Phase transitions for polyadic epidemic and voter models with multiscale groups

Abstract

Polyadic (or higher-order) interactions can significantly impact the dynamics of interacting particle systems. However, previous studies have often assumed group sizes to be relatively small. In this work, we examine the influence of multiscale polyadic group interactions, where some groups are small and others are very large. We consider two paradigmatic examples, an SIS-epidemic and the adaptive voter model. On the level of the mean field, we specifically discuss the impact of the multiscale polyadic interactions on equilibrium dynamics and phase transitions. For the SIS-epidemic model, we find a region of bistability that protects the disease-free state over a wide range beyond the classical epidemic threshold from a significant outbreak. For the adaptive voter model, we show that multiscale polyadic interactions can stabilize the network or increase the convergence rate to an unbiased equilibrium.

Figures (4)

• Figure 1: The SIS-epidemic model with large recovery groups.(a): Phase diagrams for $g_{\mathrm{I}} = 2$ and $g_{\mathrm{I}} = 3$ ($g_{\mathrm{I}} > 3$ are qualitatively similar to $g_{\mathrm{I}} = 3$). The transcritical bifurcation occurs at $\lambda N = 1$. As $r g_{\mathrm{I}} N^{g-1}$ is increased, its slope becomes shallower in the case of $g_{\mathrm{I}} = 2$ while remaining constant in the case of $g_{\mathrm{I}} = 3$ (inset, showing a neighborhood of the bifurcation point within a radius of $3 \cdot 10^{-2}$), but remains supercritical. Globally, a double saddle-node bifurcation forms. (b): Bifurcation diagrams for $g_{\mathrm{I}} = 1$. As $r g_{\mathrm{I}} N^{g-1}$ is increased, the transcritical bifurcation point shifts to the right, becomes shallower, and eventually turns subcritical. (c): Threshold value of $r g_{\mathrm{I}} N^{g-1}$ as a function of the group composition above which the double saddle-node bifurcation forms. This value increases rapidly as $g_{\mathrm{S}}$ or $g_{\mathrm{I}}$ are increased (see Appendix \ref{['sec:appendix-SIS-double-saddle-node']}).
• Figure 2: The adaptive voter model in the presence of large groups.(a): Schematic of the local topology: The vertices of an underlying network are divided into $M_{g}$ groups of size $g$. (b): The probability that a randomly chosen (2-)edge is contained in none of the $M_{g}$$g$-hyperedges, $\mathfrak{q}_{g,M_{g}}$, is decreasing in both $M_{g}$ and $g$. (c): When the containment in a hyperedge prevents the rewiring of an active edge, as $\mathfrak{q}_{g,M_{g}}$ decreases, i.e., the number $M_{g}$ or size $g$ of groups increases, the transcritical bifurcation point $p_{*}$ moves to the right ($\langle{k}\rangle = 5$, $\mu = -0.5$), stabilizing the topology. (d): For that same uniform initial conditions (dotted line), the dynamics follow different trajectories depending on the value $(1-\mathfrak{q}_{g,M_{g}}) \beta$ (colored lines) to the nontrivial equilibrium (left; $\langle{k}\rangle = 5$, $p = 0.8$, $\mu_{0} = 0.8$, $(1-\mathfrak{q}_{g,M_{g}}) \beta \in \{-1.0, -0.8, \ldots -0.2\}$). When the containment in a hyperedge introduces a bias in the propagation along an active hyperedge, as $\mathfrak{q}_{g,M_{g}}$ decreases, the convergence to the nontrivial equilibrium is accelerated (right; $\langle{k}\rangle = 5$, $p = 0.8$, $\epsilon = 10^{-2}$). For initial $|\mu|$ large and $(1-\mathfrak{q}_{g,M_{g}}) \beta$ small, the dynamics do not converge to the nontrivial equilibrium (see Appendix \ref{['sec:appendix-AVM-bistability']}). The numerical integration was performed using SciPy's LSODA integrator virtanen2020scipy.
• Figure A1: Formation of the double saddle-node bifurcation.(a): The existence of a solution to $\xi^{2} - \gamma \phi_{(g_{\mathrm{S}},g_{\mathrm{I}})}(\xi) = 0$ for $0 < \xi < 1$ (or rather $\xi_{0} < \xi < 1$) characterizes the presence of double saddle-node bifurcation ($g_{\mathrm{S}} = 3$, $g_{\mathrm{I}} = 2$). As $\gamma$ is increased, solutions eventually exist. Hence, the minimal value for $\gamma$, $\gamma_{*}$, such that a solution exists characterizes its formation. (b): In comparison with the true critical values $\gamma_{*}$, the asymptotic value provides an excellent estimate for $g_{\mathrm{S}}$ sufficiently large.
• Figure A2: Emergence of a bistability region in the adaptive voter model. Sufficiently close to the bifurcation point, along the manifold of trivial equilibria, (linear) stability changes as indicated by the value of the only nonvanishing eigenvalue of the linearization (colorbar). Consequently, it may happen that both the nontrivial and some subset of the trivial equilibria are stable. The set of uniform initial conditions forms a parabola in the phase space (dotted line). (a): Phase space of the adaptive voter model without bias and fixed magnetization. For uniform initial conditions, below the bifurcation point, the dynamics converge to the nontrivial equilibrium ($\langle{k}\rangle = 5$, $p = 0.88$, $\mu_{0} = -0.2$). The numerical integration was performed using SciPy's LSODA integrator virtanen2020scipy. (b): Full phase space of the adaptive voter model with bias. For different uniform initial conditions (dotted line) and below the bifurcation point, the dynamics converge either to a trivial or the nontrivial equilibrium ($\langle{k}\rangle = 5$, $p = 0.88$, $\hat{\beta} = -0.2$, $\mu_{0} \in \{-0.9, -0.8, \ldots -0.5\}$). The numerical integration was performed using SciPy's LSODA integrator virtanen2020scipy.