Group dynamics shape contagion onsets and multistable active phases under collective reinforcement

Abstract

Group-based reinforcement can induce discontinuous transitions from inactive to active phases in higher-order contagion models. However, these results are typically obtained on static interaction structures or within mean-field approximations that neglect temporal changes in group composition. Here, we show that group dynamics is not a secondary effect but a central aspect that determines the macroscopic transition class of higher-order contagion processes. We develop an analytically tractable approximate master equation model that effectively interpolates between quenched and mean-field limits through a group composition turnover rate. Our results reveal the rich impact of time-varying structures: it can induce discontinuous phase transition, broaden the bistable region, and at the same time promote or suppress contagion near criticality. Moreover, when real-world turnover rates and group-size heterogeneity are taken into account, the system exhibits a qualitatively richer phase diagram with four distinct dynamical phases, combining continuous or discontinuous transitions with localized or delocalized activity. In localized regimes, we uncover multistable active phases with multiple coexisting active states, which are observed in neither the annealed nor the quenched limits, and extend classical absorbing-active bistability. Finally, we demonstrate that the emergence of discontinuous transitions in real-world systems requires stronger nonlinear reinforcement than previously thought, indicating that simulations in static structures can yield qualitatively misleading predictions.

Figures (9)

• Figure 1: Temporal higher-order contagion model. In our model, a susceptible individual in a group of size $n$ with other $i$ adopters becomes infected at rate $\beta(n,i)=\lambda i^\nu$. The synergy exponent $\nu$ controls the group reinforcement. Adopters recover at rate $\mu$. While contagion unfolds within each group, the individuals move between groups at a at rate $\omega$.
• Figure 1: Interplay between membership, group size, group switching, and synergy reshapes the phase diagram. Phase portraits showing the critical adoption threshold $\lambda_c$ as a function of the group switching rate $\omega$ for different synergy factors, group sizes, and membership distributions. Panels a–c (d–f) correspond to fixed group size $p(n)=\delta_{n,3}$ ($p(n)=\delta_{n,5}$), while the left, central, and right columns correspond to $\langle k \rangle = 1.00$, $1.15$, and $3.00$, respectively. Each panel highlights the tricritical lines separating regions with continuous and discontinuous transitions. The inset in panel a shows the critical membership required to erase the non–monotonicity in the invasion threshold according to Eq. (\ref{['eq:kc']}).
• Figure 2: Group dynamics reshapes the critical behavior of higher-order contagion.a–b Stationary prevalence $I^\star$ and corresponding phase diagram for linear contagion ($\nu=1$) as a function of the spreading rate $\lambda$ and of the group switching rate $\omega$. Increasing the rate at which individuals switch groups lowers the invasion threshold, and the transition remains continuous. c–d Same as in previous panels for nonlinear contagion ($\nu=4$). Here, group dynamics fundamentally alters the transition: the invasion threshold becomes non-monotonic in $\omega$, and sufficiently strong synergy produces a discontinuous transition. Analytical predictions in Eqs. (\ref{['eq:inv_thres_compact']}) and (\ref{['eq:lambda-bi-final']}) for the invasion threshold $\lambda_c(\omega)$ and the persistence threshold $\lambda_p(\omega)$ are shown throughout, the tricritical point derived using Eq. (\ref{['eq:bi_threshold']}) in Methods is marked with a star in panel d, and the minimum of the invasion threshold according to Eqs. (\ref{['eq:wstar_general']})-(\ref{['eq:lamb_star_ge']}) is marked with a cross in panel d. e Width of the bistable region, $\Delta\lambda(\omega)=\lambda_c(\omega)-\lambda_p(\omega)$, as a function of the group switching rate for different values of the synergy exponent $\nu$. f Bifurcation diagram showing the analytically derived tricritical line separating continuous and discontinuous transitions, together with the corresponding invasion and persistence thresholds for the nonlinearities considered in panel e. All analytical curves closely match the steady-state solutions of the AMEs. In all panels, each individual belongs to one group only ($g_k=\delta_{k,1}$), the groups have homogeneous size ($p_n=\delta_{n,3}$), and $\mu=1$.
• Figure 2: Empirical temporal structure of real datasets. Time series of the number of groups $N(t)$, distribution of group sizes $p(n)$, and distribution of memberships per snapshot $g(k)$. The datasets belong to the different social contexts reported in Table \ref{['table:t1']}.
• Figure 3: Interplay between group dynamics and heterogeneity yields multistability. Using the empirically observed group-size and membership distributions of a real-world system, namely the high school dataset of Ref. mastrandrea2015contact, the panels show: a Tricritical line in the $(\omega,\lambda)$ plane with the color corresponding to the value of the synergy exponent $\nu$. b Stationary prevalence $I^\star$ as a function of the adoption rate $\lambda$ for selected values of the synergy exponent $\{\nu\}=\{1, 3.2,4.5,5.75,7,8.25,9.5\}$ and $\omega=5$. Solid (dashed) lines denote stable (unstable) stationary states. c-f Representative phase diagrams illustrating four distinct dynamical regimes: continuous transition in panel c for $\nu=1$, discontinuous transition with absorbing–active bistability in panel d for $\nu=3.2$, continuous hybrid transition with active bistability in panel e for $\nu=7$, and discontinuous hybrid transition with three coexisting stable states in panel f for $\nu=9.5$. Bottom rows show the effective participation ratio $P$ quantifying the degree of localization of activity across groups. In all panels, $\mu=1$.