Higher-Order Contagion Dynamics

• Higher-order contagion processes are characterized by group interactions that require simultaneous exposure to multiple sources to trigger state changes.
• Models employ hypergraphs and simplicial complexes to capture non-linear phenomena such as explosive transitions, bistability, and critical mass effects.
• Mean-field and heterogeneous methods reveal diverse phase transitions and guide optimal control strategies in social, biological, and technological systems.

Higher-order contagion processes are dynamical spreading phenomena in which transmission occurs not only through pairwise (dyadic) contacts but also via group interactions—i.e., the infection, activation, or adoption of a state (such as disease, behavior, or information) requires simultaneous or cumulative exposure to multiple sources, often mediated by complex group structures such as hyperedges, simplices, or communities. This framework generalizes classical network models to account for the rich micro- and mesoscopic interaction patterns present in real-world systems, capturing reinforcement, nonlinearity, and multistability phenomena absent in purely pairwise models (Iacopini et al., 2018, Barrat et al., 2021, Arruda et al., 2024). The field is motivated by empirical evidence in social, biological, and technological domains, which reveals that many collective phenomena—opinion formation, social norm adoption, viral marketing, cooperative behavior, biological contagion—cannot be fully captured by simple contagion models.

1. Mathematical Formalism and Model Classes

At the foundation of higher-order contagion modeling lies the extension from network graphs (nodes and edges) to higher-order network topologies, such as hypergraphs and simplicial complexes. A hypergraph $\mathcal H = (\mathcal V, \mathcal E)$ consists of nodes and arbitrary-size hyperedges, while a $d$-dimensional simplicial complex encodes all $k$-simplices $(k=0,\ldots,d)$, generalizing cliques and encoding inclusion relations (Iacopini et al., 2018). In these structures, a "higher-order contagion" process is defined by two components:

• State variables: Nodes typically have binary (S, I) or (S, I, R) states.
• Transmission kernels: The infection, adoption, or transition rates depend not only on pair activities but on the configuration of an entire group—e.g., a susceptible node becomes infected only if $m$ or more group-mates are infected, or if all co-members of a simplex are infected.

Abstractly, the equation for the expected state evolves as

$$\frac{d\langle Y_i\rangle}{dt} = -\delta\,\langle Y_i\rangle + \lambda\sum_{j: i\in e_j} \lambda^*(|e_j|)\langle X_i\,f_j^i(\{Y\})\rangle$$

where $f_j^i$ encodes the group-level contagion rule (e.g., social reinforcement, threshold, power-law, or "all-infected" activation) (Arruda et al., 2024).

Key functional types of $f_j^i$ documented in the literature include:

Kernel Type	$f_j^i(\{Y\})$	Transition Phenomenology
Pairwise SIS	$Y_k$	Continuous (2nd order), classic threshold
Simplicial (all-infected)	$\prod_{k\in e_j\setminus\{i\}} Y_k$	Discontinuous (1st order), bistability, critical mass effects
Power-law reinforcement	$(m_j^i)^\nu$	Hybrid/hysteresis if $\nu>1$, otherwise continuous
Threshold (critical mass)	$H(m_j^i - \Theta_j)$	Hybrid, multistable, intermittency in modular structures
Saturation/limited group size	$\min\{m_j^i, c\}$	Typically continuous, saturating threshold

(Arruda et al., 2024)

2. Mean-Field Theory, Bifurcation Structure, and Critical Phenomena

Mean-field (MF) and heterogeneous mean-field (HMF) approximations have been developed for higher-order contagion processes on hypergraphs and simplicial complexes (Barrat et al., 2021, Iacopini et al., 2018, Arruda et al., 2024). In the prototypical case of an SIS process with pairwise and 3-body (triangular) group transmission, the (homogeneous) MF equation is

$$\frac{d\rho}{dt} = -\mu\rho + \beta_1 \langle k_1 \rangle \rho(1-\rho) + \beta_2 \langle k_2 \rangle \rho^2 (1-\rho)$$

with recovery rate $\mu$, pairwise infection $\beta_1$, group infection $\beta_2$, and $\langle k_m \rangle$ the average number of $m$-sets per node (Iacopini et al., 2018, Barrat et al., 2021, Chowdhary et al., 2021). This equation yields a cubic polynomial in $\rho$ at steady-state, with the following generic features:

• For small $\beta_2$, the model reduces to SIS with a continuous (transcritical) phase transition at $\lambda_c^{\rm pairwise}=1$.
• For sufficiently large $\beta_2$, a discontinuous (saddle-node) transition appears at lower threshold $\lambda_c<1$, producing a region of bistability and hysteresis, with a critical-mass phenomenon: contagion can only survive if initial prevalence exceeds $\rho_{-}>0$.
• The precise threshold criteria and bifurcation points can be calculated analytically (see (Iacopini et al., 2018), Eq. (3)), with the hybrid regime set by $$\lambda_\Delta\equiv \beta_2 \langle k_2 \rangle/\mu>1$$.

Heterogeneous (degree-based) mean-field models preserve this scenario and show that only the linear pairwise term determines the epidemic threshold, but higher-order terms drive bistability and discontinuity. In systems with $m$-body interactions ($m>2$), the degree of the mean-field polynomial increases and, correspondingly, multistability and cusp bifurcations become possible (Kiss et al., 2023).

Field-theoretic analyses (Langevin description, renormalization group) reveal that higher-order terms up to order $\omega=2$ (triadic) are "relevant" at the mean-field and upper-critical-dimension levels: only $O(\rho^2)$ terms can turn the transition discontinuous. Beyond that, higher-order corrections are perturbative, and do not alter universality, unless strong heterogeneities or low spectral dimension intervene (Meloni et al., 25 Feb 2025).

3. Role of Network Structure, Overlap, and Correlations

Higher-order contagion processes are fundamentally dependent on the structure of higher-order components.

• Higher-Order Components (HOCs): The existence of a giant $m$-th–order component (i.e., a maximal set of hyperedges connected via shared nodes of cardinality at least $m$) is a necessary (and essentially sufficient) structural condition for global outbreaks. In real-world hypergraphs, only systems with a macroscopic HOC enable invasion from a single seed (Kim et al., 2022).
• Hyperedge Overlap: The interplay between pairwise and higher-order links, measured by overlap parameters (e.g., the fraction of triangles that are also 3-cliques), modulates both the epidemic threshold and prevalence. Increased overlap generally lowers the invasion threshold but can decrease eventual prevalence due to redundancy in exposure (Burgio et al., 2023).
• Degree Heterogeneity and Cross-Order Correlations: Degree distributions and correlations between pairwise and group degrees shape transition order, critical points, and dynamical desynchronization. Positive cross-order correlation synchronizes pairwise and higher-order spreading and lowers thresholds, while anti-correlation desynchronizes pathways and raises epidemic thresholds, altering optimal control strategies (Guzmán et al., 21 Jan 2026, Landry et al., 2020).

4. Temporal and Adaptive Effects

The temporal dimension and adaptive behaviors introduce additional phenomenology:

• Time-Varying Structures: In dynamically rewired or memoryful simplicial complexes, temporality suppresses or delays higher-order-induced discontinuities. For rapidly fluctuating group memberships, higher-order transitions may revert to pairwise-like behavior, while persistent group structure preserves explosive and critical-mass effects (Chowdhary et al., 2021).
• Adaptive Behaviors: Risk-aware adaptation via local or group-based awareness—reducing transmission rates or connections upon increased perceived risk—suppresses or eliminates bistability and explosive transitions. In higher-order processes, adaptation triggered by group-level status is highly effective, restoring the system to classic, continuous SIS transition order and yielding low social cost (activity reduction localized to superspreading nodes/groups) (Mancastroppa et al., 9 Jan 2026, Mancastroppa et al., 5 Feb 2026).

5. Inference, Classification, and Empirical Consequences

• Model Distinguishability: Higher-order contagion processes imprint distinct dynamical signatures, such as infection-order correlations with local topology (e.g., number of group participations). Simple, threshold, and higher-order processes can be algorithmically distinguished—from a single outbreak observation—using local network features and rank correlations (Cencetti et al., 2023).
• Parameter Estimation: The tensor-based and regression frameworks allow efficient parameter learning for higher-order interactions directly from discrete-time group-level exposure time series, matching the $2^n$-state Markov chain in accuracy on moderate-size systems (Liang et al., 2024).
• Intervention Strategies: The optimal targeting of nodes for vaccination or control depends on both their pairwise and group centrality, and the level of cross-order degree correlation in the underlying hypergraph (Guzmán et al., 21 Jan 2026).

6. Phase Transitions, Universality, and Analytical Insights

• Generic Transition Landscape: Higher-order contagion models generate a broad menagerie of non-equilibrium critical phenomena: continuous transitions, first-order transitions with hysteresis, hybrid transitions (both transcritical and fold), multistability with up to $m$ endemic states (for $m$-body models), and critical-mass–driven thresholds (Iacopini et al., 2018, Barrat et al., 2021, Kiss et al., 2023).
• Loops vs. Groups in Discontinuity: Long cycles (loops where multiple transmission chains meet within a group) are the necessary structural driver of discontinuous phase transitions. Pure group synergy (within-group reinforcement) is alone insufficient; only loop-mediated reinforcement contributes a saddle-node bifurcation yielding a jump in epidemic size (Keating et al., 19 Nov 2025).
• Critical Exponents and Scaling: At continuous onsets, prevalence scales as $(\lambda-\lambda_c)^{1/2}$ in mean field; at a saddle-node, the approach to the fold follows a square-root law, and for $m$-body models, the polynomial form admits classical root-counting and cusp analysis (Kiss et al., 2023, Kemmeter et al., 2023).

7. Open Questions and Research Directions

Key unresolved scientific avenues include:

• Systematic identification of universality classes for hybrid and discontinuous transitions, including the precise role of kernel shape and network topology (Arruda et al., 2024).
• Analytical extensions of pair and higher-moment closure techniques, going beyond current MF/HMF treatments, to capture dynamical correlations and temporal effects (Malizia et al., 2023, Burgio et al., 20 May 2025).
• Empirical design of experiments and inference pipelines to validate higher-order models, especially in social-behavioral spreading, and recovery of latent group structure from observed pairwise contact data (Arruda et al., 2024).

In conclusion, higher-order contagion processes represent a rigorous, structurally grounded extension of classical contagion models, elucidating mechanisms of reinforcement, critical mass, explosive transitions, hysteresis, and network-induced complexity. The field now couples rich mathematical theory—mean-field, master equation, spectral, field-theoretic, and branching process approaches—with computational simulations and emerging data-science methodologies to address concrete phenomena in epidemic, social-behavioral, and technological spreading contexts (Iacopini et al., 2018, Kim et al., 2022, Malizia et al., 29 Jan 2025, Mancastroppa et al., 9 Jan 2026).