Explosive transitions induced by interdependent contagion-consensus dynamics in multiplex networks

D. Soriano-Paños, Q. Guo, V. Latora, and J. Gómez-Gardeñes

Phys. Rev. E 99, 062311 – Published 25 June 2019

Abstract

We introduce a model to study the interplay between information spreading and opinion formation in social systems. Our framework consists in a two-layer multiplex network where opinion dynamics takes place in one layer, while information spreads on the other one. The two dynamical processes are mutually coupled in such a way that the control parameters governing the dynamics of the node states at one layer depend on the dynamical states at the other layer. In particular, we consider the case in which consensus is favored by the common adoption of information, while information spreading is boosted between agents sharing similar opinions. Numerical simulations of the model point out that, when the coupling between the dynamics of the two layers is strong enough, a double explosive transition, i.e., a discontinuous transition both in consensus dynamics and in information spreading appears. Such explosive transitions lead to bi-stability regions in which the consensus-informed states and the disagreement-uninformed states are both stable solutions of the intertwined dynamics.

Received 20 December 2018
Revised 22 April 2019

DOI:https://doi.org/10.1103/PhysRevE.99.062311

Physics Subject Headings (PhySH)

Dynamics of networks

Multilayer & multiplex networks

Interdisciplinary PhysicsNonlinear DynamicsNetworks

Authors & Affiliations

D. Soriano-Paños^1,2, Q. Guo^3,4,*, V. Latora^5,6,7,†, and J. Gómez-Gardeñes ^1,2,‡

¹GOTHAM Laboratory, Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain
²Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain
³School of Mathematics and Systems Science, Beihang University & Key Laboratory of Mathematics Informatics Behavioral Semantics (LMIB), Beijing 100191, China
⁴China Construction Bank, Beijing 100033, China
⁵School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
⁶Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, Catania I-95123, Italy
⁷Complexity Science Hub Vienna (CSHV), 1080 Vienna, Austria

^*quantongg@buaa.edu.cn
^†v.latora@qmul.ac.uk
^‡gardenes@unizar.es

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Vol. 99, Iss. 6 — June 2019

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Images

Figure 1
Left: Schematic representation of our model on a multiplex network with $M = 2$ layers and $N = 5$ nodes. The first (top) layer accounts for the consensus dynamics, which is modeled by a Kuramoto model as in Eq. (3), whereas the second (bottom) layer describes the spreading of information according to the SIS model as in Eq. (4). Right: The coupling strength $λ$ between opinions (top) as well as the contagion rate $β$ (bottom) have been modified as in Eqs. (7) and (6), respectively, to mutually couple the synchronization process to the spreading of information.
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Figure 2
Average global consensus $r$ (top) and fraction of information spreaders $I$ (bottom) as a function of the coupling constant $λ$ for the SF-ER multiplex configuration. These order parameters have been computed adiabatically by increasing the value of the coupling constant $λ$ (Forward) from $λ = 0$ or by decreasing it (Backward) from $λ = 1.5$ . The contagion rate values used are for (a) and (b) $β = 0.70$ and for (c) and (d) $β = 1$ . The rest of the model parameters are set to $α = 10$ and $μ = 1.0$ . All the layers are composed by $N = 500$ nodes with average degree $〈 k 〉 = 4$ . The SF network follows a power law distribution with exponent $γ = 3$ .
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Figure 3
Diagrams of the average global consensus $r$ (top) and of the fraction of spreaders nodes $I$ (bottom) as a function of the infectivity $β$ and the coupling parameter $λ$ for SF-ER (left) and SF-SF (right) multiplex networks. The color code encodes the magnitudes of the two order parameters. The striped regions in the panels highlight the parameter region ( $λ, β$ ), where hysteresis cycles appear due to the coexistence of two stable solutions: total consensus-disagreement in the synchronization layer and active-inactive spreaders at the information spreading layer. The black solid lines denote the critical values of the coupling $λ$ separating the two solutions. All the layers consists of $N = 500$ nodes with average degree $〈 k 〉 = 4$ . The SF networks follow a power law distribution with exponent $γ = 3$ .
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Figure 4
Fraction of spreaders in the nonsynchronized regime ( $λ = 0$ ) as a function of the transmissibility $β$ for both SF-ER (solid lines) and SF-SF configurations (dashed lines). The line color denotes the value of the social pressure $α$ . Regarding the underlying topologies, all layers are composed by $N = 500$ nodes with average degree $〈 k 〉 = 4$ . The degree distribution of the SF layer follows a power law distribution with exponent $γ = 3$ .
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Figure 5
Diagrams of the fraction of spreaders nodes $I$ (color code) as a function of the infectivity $β$ and of the coupling parameter $λ$ for SF-ER multiplex networks. The social pressure $α$ is set to $α = 5$ (left panels) and $α = 20$ (right panels). The average degree of the ER spreading layer is fixed to $〈 k 〉 = 6$ (top row) and to $〈 k 〉 = 10$ (bottom row). In all panels, the fraction of spreaders has been obtained by averaging their values over $T = 800$ steps. The striped regions in the panels highlight the parameter region ( $λ, β$ ) where hysteresis cycles appear due to the coexistence of two stable solutions: total consensus-disagreement in the synchronization layer and active-inactive spreaders at the information spreading layer. The black solid lines denote the critical values of the coupling $λ$ separating the two solutions.
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Figure 6
(a) Diffusion threshold $β_{c}$ (color code) as a function of the social pressure $α$ and the average degree $〈 k 〉$ of the diffusion layer for the SF-ER configuration. In panels (b) and (c) we show the diffusion threshold $β_{c}$ as a function of the average degree $〈 k 〉$ and the social pressure $α$ , respectively. In both cases we show these functions for several values of the social pressure $α$ (b) and the average degree $〈 k 〉$ (c).
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