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The dynamics of polarized beliefs in networks governed by viral diffusion and media influence

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Abstract

This paper studies the evolution of polarized beliefs governed by the intertwined dynamics of viral diffusion and media influence in influence networks. It addresses the question of how different forms of influence interact with each other. First, we propose a Markov chain to model the dynamics of individuals as they transition between three belief states (neutral, positive and negative) based on the states of their neighbors. This stochastic system assumes that individuals are influenced via the links of the network or through the global effect of mass media. For exponential and scale-free networks, we approximate this stochastic system by deterministic differential equations and define the homogeneous mean-field system and heterogeneous mean-field system, respectively. Studying stability conditions for these deterministic dynamical systems, we analyze the fraction of neutral, positive and negative individuals in the population. Also, we determine the conditions under which desired dynamical transitions happen for the targeted population. These conditions allow us to predict macroscopic measures of dynamics of adoption in influence networks. Finally, the derived analytical results are validated using simulations of four synthetic networks: Watts–Strogatz, random regular, Barabasi–Albert and small-world forest-fire, as well as five real-world networks: ego-Facebook, Deezer, Livemocha, a Facebook interaction network and Douban. Also, we demonstrate how the proposed model can be leveraged by marketing campaigns for optimal resource allocations between viral marketing and media marketing to minimize the number of final negative individuals in different network settings.

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Acknowledgements

The contents of this paper were partially presented in Sanatkar (2016) with the following differences. This paper includes a new theorem that provides a sufficient condition for the global stability of the negative-free equilibrium point of the HET-MEAN system. Also, regarding the simulation results, different from Sanatkar (2016), initial fractions of neutral, positive and negative nodes are randomly drawn from the probability simplex for each network realization, and in order to make the initial states of the individuals realistic, their initial polarities are designed to depend on their neighbors using the Breadth First Search algorithm. Simulation results are extended for five new networks: the Deezer network, Livemocha network, a Facebook interaction network, the Douban network and the small-world forest-fire network. Moreover, for the study of the optimal resource allocation between viral marketing and media marketing, we present new results for the Deezer network.

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Correspondence to Mohammad Reza Sanatkar.

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Appendix

Appendix

Proof of Theorem 3

If \(({\bar{n}}_{2},0)\) is locally exponentially stable, the Jacobian matrix A in (16), computed at \(({\bar{n}}_{2},0)\), is negative definite. We define \({\hat{n}}\) and \({\hat{u}}\) as follows:

$$\begin{aligned} \begin{aligned} {\hat{n}}&= n - {\bar{n}}_{2},\\ {\hat{u}}&=u. \end{aligned} \end{aligned}$$
(59)

Replacing n and u with \({\hat{n}}\) and \({\hat{u}}\) in (3), we can write

$$\begin{aligned} \begin{aligned} {\hat{n}}^{.}&=k\beta {\hat{n}}^{2} + (2k\beta n^{*}-(k\beta + \alpha + \gamma )) {\hat{n}} + (k\beta - k\mu ) {\hat{n}} {\hat{u}}\\&\quad +((k\beta - k\mu )n^{*}+\theta - \gamma ){\hat{n}},\\ {\hat{u}}^{.}&=k\mu {\hat{n}} {\hat{u}} +(k\mu n^{*}-\theta - \delta \alpha ){\hat{u}}. \end{aligned} \end{aligned}$$
(60)

The origin of the transformed dynamical system in (60) is corresponding to \(({\bar{n}}_{2},0)\) for the HOM-MEAN system in (3). We use Lyapunov functions to estimate the region of attraction for the origin of the transformed dynamical system. A Lyapunov function can be found by solving the Lyapunov equation

$$\begin{aligned} PA+A^{T}P=-Q, \end{aligned}$$
(61)

where Q is a positive definite matrix. Taking \(Q=I\), the unique solution is the positive definite matrix

$$\begin{aligned} P=\left[ \begin{array}{cc}p_{1}&{}\quad p_{2}\\ p_{2}&{}\quad p_{4}\end{array}\right] , \end{aligned}$$
(62)

where

$$\begin{aligned} \begin{aligned} p_{1}&=-\frac{1}{2A_{11}},\\ p_{2}&=\frac{A_{12}}{2A_{11}(A_{11}+A_{22})},\\ p_{4}&=\frac{-A_{12}^{2}}{2A_{11}A_{22}(A_{11}+A_{22})}. \end{aligned} \end{aligned}$$
(63)

The quadratic function \(V({\hat{n}},{\hat{u}})=p_{1}{\hat{n}}^{2}+2p_{2} {\hat{n}}{\hat{u}}+p_{4}{\hat{u}}^{2}\) is a Lyapunov function for the transformed dynamical system in a certain neighborhood of the origin (Khalil 2000). To find the region of attraction of the origin, we need to determine a domain D around the origin where V is positive definite and \(V^{.}\) is negative definite. Since P is positive definite, D can be determined by studying negative definiteness of \(V^{.}\) around the origin. We can write

$$\begin{aligned} \begin{aligned} V^{.}({\hat{n}},{\hat{u}})&=-{\hat{n}}^{2}+(2p_{4}{\hat{u}} +2p_{2}{\hat{n}})k\mu {\hat{n}}{\hat{u}}\\&\quad -{\hat{u}}^{2} +(2p_{1}{\hat{n}}+2p_{2}{\hat{u}})(k\beta {\hat{n}}^{2} +(k\beta -k\mu ){\hat{n}}{\hat{u}})\\&=-{\hat{n}}^{2}-{\hat{u}}^{2}+f{\hat{n}}^{3}+g{\hat{n}}^{2}{\hat{u}} +h{\hat{n}}{\hat{u}}^{2}, \end{aligned} \end{aligned}$$
(64)

where \(f=2p_{1}k\beta \), \(g=2(p_{1}(k\beta -k\mu )+p_{2}(k\beta +k\mu ))\) and \(h=2(p_{2}(k\beta -k\mu )+p_{4}k\mu )\). Let \(X:=[\begin{array}{cc}{\hat{n}}&{\hat{u}}\end{array}]^{T}\). Hence, it can be written

$$\begin{aligned} \begin{aligned} V^{.}&=-||X||^{2}+f{\hat{n}}^{3}+{\hat{n}}{\hat{u}}(g{\hat{n}}+h{\hat{u}})\\&=-||X||^{2}+f{\hat{n}}^{3}+{\hat{n}}{\hat{u}}[\begin{array}{cc}g&h\end{array}]X\\&\le -||X||^{2}+f||X||^{3}+\frac{1}{2}||X||^{2}\sqrt{g^{2}+h^{2}}||X||\\&\le -||X||^{2}+(f+\frac{1}{2}\sqrt{g^{2}+h^{2}})||X||^{3}.\\ \end{aligned} \end{aligned}$$
(65)

Therefore, \(V^{.}\) is negative if \(||X||<1/(f+\frac{1}{2}\sqrt{g^{2}+h^{2}})\). Let \(r:=1/(f+\frac{1}{2}\sqrt{g^{2}+h^{2}})\). And, we have

$$\begin{aligned} \eta <\min _{||x||=r}{x^{T}Px}=\lambda _{min}(P)r^{2} \end{aligned}$$
(66)

Therefore, we can write

$$\begin{aligned} R_{D}=[0,1]^{2}\cap {\hat{R}}_{D}, \end{aligned}$$
(67)

where

$$\begin{aligned} {\hat{R}}_{D}=\left\{ (n,u)| p_{1}(n-n^*)^2+2p_{2}(n-n^*)u+p_{4}u^2<\eta \right\} . \end{aligned}$$
(68)

\(\square \)

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Sanatkar, M.R. The dynamics of polarized beliefs in networks governed by viral diffusion and media influence. Soc. Netw. Anal. Min. 10, 17 (2020). https://doi.org/10.1007/s13278-020-0627-1

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