Adaptive voter model on simplicial complexes

Leonhard Horstmeyer and Christian Kuehn

Phys. Rev. E 101, 022305 – Published 12 February 2020

Abstract

Collective decision making processes lie at the heart of many social, political, and economic challenges. The classical voter model is a well-established conceptual model to study such processes. In this work, we define a form of adaptive (or coevolutionary) voter model posed on a simplicial complex, i.e., on a certain class of hypernetworks or hypergraphs. We use the persuasion rule along edges of the classical voter model and the recently studied rewiring rule of edges towards like-minded nodes, and introduce a peer-pressure rule applied to three nodes connected via a 2-simplex. This simplicial adaptive voter model is studied via numerical simulation. We show that adding the effect of peer pressure to an adaptive voter model leaves its fragmentation transition, i.e., the transition upon varying the rewiring rate from a single majority state into a fragmented state of two different opinion subgraphs, intact. Yet, above and below the fragmentation transition, we observe that the peer pressure has substantial quantitative effects. It accelerates the transition to a single-opinion state below the transition and also speeds up the system dynamics towards fragmentation above the transition. Furthermore, we quantify that there is a multiscale hierarchy in the model leading to the depletion of 2-simplices, before the depletion of active edges. This leads to the conjecture that many other dynamic network models on simplicial complexes may show a similar behavior with respect to the sequential evolution of simplices of different dimensions.

Received 14 September 2019
Revised 30 November 2019
Accepted 7 January 2020

DOI:https://doi.org/10.1103/PhysRevE.101.022305

Physics Subject Headings (PhySH)

Collective dynamics Evolving networks Rumor evolution

Network evolution Social dynamics Stochastic dynamical systems Stochastic networks

Network Models Networks Analysis Tools Topological tools

Nonlinear DynamicsStatistical Physics & ThermodynamicsNetworks

Authors & Affiliations

Leonhard Horstmeyer ^1,2 and Christian Kuehn ^3,1

¹Complexity Science Hub Vienna, Josefstädterstrasse 39, A-1090 Vienna, Austria
²Basic Research Community for Physics, Mariannenstrasse 89, 04315 Leipzig, Germany
³Faculty of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching München, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 101, Iss. 2 — February 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
We show two sample paths of the dynamics in the $(m, ρ)$ space at a rewiring probability of $p = 0.55$ : The black path shows the scenario where peer pressure is absent, i.e., $q = 0$ . The green and cyan paths show the scenario where the peer pressure is at $q = 3 / 4$ , respectively before the depletion of triangles at time $τ$ and thereafter. The dotted grey lines are best fits of the paths to the parametrized parabola $ρ = ξ_{p} (1 - m^{2})$ , respectively for $q = 0$ and $q = 0.75$ before the depletion of triangles. The horizontal lines indicate the respective values of their apexes $ξ_{p}$ . The size of the network is $N = 500$ , its mean degree is $μ = 8$ , and the simplex-per-edge degree is $s = 0.2$ . There is an initial population of triangles, such that the triangles-per-edge degree is $t = 0.8$ .
Reuse & Permissions
Figure 2
We compare the curves of the order parameter for various peer pressures. The estimates of the order parameters $ξ_{p}$ , i.e., the apexes of the parabolic regions (cf. Fig. 1), are plotted for a range of rewiring probabilities between zero and 1 and peer pressures $q \in {0, 0.25, 0.5, 0.75}$ , which are respectively color coded with black, red, blue, and green. Each data point is generated from 200 runs and shows the mean and the standard deviations of the fitted values for $ξ_{p}$ . Here, the size of the network is $N = 500$ , its mean degree is $μ = 8$ , and the simplices-per-edge degree is $s = 0.2$ . We also have an initial population of triangles such that the triangles-per-edge degree is $t = 0.8$ . However, the parameter $t$ does not directly affect the curves for $ξ_{p}$ , but only the triangle depletion time.
Reuse & Permissions
Figure 3
We show the average inverse depletion time of triangles for rewiring probabilities in the entire range between zero and 1 and for peer pressures $q \in {0, 0.25, 0.5, 0.75}$ . The initial triangles-per-edge degree $t$ is 0.8 and all other parameters $N, μ$ , and $s$ are as above.
Reuse & Permissions
Figure 4
We show the average initial depletion rate of active edges for rewiring probabilities in the entire range between zero and 1 and for peer pressures $q \in {0, 0.25, 0.5, 0.75}$ . All other parameters, i.e., $N, μ$ , and $s$ , are as above.
Reuse & Permissions
Figure 5
This plot shows the diffusivity of the dynamics along the direction of $m$ at $m = 0$ . It is measured in terms of the mean rate of change of $m^{2}$ for rewiring probabilities in the range between zero and 1 and for peer pressures $q \in {0, 0.25, 0.5, 0.75}$ . All other parameters, i.e., $N, μ, s$ , and $t$ , are as above.
Reuse & Permissions
Figure 6
We show the average radial drift velocity of $m$ at $m = \pm 0.2$ for rewiring probabilities in the range between zero and 1 and for peer pressures $q \in {0, 0.25, 0.5, 0.75}$ . There are no data points for rewiring rates above the fragmentation transition because the dynamics does not enter the slow manifold on which it can explore the regions of higher majority disparity $m$ . All other parameters, i.e., $N, μ, s$ , and $t$ , are as above.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics