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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.

The Internet [

Based on complex network theory [

Moreover, in social networks, because of multiple confirmations of the credibility and legitimacy of information, information spreading exhibits the effect of social reinforcement [

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

The remainder of the paper is organized as follows: In Section

For investigating information spreading on a weighted multiplex social network, we constructed an information spreading model based on an

Illustration of social information spreading with transformation of three states (susceptible, adopted, and recovered) on a weighted multiplex network. Two different independent social networks

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 [

Furthermore, the detailed information spreading process is as follows: Initially, randomly selecting a fraction

To include the social reinforcement effect, the mean-filed theory [

In the edge-weight-based compartmental theory, a node in the cavity state [

An arbitrary S-state node

As mentioned in Section

We define the probability that a susceptible node with any degree in the layer

Furthermore, the fraction of nodes in the S-state at time

If we wish to obtain

If neighbor

Analogously, neighbor

Accordingly, given the joint degree distribution

Next, we proceed with the evolution of

If neighbor

The initial conditions are

After inserting equations (

In equation (

When

Inserting

For a convenient expression, we rewrite

A discontinuous growth pattern will occur [

From the theoretical analysis above, the general solutions for adoption thresholds

Owing to independence, we obtain

Based on Erdös–Rényi (ER) [

To obtain the critical condition from the simulations, the

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 Section

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

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 exponents

Next, given the adoption thresholds

Growth pattern of the final adoption size

As shown in Figure

Effect of fraction of initial seeds on the growth pattern of the final adoption size

For a more comprehensive study, we obtained the results of the final adoption size on the

Illustrations of the growth pattern of

For cases of heterogeneous degree distributions, we investigate information spreading on a two-layered weighted ER-SF network with Poisson degree distribution

On the weighted ER-SF networks of distinct degree distribution exponents, the growth pattern is shown in Figure

Effect of degree distribution heterogeneity on the final adoption size

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.

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

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (nos. 61602048, and 61673086) and the Fundamental Research Funds of the Central Universities.