Kinetics of Social Contagion

Zhongyuan Ruan, Gerardo Iñiguez, Márton Karsai, and János Kertész

Phys. Rev. Lett. 115, 218702 – Published 20 November 2015

Abstract

Diffusion of information, behavioral patterns or innovations follows diverse pathways depending on a number of conditions, including the structure of the underlying social network, the sensitivity to peer pressure and the influence of media. Here we study analytically and by simulations a general model that incorporates threshold mechanism capturing sensitivity to peer pressure, the effect of “immune” nodes who never adopt, and a perpetual flow of external information. While any constant, nonzero rate of dynamically introduced spontaneous adopters leads to global spreading, the kinetics by which the asymptotic state is approached shows rich behavior. In particular, we find that, as a function of the immune node density, there is a transition from fast to slow spreading governed by entirely different mechanisms. This transition happens below the percolation threshold of network fragmentation, and has its origin in the competition between cascading behavior induced by adopters and blocking due to immune nodes. This change is accompanied by a percolation transition of the induced clusters.

Received 30 May 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.218702

Authors & Affiliations

Zhongyuan Ruan^1,2, Gerardo Iñiguez^3,4, Márton Karsai⁵, and János Kertész^1,2,4,*

¹Center for Network Science, Central European University, H-1051 Budapest, Hungary
²Institute of Physics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary
³Centro de Investigación y Docencia Económicas, Consejo Nacional de Ciencia y Tecnología, 01210 México D.F., Mexico
⁴Department of Computer Science, Aalto University School of Science, FI-00076 AALTO, Finland
⁵Laboratoire de l’Informatique du Parallélisme, INRIA-UMR 5668, IXXI, ENS de Lyon, 69364 Lyon, France

^*janos.kertesz@gmail.com

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Vol. 115, Iss. 21 — 20 November 2015

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Images

Figure 1
(a) Numerical simulation of a general threshold model over an empirical network, with adoption threshold $ϕ = 0.2$ , rate of spontaneous adopters $p = 0.0005$ , and fraction of blocked nodes $r = 0.1$ . The network is an ego sample of Facebook friendships with size $N = 96$ and average degree $z = 10.63$ [25]. Susceptible nodes adopt spontaneously with rate $p$ or after a fraction $ϕ$ of their neighbors has adopted, while blocked nodes never adopt. (b) Schematic illustration of the spreading process. At $t = t_{0}$ node $P$ spontaneously becomes an adopter, “infecting” nodes $a$ and $b$ . When $Q$ adopts, it induces the adoption of nodes c–f. Nodes inside the ellipse constitute an induced cluster of adoption.
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Figure 2
Frequency $F_{g}$ of global cascades as a function of node threshold $ϕ$ and network average degree $z$ at an intermediate time $t = 100$ , for varying $p$ and $r$ . (a)–(b) As $p$ increases, the region that allows global cascades of adoption grows in size. Its boundary is well approximated by a cutoff in the fraction of adopters $ρ$ as calculated by Eq. (5). (c)–(d) Conversely, an increasing fraction of blocked nodes shrinks the global cascade regime. Dots show the boundary of this regime according to Eq. (1). Simulations correspond to an ER network with $N = 10^{4}$ and are averaged over $10^{4}$ realizations.
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Figure 3
Numerical simulations and analytical approximation of the threshold model for $z = 7$ and $ϕ = 0.2$ (continuous and dotted lines, respectively). (a) Fraction of adopters $ρ$ as a function of time for varying $p$ and fixed $r$ . (b) Time evolution of the normalized adoption density $ρ / (1 - r)$ for different values of $r$ and fixed $p$ . (c) Final relative density $ρ_{i}^{\infty} / ρ^{\infty}$ as a function of $r$ , for both spontaneous and induced adopters ( $i = 0$ , 1, respectively). (d) Final fraction of spontaneous adopters $ρ_{0}^{\infty}$ as a function of $r$ . (e) Normalized maximum speed of spreading ${\dot{ρ}}_{m} / [p (1 - r)]$ , calculated from the derivative of $ρ (t)$ . Shaded areas signal the regime of slow contagion $r > r_{\times}$ . Curves correspond to $N = 10^{4}$ and are averaged over $10^{3}$ realizations.
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Figure 4
(a) Size distribution $P (s)$ of induced clusters in the regimes of fast ( $r < r_{\times}$ ) and slow ( $r > r_{\times}$ ) spreading, at several stages in the adoption process. In early times, $P (s)$ is unimodal and qualitatively similar in both regimes. As $t$ increases, the distribution becomes bimodal only for $r < r_{\times}$ , indicating the presence of global cascades. (b) Asymptotic size distribution $P_{\infty} (s)$ of induced clusters, after $t = 5000$ and for varying $r$ . For $r < r_{\times}$ global cascades may still develop and make $P_{\infty} (s)$ bimodal. As $r$ increases, the distribution becomes unimodal and global cascades disappear. Simulations correspond to $z = 7$ , $ϕ = 0.2$ , $p = 0.0005$ , $N = 10^{4}$ , and are averaged over $10^{4}$ realizations.
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