Higher-order non-Markovian social contagions in simplicial complexes

The study investigates how higher-order interactions in networks, captured by simplicial complexes (e.g., triangles and tetrahedra), shape social contagion dynamics. Traditional pairwise models and threshold-based contagion models overlook group effects that are prevalent in real systems such as social, collaboration, brain, email, and human contact networks. Concurrently, many real spreading processes are non-Markovian: inter-event times for infection and recovery deviate from exponential distributions and often exhibit peaked or heavy-tailed forms. The research question is how non-Markovian dynamics interact with higher-order structures to influence contagion, and whether non-Markovian higher-order processes can be related to Markovian counterparts. The paper proposes a higher-order non-Markovian SIS contagion model on simplicial complexes, develops a mean-field framework to capture its temporal evolution, examines equivalence with a higher-order Markovian model, and studies implications for system resilience against outbreaks.

Prior work has shown that pairwise models can miss group effects; higher-order interactions (e.g., 2-simplices/triangles) can change contagion phenomenology, including inducing discontinuous phase transitions in social contagion, affecting dominance in competing epidemics, constraining spreading in temporal higher-order networks, and suppressing outbreaks in multilayer settings. Explosive synchronization has been linked to coupling of node- and edge-defined signals in higher-order structures. On non-Markovian processes, studies have highlighted non-exponential inter-event times in human and epidemic dynamics, their effects on epidemic thresholds, and conditions yielding equivalence between non-Markovian and Markovian spreading. Non-Markovian recovery has been shown to enhance resilience against failures, and transient-state equivalence conditions have been identified for memory-dependent dynamics. Despite these advances, the combined role of non-Markovianity and higher-order structures in contagion has remained less explored, motivating the present work.

Model and definitions: The authors consider SIS dynamics on simplicial complexes up to dimension z = 2 (nodes, edges, triangles). They introduce z-dimensional virtual nodes f^z (each is a set of z real nodes within a z-simplex), virtual infected nodes, virtual edges, and ages. A virtual node f^z is infected if all its z real nodes are infected; its infection age is the minimum of its constituent nodes’ infection ages. A virtual edge connects f^z to the remaining node in the simplex, becoming active when f^z is infected. Infection process: A z-dimensional virtual infected node attempts to infect the remaining node in the simplex at a time-dependent rate η(τ) that depends on its infection age τ. This produces a non-Markovian infection process; in each simplex, infection attempts follow a renewal process with inter-event time distribution ψ(t). Recovery process: An infected real node recovers at age-dependent rate ω_rec(τ) with inter-event time distribution r_rec(τ). The framework thus captures both simple (edges) and complex (triangles) propagation.

Theory: A non-Markovian mean-field description is developed using age-structured densities I_i(τ; t) and S_i(τ; t) for infected and susceptible states, with evolution equations that account for recovery and infection hazard functions. The infection pressure Φ(t) on a susceptible node aggregates contributions from all z-simplices via integrals over η^z(τ) weighted by the age distribution of infected virtual nodes B^{(z)}(τ; t), where B^{(z)} encodes the minimum-age constraint among involved real nodes. Under homogeneous assumptions (uniform z-simplicial degree), the system reduces to coupled integro-differential equations for I(τ; t), S(τ; t), Φ(t). For the Markovian limit (constant rates), the equations reduce to a higher-order homogeneous mean-field ODE dI/dt = Σ_z ⟨k^{(z)}⟩ β^{(z)} S^{z-1} I^z − μ I. To incorporate correlations for z ≤ 2, a pair approximation (PA) is formulated in terms of edge-type densities UV and triangle closures UVW, yielding closed ODEs for I(t) and II.

Equivalence at steady state: Analytic derivations show that at steady state, the non-Markovian higher-order dynamics can be mapped to a Markovian form via an effective infection rate λ_eff defined by integrals of η(τ) and the recovery survival function Ψ_rec(τ). For z = 1 this reduces to the classic non-Markovian SIS λ_eff = ∫_0^∞ η(τ)Ψ_rec(τ) dτ.

Simulations: Monte Carlo simulations are performed on random simplicial complexes (RSC) of dimension z = 2. Construction: For N nodes, edges are added with probability ζ_1 and triangles with probability ζ_2, tuning to target average degrees ⟨k⟩ and ⟨k^{(2)}⟩ (both Poisson distributed). Unless otherwise specified: N = 2000, ⟨k⟩ = 14, ⟨k^{(2)}⟩ = 6; initial infected fraction I(0) = 0.02 with age 0. Non-Markovian inter-event times are modeled by Weibull distributions for infection attempts ψ(t) and recovery r_rec(τ), with shape/scale parameters varied to adjust concentration and mean durations; the exponential (Markovian) case occurs at shape parameter = 1. Parameters are chosen to compare Markovian and non-Markovian cases at matched average times or matched λ_eff. Time evolution and steady states are compared to the non-Markovian mean-field theory and, in Markovian cases, to mean-field and pair-approximation predictions. Hysteresis is probed by adiabatically sweeping effective infection rate and by varying initial conditions to assess resilience thresholds.

• The proposed non-Markovian mean-field theory accurately predicts steady-state infection densities; transient deviations arise due to nodal-pair dynamical correlations, especially when 2-simplices are included.
• Equivalence result: At steady state, higher-order non-Markovian contagion is equivalent to a higher-order Markovian process when the non-Markovian dynamics are summarized by an effective infection rate λ_eff (derived analytically). Simulation results collapse onto Markovian predictions when plotted versus λ_eff, confirming the equivalence for z = 1 and z ≤ 2.
• Higher-order interactions (triangles) promote spreading and can induce a transition from continuous to discontinuous (abrupt) phase transitions. Including 2-simplices shifts behavior compared with edge-only propagation and strengthens correlations.
• As the average 2-simplicial degree ⟨k^{(2)}⟩ increases (e.g., from 0 to 7 in simulations), steady-state infection density increases and the spreading threshold increases; a transition from continuous to discontinuous phase transition is observed with rising ⟨k^{(2)}⟩.
• Non-Markovian recovery enhances system resilience under different regimes: when the recovery-time distribution is more concentrated and longer (shape > 1 with large parameter), networks are more resilient to large initial infections (higher critical effective infection rate needed to trigger high-infection state). When recovery times are faster and more concentrated at short durations (shape < 1), resilience against small initial infections improves, as early recoveries suppress growth.
• For z ≤ 2, Markovian mean-field is less accurate than pair approximation due to strong pair correlations; PA aligns better with simulations.

The work addresses the gap in understanding how memory effects (non-Markovianity) combine with higher-order interactions to shape contagion. By extending SIS dynamics to simplicial complexes with age-dependent hazards, the authors show that steady-state behavior is governed by an effective infection rate, enabling a unifying mapping to Markovian models. This equivalence clarifies why some phenomenology can be captured by Markovian models once non-Markovian timings are encoded into λ_eff, while also emphasizing that transient dynamics and correlation effects can differ. The inclusion of triangles demonstrates that group interactions can promote spreading and precipitate discontinuous transitions and hysteresis, highlighting the importance of higher-order structures in realistic contagion processes. The resilience analysis shows practical implications: tailoring recovery-time distributions (e.g., via interventions that alter recovery heterogeneity) can enhance robustness against either large-scale or small-scale outbreaks, depending on the parameter regime. The findings are relevant for understanding and controlling social propagation (e.g., public opinion and rumor dynamics) on higher-order networks.

The study introduces a higher-order non-Markovian social contagion model on simplicial complexes, develops an age-structured mean-field framework (and PA for z ≤ 2), and proves a steady-state equivalence to a higher-order Markovian model via an effective infection rate. Simulations on random simplicial complexes validate the theory and reveal that higher-order interactions promote spreading and can produce discontinuous phase transitions. Non-Markovian recovery can increase system resilience, helping networks withstand both large-scale and small-scale infections under different conditions. Future work includes developing more precise theories that account for strong dynamical correlations in higher-dimensional simplices, extending beyond z = 2, and exploring broader classes of inter-event time distributions and network structures.

• Strong nodal-pair dynamical correlations in networks with 2-simplices lead to deviations between mean-field predictions and simulations; accurate prediction requires more complex correlation-aware theories (e.g., pair approximation or higher-order closures).
• Theoretical exploration beyond pairwise correlations is computationally intensive, and the study did not pursue more complex closures due to high computational cost.
• Simulations and theory focus on simplicial complexes up to z = 2 and on random simplicial complex models with Poisson degree distributions; generalizability to heterogeneous or real-world simplicial structures and higher z remains to be tested.
• Non-Markovian processes are modeled with Weibull distributions; other empirical inter-event time forms were not systematically explored.