper, we assume that there is no correlation between the degree and weight distributions and the joint degree distribution is Pki→=PAkiAPBkiB. Self-loop 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 TA and TB 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. Therefore, we further construct the weighted multiplex network with two dependent edge-weight distributions, gAw and gBw. Subsequently, if an edge with weight wijX 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(1)λXwijX=1−1−βXwijX,where βX is the unit transmission probability in the subnetwork X∈A,B. Given βX, enlarging wijX can monotonically increase λXwijX, which coincides with the positive correlation between relationship weight and transmission preference.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 networksA 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.
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(d)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). The S-state implies that an individual has not adopted the information. The A-state indicates that an individual has adopted the information and can transmit the information to neighbors. The 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 fractionρ0 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 wijA with probability λAwijA in layer A and through an edge of weight wijB with probability λBwijB 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 mjX of information pieces in the layer X∈A,B will increase by one, i.e., mjX⟶mjX+1. Then, node j will compare the accumulative number mjX with its adoption threshold TX. 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. Therefore, the susceptible node j can enter the A-state when mjA≥TA and mjB≥TB 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 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.


## 3. 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. The 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 θXtX∈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 θwXt 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 gAw in layer A and gBw in layer B, we can obtain(2)θAt=∑wgAwθwAt,(3)θBt=∑wgBwθwBt.An arbitrary S-state nodei with degree vector ki→=kiA,kiB receives mA and mB pieces of information by time t with the probabilities(4)ϕmAAkiA,t=kiAmAθAtkiA−mA1−θAtmAϕmBBkiB,t=kiBmBθBtkiB−mB1−θBtmBrespectively.As mentioned in Section2, an S-state node with degree vector k→=kA,kB can adopt the information only if the cumulative numbers mA≥TA and mB≥TB occur simultaneously; inversely, the S-state node having received mA and mB pieces of information by time t will remain in the S-state with probability(5)sk→,t=1−ρ01−1−∑mA=0TA−1ϕmAAkA,t×1−∑mB=0TB−1ϕmBBkB,t.We define the probability that a susceptible node with any degree in the layerX∈A,B remaining in the S-state if it receives mX pieces of information by time t as(6)ηXt=∑kXPXkX∑mX=0TX−1ϕmXXkX,t.Furthermore, the fraction of nodes in the S-state at timet can be expressed as(7)St=∑k→Pk→sk→,t=1−ρ0ηA+ηB−ηAηB.If we wish to obtainSt, we should solve θXt, which depends on θwXt in a weighted network. Therefore, we focus on θwXt 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, θwXtX∈A,B comprises the following three parts:(8)θwXt=ξS,wXt+ξA,wXt+ξR,wXt,where ξS,wXt, ξA,wXt, and ξR,wXt 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 neighborj with degree vector kj→=kjA,kjB is linked by an edge of weight w to the node i in the cavity state in layer AB, then neighbor j can only receive information from other kjA−1kjB−1 edges in layer AB. Therefore, neighbor j receives nA pieces of information in layer A with probability ϕnAkjA−1,t and, with no constraint in layer B, receives nB pieces of information in layer B with probability ϕnVkjB,t. Subsequently, neighbor j remains in the S-state in layer A with probability(9)ΘAkj→,t=1−1−∑nA=0TA−1ϕnAkjA−1,t×1−∑nB=0TB−1ϕnBkjB,t.Analogously, neighborj is connected with node i by an edge with weight w in layer B and remains in the S-state with probability(10)ΘBkj→,t=1−1−∑nA=0TA−1ϕnAkjA,t×1−∑nB=0TB−1ϕnBkjB−1,t.Accordingly, given the joint degree distributionpk→, in layer X, an edge with weight w connects to an S-state neighbor j with probability(11)ξS,wXt=1−ρ0∑kj→kjXpkj→ΘXkj→,tkjX,where the expression kjXpkj→/kjX denotes the probability an edge connecting to a neighbor with degree kjX in layer X.Next, we proceed with the evolution ofξA,wXt and ξR,wXt in the layer X∈A,B. If, through an edge with weight w, the information is successfully transmitted to the neighbor in layer X with probability λwX, the change in θwXt is calculated as(12)dθwXtdt=−λwXξA,wXt.If neighborj in the A-state has not transmitted information to node i through an edge of weight w with probability 1−λwX in layer X but enters into the R-state with probability γ at time t, the value of ξR,wXt will increase, and thus the dynamical differential value of ξR,wXt can be obtained by(13)dξR,wXtdt=γ1−λwXξA,wXt.The initial conditions areθwX0=1 and ξR,wX0=0; therefore, combining equation (12) with (13), we can obtain(14)ξR,wXt=γ1−λwX1−θwXtλwX.After inserting equations (11) and (14) into equation (8), we can obtain(15)ξA,wXt=θwXt−1−ρ0∑kj→kjXPkj→ΘXkj→,tkjX−γ1−λwX1−θwXtλwX.In equation (15), substituting ξA,wXt with 1/−λwXdθwXt/dt according to equation (8), we further deduce(16)dθwXtdt=λwX1−ρ0∑kj→kjXPkj→ΘXkjX,tkjX+γ1−λwX−γ+1−γλwXθwXt.Whent⟶∞, dθwXt/dt=0, and we have the steady-state value θwX∞ as(17)θwX∞=λwX1−ρ0∑kj→kjXPkj→ΘXkj→,∞+kjXγ1−λwXkjXγ+1−γλwX.InsertingθwX∞ into equations (2) and (3), we thus obtain(18)θX∞=∑wgXwθwX∞=fXθA∞,θB∞.For a convenient expression, we rewriteθA∞=fAθA∞,θA∞ as(19)θA=FAθB,and θB∞=fBθA∞,θB∞ as(20)θB=FBθA.A discontinuous growth pattern will occur [68] when equation (19) is tangent to equation (20) with θA<1 and θB<1. Moreover, critical conditions can be acquired from the following equation:(21)∂fAθA∞,θB∞∂θB∞∂fBθA∞,θB∞∂θA∞=1.From the theoretical analysis above, the general solutions for adoption thresholdsTA and TB can be derived, but the theoretical analysis itself may not be easily comprehended. Therefore, an intuitive illustration exemplifies the general theory with a particular case, i.e., TA=TB=1. In this case, equations (9) and (10) are rewritten as(22)ΘAkj→,t=θAtkjA−1+θBtkjB−θAtkjA−1θBtkjB,ΘBkj→,t=θAtkjA+θBtkjB−1−θAtkjAθBtkjB−1.Owing to independence, we obtainPkj→=PkjAPkjB and substitute (23) and (24) into equation (17) as follows:(23)θwA∞=λwA1−ρ0HA1θA∞+HB0θB∞−HAB1,0θA∞,θB∞γ+1−γλwA+γ1−λwAγ+1−γλwA,(24)θwB∞=λwB1−ρ0HA0θB∞+HB1θA∞−HAB0,1θA∞,θB∞γ+1−γλwB+γ1−λwBγ+1−γλwB,where HX0x=∑kXPXkXxkX represents the generation function of degree distribution PXkX,X∈A,B, and HX1x=∑kXkXPkXxkX−1/kX,X∈A,B represents the generation function of excess degree distribution PXkX. Additionally, the generation functions of degree distribution Pk→ are expressed as follows:(25)HAB1,0x,y=1HA0′1∂HAB0,0x,y∂x,HAB0,1x,y=1HB0′1∂HAB0,0x,y∂y,HAB0,0x,y=∑kA,kBPk→xkAykB,where k→=kA,kB. Subsequently, substituting equations (23) and (24) into equation (21), we can deduce the critical conditions.


## 4. 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=104 and mean-degree kA=kB=k=10. For convenience, the information transmission unit probability in the two layers is set as βA  =  βB  = β. Additionally, the weight distribution of the network layer X∈A,B follows gXω∼ω−αωX with ωXmax∼N1/αωX−1 and mean weight ωA=ωB=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∞. The simulation results are obtained by averaging over at least 104 independent dynamical realizations on 30 artificial networks.To obtain the critical condition from the simulations, therelative variance χ [70] is applied in our numerical verification, using the following method:(26)χ=NR∞2−R∞2R∞,where … is the ensemble average, and R∞ and R∞2 are the first and secondary moments of R∞, respectively. The peaks of the curve of relative variance χ versus the information transmission probability correspond to the critical points.


## 5. 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. Therefore, 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 Section4.


### 5.1. ER-ER Weighted Multiplex Networks

First, we investigate information spreading on a two-layered weighted ER-ER network with Poisson degree distributionpAkA=e−kAkAkA/kA! in layer A and pBkB=e−kBkBkB/kB! in layer B.We first explore the time evolutions of node densities in three states (S-state, A-state, and R-state) under different weight distribution exponentsαωA and αωB, as shown in Figure 2. Under the same information transmission unit probability β but different adoption thresholds and seed sizes, Figure 2 shows the plots of curves with TA=1;TB=2;ρ0=0.03 in the top panel and TA=1;TB=3;ρ0=0.1 in the bottom panel. Regardless of the top or the bottom panel, with time t, the size of St decreases gradually from 1−ρ0 to 0, At changes from ρ0 to 0 eventually, and Rt indicating the final adoption size increases gradually from 0 to 1 at the end. With the increase in the weight distribution exponent from αωA=3,αωB=2.5 in Figures 2(a) and 2(e), αωA=3,αωB=3 in Figures 2(b) and 2(f), and αωA=3,αωB=3.5 in Figures 2(c) and 2(g), the evolution time consumed decreases from 19 to 13 in the top panel and 11 to 9 in the bottom panel. Additionally, the final adoption size R∞ augments at the same time t (see the comparison in Figures 2(d) and 2(h)). The 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.Figure 2
Time evolutions of node densities in three states (S-state, A-state, and R-state) on the ER-ER weighted multiplex network, under different weight distribution exponentsαωA and αωB. To unveil the effect of the threshold and seed size, two groups of subgraphs with TA=1;TB=2;ρ0=0.03 and TA=1;TB=3;ρ0=0.1 are shown in the top and bottom panels, respectively. Under the same information transmission unit probability β, subgraphs (a), (b), and (c) from the top panel have similar parameters of αωA=3, αωB=2.5; αωA=3, αωB=3; and αωA=3, αωB=3.5 with (e), (f), and (g), respectively. Particularly, subgraphs (d) and (h) emphasize the time evolution of the R-state node density Rt 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 αωA=3,αωB=3.5 exceeds other densities under αωA=3,αωB=2.5 and αωA=3,αωB=3. These phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading. The basic parameters are N=104, k=10, ω=8, γ=1.0, and β=0.3.
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(h)Next, given the adoption thresholdsTA, TB, and seed size ρ0, we investigate the growth pattern of the final adoption size R∞ versus information transmission unit probability β. In Figure 3(a), with TA=1, TB=1, and ρ0=0.005, increasing the weight distribution exponent such as from αωA=3,αωB=2.5 to αωA=3,αωB=3.5 does not change the growth pattern of the final adoption size R∞ versus β but reduces the critical information transmission unit probability βc, 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 TA=1;TB=2;ρ0=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 (ρ0=0.1) in Figure 3(c) with the same adoption threshold of Figure 3(b), the final adoption size R∞ versus β increases in the same pattern regardless of the increase in weight distribution. Besides, keeping the seed size ρ0 as the same of Figure 3(c) (ρ0=0.1), Figure 3(d), with further increase of the adoption threshold TA=1,TB=3, also shows similar phenomena as those in Figures 3(a)–3(c), verified by the peaks of χ in Figure 3(h). The above phenomena suggest that increasing the weight distribution exponent (decreasing weight distribution heterogeneity) will 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).Figure 3
Growth pattern of the final adoption sizeR∞ versus information transmission unit probability β on the ER-ER weighted multiplex network, under different weight distribution exponents αωA and αωB, given adoption thresholds (TA, TB) and initial fraction ρ0 of seeds. The final adoption size R∞ versus information transmission probability β with (a) TA=1; TB=1; ρ0=0.005, (b) TA=1;TB=2;ρ0=0.03, (c) TA=1;TB=2;ρ0=0.1, and (d) TA=1;TB=3;ρ0=0.1. The lines denoting theoretical solutions agree exceptionally well with the symbols representing numerical simulations. Moreover, in subgraphs (e), (f), (g), and (h), the relative variance χ in terms of the peaks numerically indicates the critical spreading probabilities in (a), (b), (c), and (d), respectively, showing that increasing the weight distribution exponent (decreasing weight distribution heterogeneity) can decrease the critical spreading probability. The phenomena above suggest that decreasing weight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versus information transmission probability. The basic parameters are N=104, k=10, ω=8, and γ=1.0.
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(h)As shown in Figure3, 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) (TA=1;TB=1;αwA=3,αwB=2.5), increasing the seed size from ρ0=0.005 to ρ0=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 αwA=3,αwB=3 in Figure 4(d) and αwA=3,αwB=3.5 in Figure 4(g), increasing the seed size ρ0 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 TA=1,TB=2 in Figures 4(b), 4(e), and 4(h), or TA=1,TB=3 in Figure 4(c), 4(f), and 4(i) renders information adoption more difficult, it is clear that increasing ρ0 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 TA=1,TB=1. Additionally, changes in adoption thresholds and seed size do not qualitatively alter phenomena.Figure 4
Effect of fraction of initial seeds on the growth pattern of the final adoption sizeR∞ versus β on the ER-ER weighted multiplex network, under different weight distribution exponents αωA and αωB. The values of weight distribution exponent αωA of all subgraphs are 3, and the subgraphs in each row have the same αωB, i.e., subgraphs (a), (b), and (c) correspond to αωB=2.5; (d), (e), and (f) correspond to αωB=3; and (g), (h), and (i) correspond to αωB=3.5. Additionally, the subgraphs in each column have the same adoption threshold, i.e., subgraphs (a), (d), and (g) are set with TA=1,TB=1; (b), (e), and (h) with TA=1,TB=2; and (c), (f), and (i) with TA=1,TB=3. The theoretical solutions (lines) coincide well with the numerical results (symbols). 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. The 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. The basic parameters are N=104, k=10, ω=8, and γ=1.0.
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(i)For a more comprehensive study, we obtained the results of the final adoption size on theβ,ρ0 parameter plane, as illustrated in Figure 5. Here, we set the subgraphs in column 1 with TA=1,TB=1; in column 2 with TA=1,TB=2; in column 3 with TA=1,TB=3; in row 1 with αwA=3,αwB=2.5; in row 2 with αwA=3,αwB=3; and in row 3 with αwA=3,αwB=3.5. In the following, we discuss the findings based on Figure 5 in detail. In Figure 5(a) (TA=1;TB=1;αwA=3,αwB=2.5), a critical ρ0∗=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(d) (αwA=3,αwB=3) and Figure 5(h) (αwA=3,αwB=3.5). The results indicate that the three subgraphs have the same critical seed size ρ0∗=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 αwA=3,αwB=2.5 and increase the adoption thresholds in Figure 5(b) (TA=1,TB=2) and Figure 5(c) (TA=1,TB=3). Under these circumstances, we discovered that the critical seed size ρ0∗ is enlarged from 0.003 to 0.06, which implies increasing the difficulty in information adoption. Analogously, other subgraphs exhibit similar findings. The 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.Figure 5
Illustrations of the growth pattern ofR∞ on the β,ρ0 parameter plane on the ER-ER weighted multiplex network. According to the theoretical method, the final adoption size R∞ versus the parameter pair β,ρ0 was obtained. The results are plotted in subgraphs (a), (b), and (c) from row 1 with TA=1,TB=1; (d), (e), and (f) from row 2 with TA=1,TB=2; and (g), (h), and (i) from row 3 with TA=1,TB=3. The subgraphs have weight distribution exponents of αωA=3, αωB=2.5 in column 1; αωA=3, αωB=3 in column 2; and αωA=3, αωB=3.5 in column 3. All subgraphs show that a critical seed size ρ0∗ 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. The basic parameters are N=104, k=10, ω=8, ρ0=0.001, and γ=1.0.
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### 5.2. ER-SF Weighted Multiplex Networks

For cases of heterogeneous degree distributions, we investigate information spreading on a two-layered weighted ER-SF network with Poisson degree distributionpAkA=e−kAkAkA/kA! in layer A and with power-law degree distribution pBkB=ζBkB−vB in layer B, where ζB=1/∑kBkB−vB and parameter vB denotes the degree exponent of layer B. Additionally, we set the minimum degree kBmin=4 and maximum degree kBmax=N.On the weighted ER-SF networks of distinct degree distribution exponents, the growth pattern is shown in Figure6, with the common setting TA=1,TB=3 in all subgraphs. In Figure 6(a) with degree distribution exponent vB=2.1, we discovered that enlarging the weight distribution exponent, such as from αwA=3,αwB=2.5 to αwA=3,αwB=3.5, 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) (vB=3) and Figure 6(c) (vB=4). Consequently, from Figure 6(a) to 6(c), given αwA and αwB, the growth pattern changes from continuous to the discontinuous growth pattern. However, at any given vB, changing the weight distribution exponent will not alter the growth pattern of R∞. The 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.Figure 6
Effect of degree distribution heterogeneity on the final adoption sizeR∞ versus β on the ER-SF weighted multiplex network with different degree distribution exponents vB. Subgraphs (a) (vB=2.1), (b) (vB=3), and (c) (vB=4) illustrate the growth patterns of R∞ versus β under different weight distribution exponents (αωA=3, αωB=2.5), (αωB=3, αωB=3), and (αωA=3, αωB=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.
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## 5.1. ER-ER Weighted Multiplex Networks

First, we investigate information spreading on a two-layered weighted ER-ER network with Poisson degree distributionpAkA=e−kAkAkA/kA! in layer A and pBkB=e−kBkBkB/kB! in layer B.We first explore the time evolutions of node densities in three states (S-state, A-state, and R-state) under different weight distribution exponentsαωA and αωB, as shown in Figure 2. Under the same information transmission unit probability β but different adoption thresholds and seed sizes, Figure 2 shows the plots of curves with TA=1;TB=2;ρ0=0.03 in the top panel and TA=1;TB=3;ρ0=0.1 in the bottom panel. Regardless of the top or the bottom panel, with time t, the size of St decreases gradually from 1−ρ0 to 0, At changes from ρ0 to 0 eventually, and Rt indicating the final adoption size increases gradually from 0 to 1 at the end. With the increase in the weight distribution exponent from αωA=3,αωB=2.5 in Figures 2(a) and 2(e), αωA=3,αωB=3 in Figures 2(b) and 2(f), and αωA=3,αωB=3.5 in Figures 2(c) and 2(g), the evolution time consumed decreases from 19 to 13 in the top panel and 11 to 9 in the bottom panel. Additionally, the final adoption size R∞ augments at the same time t (see the comparison in Figures 2(d) and 2(h)). The 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.Figure 2
Time evolutions of node densities in three states (S-state, A-state, and R-state) on the ER-ER weighted multiplex network, under different weight distribution exponentsαωA and αωB. To unveil the effect of the threshold and seed size, two groups of subgraphs with TA=1;TB=2;ρ0=0.03 and TA=1;TB=3;ρ0=0.1 are shown in the top and bottom panels, respectively. Under the same information transmission unit probability β, subgraphs (a), (b), and (c) from the top panel have similar parameters of αωA=3, αωB=2.5; αωA=3, αωB=3; and αωA=3, αωB=3.5 with (e), (f), and (g), respectively. Particularly, subgraphs (d) and (h) emphasize the time evolution of the R-state node density Rt 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 αωA=3,αωB=3.5 exceeds other densities under αωA=3,αωB=2.5 and αωA=3,αωB=3. These phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading. The basic parameters are N=104, k=10, ω=8, γ=1.0, and β=0.3.
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(h)Next, given the adoption thresholdsTA, TB, and seed size ρ0, we investigate the growth pattern of the final adoption size R∞ versus information transmission unit probability β. In Figure 3(a), with TA=1, TB=1, and ρ0=0.005, increasing the weight distribution exponent such as from αωA=3,αωB=2.5 to αωA=3,αωB=3.5 does not change the growth pattern of the final adoption size R∞ versus β but reduces the critical information transmission unit probability βc, 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 TA=1;TB=2;ρ0=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 (ρ0=0.1) in Figure 3(c) with the same adoption threshold of Figure 3(b), the final adoption size R∞ versus β increases in the same pattern regardless of the increase in weight distribution. Besides, keeping the seed size ρ0 as the same of Figure 3(c) (ρ0=0.1), Figure 3(d), with further increase of the adoption threshold TA=1,TB=3, also shows similar phenomena as those in Figures 3(a)–3(c), verified by the peaks of χ in Figure 3(h). The above phenomena suggest that increasing the weight distribution exponent (decreasing weight distribution heterogeneity) will 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).Figure 3
Growth pattern of the final adoption sizeR∞ versus information transmission unit probability β on the ER-ER weighted multiplex network, under different weight distribution exponents αωA and αωB, given adoption thresholds (TA, TB) and initial fraction ρ0 of seeds. The final adoption size R∞ versus information transmission probability β with (a) TA=1; TB=1; ρ0=0.005, (b) TA=1;TB=2;ρ0=0.03, (c) TA=1;TB=2;ρ0=0.1, and (d) TA=1;TB=3;ρ0=0.1. The lines denoting theoretical solutions agree exceptionally well with the symbols representing numerical simulations. Moreover, in subgraphs (e), (f), (g), and (h), the relative variance χ in terms of the peaks numerically indicates the critical spreading probabilities in (a), (b), (c), and (d), respectively, showing that increasing the weight distribution exponent (decreasing weight distribution heterogeneity) can decrease the critical spreading probability. The phenomena above suggest that decreasing weight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versus information transmission probability. The basic parameters are N=104, k=10, ω=8, and γ=1.0.
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(h)As shown in Figure3, 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) (TA=1;TB=1;αwA=3,αwB=2.5), increasing the seed size from ρ0=0.005 to ρ0=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 αwA=3,αwB=3 in Figure 4(d) and αwA=3,αwB=3.5 in Figure 4(g), increasing the seed size ρ0 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 TA=1,TB=2 in Figures 4(b), 4(e), and 4(h), or TA=1,TB=3 in Figure 4(c), 4(f), and 4(i) renders information adoption more difficult, it is clear that increasing ρ0 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 TA=1,TB=1. Additionally, changes in adoption thresholds and seed size do not qualitatively alter phenomena.Figure 4
Effect of fraction of initial seeds on the growth pattern of the final adoption sizeR∞ versus β on the ER-ER weighted multiplex network, under different weight distribution exponents αωA and αωB. The values of weight distribution exponent αωA of all subgraphs are 3, and the subgraphs in each row have the same αωB, i.e., subgraphs (a), (b), and (c) correspond to αωB=2.5; (d), (e), and (f) correspond to αωB=3; and (g), (h), and (i) correspond to αωB=3.5. Additionally, the subgraphs in each column have the same adoption threshold, i.e., subgraphs (a), (d), and (g) are set with TA=1,TB=1; (b), (e), and (h) with TA=1,TB=2; and (c), (f), and (i) with TA=1,TB=3. The theoretical solutions (lines) coincide well with the numerical results (symbols). 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. The 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. The basic parameters are N=104, k=10, ω=8, and γ=1.0.
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(i)For a more comprehensive study, we obtained the results of the final adoption size on theβ,ρ0 parameter plane, as illustrated in Figure 5. Here, we set the subgraphs in column 1 with TA=1,TB=1; in column 2 with TA=1,TB=2; in column 3 with TA=1,TB=3; in row 1 with αwA=3,αwB=2.5; in row 2 with αwA=3,αwB=3; and in row 3 with αwA=3,αwB=3.5. In the following, we discuss the findings based on Figure 5 in detail. In Figure 5(a) (TA=1;TB=1;αwA=3,αwB=2.5), a critical ρ0∗=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(d) (αwA=3,αwB=3) and Figure 5(h) (αwA=3,αwB=3.5). The results indicate that the three subgraphs have the same critical seed size ρ0∗=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 αwA=3,αwB=2.5 and increase the adoption thresholds in Figure 5(b) (TA=1,TB=2) and Figure 5(c) (TA=1,TB=3). Under these circumstances, we discovered that the critical seed size ρ0∗ is enlarged from 0.003 to 0.06, which implies increasing the difficulty in information adoption. Analogously, other subgraphs exhibit similar findings. The 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.Figure 5
Illustrations of the growth pattern ofR∞ on the β,ρ0 parameter plane on the ER-ER weighted multiplex network. According to the theoretical method, the final adoption size R∞ versus the parameter pair β,ρ0 was obtained. The results are plotted in subgraphs (a), (b), and (c) from row 1 with TA=1,TB=1; (d), (e), and (f) from row 2 with TA=1,TB=2; and (g), (h), and (i) from row 3 with TA=1,TB=3. The subgraphs have weight distribution exponents of αωA=3, αωB=2.5 in column 1; αωA=3, αωB=3 in column 2; and αωA=3, αωB=3.5 in column 3. All subgraphs show that a critical seed size ρ0∗ 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. The basic parameters are N=104, k=10, ω=8, ρ0=0.001, and γ=1.0.
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## 5.2. ER-SF Weighted Multiplex Networks

For cases of heterogeneous degree distributions, we investigate information spreading on a two-layered weighted ER-SF network with Poisson degree distributionpAkA=e−kAkAkA/kA! in layer A and with power-law degree distribution pBkB=ζBkB−vB in layer B, where ζB=1/∑kBkB−vB and parameter vB denotes the degree exponent of layer B. Additionally, we set the minimum degree kBmin=4 and maximum degree kBmax=N.On the weighted ER-SF networks of distinct degree distribution exponents, the growth pattern is shown in Figure6, with the common setting TA=1,TB=3 in all subgraphs. In Figure 6(a) with degree distribution exponent vB=2.1, we discovered that enlarging the weight distribution exponent, such as from αwA=3,αwB=2.5 to αwA=3,αwB=3.5, 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) (vB=3) and Figure 6(c) (vB=4). Consequently, from Figure 6(a) to 6(c), given αwA and αwB, the growth pattern changes from continuous to the discontinuous growth pattern. However, at any given vB, changing the weight distribution exponent will not alter the growth pattern of R∞. The 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.Figure 6
Effect of degree distribution heterogeneity on the final adoption sizeR∞ versus β on the ER-SF weighted multiplex network with different degree distribution exponents vB. Subgraphs (a) (vB=2.1), (b) (vB=3), and (c) (vB=4) illustrate the growth patterns of R∞ versus β under different weight distribution exponents (αωA=3, αωB=2.5), (αωB=3, αωB=3), and (αωA=3, αωB=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.
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## 6. 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. Therefore, information spreading on a multiplex social network should be investigated on a multilayer weighted network. Unfortunately, the mechanism of information spreading on 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 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. This 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.



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*Source: 5920187-2019-11-11.xml*