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Link overlap influences opinion dynamics on multiplex networks of Ashkin-Teller spins

Cook Hyun Kim, Minjae Jo, J. S. Lee, G. Bianconi, and B. Kahng
Phys. Rev. E 104, 064304 – Published 9 December 2021
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  • INTRODUCTION
  • MODEL AND FORMALISM
  • MEAN-FIELD SOLUTION
  • PHASE DIAGRAM I: J4/J2-DEPENDENCE
  • PHASE DIAGRAM II: λa-DEPENDENCE
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Consider a multiplex network formed by two layers indicating social interactions: the first layer is a friendship network and the second layer is a network of business relations. In this duplex network each pair of individuals can be connected in different ways: they can be connected by a friendship but not connected by a business relation, they can be connected by a business relation without being friends, or they can be simultaneously friends and in a business relation. In the latter case we say that the links in different layers overlap. These three types of connections are called multilinks and the multidegree indicates the sum of multilinks of a given type that are incident to a given node. Previous opinion models on multilayer networks have mostly neglected the effect of link overlap. Here we show that link overlap can have important effects in the formation of a majority opinion. Indeed, the formation of a majority opinion can be significantly influenced by the statistical properties of multilinks, and in particular by the multidegree distribution. To quantitatively address this problem, we study a simple spin model, called the Ashkin-Teller model, including two-body and four-body interactions between nodes in different layers. Here we fully investigate the rich phase diagram of this model which includes a large variety of phase transitions. Indeed, the phase diagram or the model displays continuous, discontinuous, and hybrid phase transitions, and successive jumps of the order parameters within the Baxter phase.

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    • Received 19 June 2021
    • Revised 20 October 2021
    • Accepted 22 November 2021

    DOI:https://doi.org/10.1103/PhysRevE.104.064304

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Network evolutionNetwork phase transitionsPercolation phase transitionPhase transitions
    NetworksStatistical Physics & Thermodynamics

    Authors & Affiliations

    Cook Hyun Kim1, Minjae Jo1, J. S. Lee2, G. Bianconi3,4, and B. Kahng5

    • 1CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
    • 2School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
    • 3School of Mathematical Sciences, Queen Mary University of London, E1 4GF, London, United Kingdom
    • 4Alan Turing Institute, The British Library, NW1 2DB, London, United Kingdom
    • 5Center for Complex Systems, KI of Grid Modernization, Korea Institute of Energy Technology, Naju, Jeonnam 58217, Korea

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    Issue

    Vol. 104, Iss. 6 — December 2021

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    Images

    • Figure 1
      Figure 1

      (a) The g-AT model on a duplex network: two species (si, σi) of Ising spins describe, respectively, the opinion of node i in layers 1 and 2. Each pair of nodes of the duplex network can be connected by a different type of multilink: multilinks (1,1) connect pair of nodes in both layers 1 and 2; multilinks (1,0) and (0,1) connect pair of nodes only in layer 1 and only in layer 2, respectively. Therefore, multilinks (1,1) describe overlapping links while multilinks (1,0) and (0,1) describe nonoverlapping links. The model can be also interpreted as a model on a colored network in which nodes are an associated pair of spin and the interactions between each pair of nodes can be distinguished in mutlilinks (1,1),(1,0), and (0,1) (b).

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    • Figure 2
      Figure 2

      The g-AT model is an Hamiltonian model combining two-body and four-body interactions. The four-body interactions characterizes the interactions between the spins of species si and σi connected by a multilink (1,1). The two-body interactions characterize the coupling between spins of a given species (either the spins si or the spins σi) connected by either a multilink (1,0) or (0,1).

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    • Figure 3
      Figure 3

      Schematic phase diagram of the g-AT model for a given set of λn=3.53 and λo=3.90. Solid and dotted curves represent continuous and discontinuous PTs, respectively. This phase diagram is mostly similar to the one of the original AT model [40].

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    • Figure 4
      Figure 4

      (a)–(d) Plot of the order parameters ma and M as a function of T/J2. (e)–(h) Plot of the free-energy density landscape as a function of mn for λn=3.53 and λo=3.90 and various interlayer interaction ratios: x=1.30 for (a) and (e); x=1.40 for (b) and (f); x=1.62 for (c) and (g); and x=1.80 for (d) and (h). The transition types are second order in region (i) for (a) and (e); successive continuous-discontinuous in region (ii) for (b) and (f); mixed-order at CE1 for (c) and (g); and discontinuous transition in region (iii) for (d) and (h).

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    • Figure 5
      Figure 5

      (a)–(d) Plot of the order parameters ma and M as a function of T/J2. (e)–(h) Plot of the free-energy density landscape as a function of mn for λn=3.53 and λo=3.90 and various interlayer interaction ratios: x=2.10 for panels (a) and (e); x=2.30 for panels (b) and (f); x=2.45 for panels (c) and (g); and x=2.60 for panels (d) and (h). The transition types are first-order in regime (iii) for panels (a) and (e); successive continuous-discontinuous in regime (iv) for panels (b) and (f); successive continuous-discontinuous in regime (v) for panels (c) and (g); and continuous transition in regime (vi) for panels (d) and (h).

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    • Figure 6
      Figure 6

      Schematic phase diagrams of the g-AT model in the parameter space [λn,λo] for (a) x≈0, (b) x=xM, and (c) x≫xM. The notations of the phases (i)−(vii) are the same as the ones presented in Figs. 3 and 10.

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    • Figure 7
      Figure 7

      Schematic phase diagrams of the g-AT model in the parameter space [x,T−1] for various values λn and λo. (λn, λo) = (a) (3.53, 3.51), (b) (3.53, 3.90), (c) (3.53, 3.49), (d) (3.35, 3.90), (e) (3.53, 3.30), and (f) (3.30, 3.90).

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    • Figure 8
      Figure 8

      For the (vii)-type of PT, schematic plots of (a) the order parameters ma and M as a function of T and (b) the free-energy density landscape as a function of mo for various Ts. The exponents of degree distributions are taken as (λn, λo) = (3.53, 3.51).

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    • Figure 9
      Figure 9

      For the (viii)- and (ix)-type of PTs, schematic plots of (a) and (b) the order parameters ma and M as a function of T, respectively. (c) and (d) schematic plots of the free-energy density landscape as a function of mo for the temperatures around Tf. The exponents of degree distributions for (a) and (c) are taken as (λn, λo) = (3.53, 3.30) and for (b) and (d) are taken as (3.53, 3.49).

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    • Figure 10
      Figure 10

      (a), (b) Plots of the order parameters ma and M at x=xM as a function of T/J2. (c), (d) Plots of the free-energy density landscape as a function of mn for various T. The exponents of degree distributions taken for panels (a) and (c) are (λn, λo) = (3.30, 3.55) and for panels (b) and (d) are (3.43, 3.55).

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    • Figure 11
      Figure 11

      Schematic contour lines of xe,M in the parameter space [λn,λo]. The rightmost (light yellow) region contains the domains (viii), (ix), (x), and (xi) as shown in Fig. 6. In this region, a continuous PT appears at Ts at xM, and thus xe,M=xM, the Coherent phase appears at xM. The regions denoted as ①−④ correspond to the regions (iii) and (vii) denoted in Fig. 6. In this region, a discontinuous PT appears at xM, and thus xe,M>xM. The contour lines represent in term of the ratio xe,M/xM. In ①, 1<xe,M/xM<1.05; in ②, 1.05<xe,M/xM<1.10; in ③, 1.10<xe,M/xM<1.15; and in ④, 1.15<xe,M/xM.

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