Contagion Dynamics on Hypergraphs

• Contagion dynamics on hypergraphs are defined by hyperedges that encode group interactions beyond pairwise links, leading to enriched epidemic phenomena.
• Unified agent-based and mean-field formulations reveal abrupt transitions, bistability, and critical thresholds influenced by hypergraph organization.
• Effective lower-order replica models with activity normalization replicate endemic states, offering practical insights for influence maximization and intervention strategies.

Contagion dynamics on hypergraphs generalize classical epidemic and information-spreading processes by encoding group interactions of arbitrary cardinality as hyperedges, rather than restricting attention to pairwise links. This framework is theoretically and practically motivated by the ubiquity of higher-order correlations in biological, social, and infrastructural systems, where collective exposure and reinforcement mechanisms cannot be reduced to dyadic contacts. Hypergraph contagion models reveal a much richer phenomenology compared to traditional network-based models, including abrupt transitions, bistability, complex critical properties, and intricate dependencies on network organization and group structure.

1. Unified Agent-Based and Mean-Field Contagion Model Formulations

A general contagion process on a hypergraph is specified by a node set $V$ ($|V|=N$) and a hyperedge set $E$, where each hyperedge $e \subset V$ can involve two or more nodes. In the agent-based discrete-time SIS/SIR model, each node’s state is updated via hyperedge-driven transitions. For each trial (with frequency $m$ per unit time), a hyperedge $e$ is sampled, and each susceptible $i \in e$ with $n_I(e)$ infected partners receives infection with probability

$$\beta_e = \min\Bigl[ f\!\left(\frac{n_I(e)}{n_S(e)-1}\right) n_I(e)\, \beta_{\rm ABM},\, 1 \Bigr],$$

where $f(\cdot)$ is a configurable activation function and $\beta_{\rm ABM}$ a base pairwise infection probability. Recovery ($I \rightarrow S$ or $I \rightarrow R$) occurs independently per node with constant rate.

Master equations and mean-field reductions have been developed to capture the high-dimensional group state space while permitting mathematical tractability. For instance, the generalized approximate master equation (GAME) framework tracks both group (hyperedge) composition distributions and node-level group activity partitions, with explicit or moment-based closures for group infection rates (Burgio et al., 20 May 2025, Burgio et al., 2023). Node-centric models capture infection by summing over all hyperedges containing a node, with possible nonlinearity in $n_I$ (Higham et al., 2021, Higham et al., 2021).

Critical to mean-field analysis is the construction of appropriate projection or co-membership matrices, e.g., $W_{ij} = \#\{e : i, j \in e\}$, whose spectral radius determines threshold phenomena for extinction and persistence (Higham et al., 2021, Higham et al., 2021).

2. Network Activity, Scaling, and Replica Reductions

A key finding is that much of the macroscopic behavior generated by arbitrary-order hypergraph interactions can, under appropriate conditions, be approximated by lower-order (often pairwise) models with rescaled infection rates. For a hypergraph $G_K$ with maximal order $K$, the mean per-hyperedge infection activity is quantified as

$$\widehat\lambda^{(K)}(G, \beta) = \frac{ \sum_{k=1}^K N_k \hat\beta^{(k)} }{ \sum_{k=1}^K N_k },$$

where $\hat\beta^{(k)}$ is the expected infection rate for a (k+1)-node hyperedge (Tan et al., 28 Aug 2025). The central result is that choosing the pairwise infection parameter so that

$$\beta_1 = \frac{\widehat\lambda^{(K)}(G_K, \beta_K)}{ \widehat\lambda^{(1)}(G_1, \beta_K)} \, \beta_K$$

allows the pairwise-only system to exactly replicate the endemic steady-state of the full higher-order system. For transient alignment, a dynamic adaptation—modifying $\beta_1(t)$ to match instantaneous network activity—achieves close adherence of the temporal infection curves, requiring only sparse parameter updates. These reductions generalize across both SIS and SIR processes (Tan et al., 28 Aug 2025).

This "activity normalization" reveals a duality: higher-order contact structure’s effect on prevalence can be replicated by time-varying effective infection rates in a simpler model, with deviations in early epidemic or peak behavior flagging possible genuinely higher-order phenomena.

3. Thresholds, Phase Transitions, and Bifurcation Structure

Epidemic thresholds and critical phenomena in hypergraph contagion models display fundamentally different behavior depending on the infection kernel’s nonlinearity and the topological features of the hypergraph.

• In linear-in-neighbors models on uniform or heterogeneous hypergraphs, the extinction threshold is set by $\beta f'(0) \lambda(W) < \delta$ (Higham et al., 2021, Higham et al., 2021).
• For threshold or collective contagion rules, higher-order terms can induce first-order (discontinuous) transitions, multiple endemic equilibria (bistability), and hysteresis (Arruda et al., 2020, Cisneros-Velarde et al., 2020, Arruda et al., 2019, Arruda et al., 2021).
• In scale-free uniform hypergraphs with "simplicial" contagion (co-infection requiring all but one group member), critical exponents and hybrid transitions emerge depending on the degree exponent, with hub structure sustaining or regularizing outbreaks (Jhun et al., 2019).
• The triadic mean-field approximation rigorously links the epidemic threshold and outbreak size to the overlap structure between lower- and higher-order interactions, with increased overlap lowering the invasion threshold but saturating or reducing final prevalence via redundant exposures (Burgio et al., 2023).

Tables (summarizing critical thresholds) are commonly used to juxtapose these dependencies:

Model/Kernel	Threshold Condition	Transition Nature
Linear SIS (pairwise)	$\beta \lambda(A)/\delta < 1$	Continuous
Hypergraph, linear	$\beta f'(0) \lambda(W)/\delta < 1$	Continuous
Collective contagion	$\beta c_2/(c_1) \lambda(W)/\delta < 1$	Discontinuous/Hysteresis
Simplicial (d-unif.)	weighted by degree and overlap	Continuous/Hybrid/Hysteresis

Discontinuous and hybrid transitions require mechanisms where multi-node exposures or group structure induce strong nonlinear feedback, as in threshold models or in cases of high-order "group reinforcement" (Arruda et al., 2020, Arruda et al., 2021).

4. Hypergraph Structure, Overlap, and Nestedness Effects

Higher-order structural features—such as hyperedge overlap, nestedness, and higher-order components—are decisive in determining both thresholds and qualitative behavior.

• Nestedness (fraction of lower-order hyperedges embedded in higher-order ones): systematically advances the invasion threshold (epidemic occurs more readily) and suppresses backward bifurcations, thereby quenching explosive transitions and narrowing bistable regimes (Malizia et al., 15 Jan 2026, Kim et al., 2023). Mechanistically, overlap reallocates contagion pathways from redundant external links to internal, group-wide channels, which—in the limit of high nesting—produces continuous transitions even in strongly nonlinear models.
• Higher-order components: The existence of a "giant higher-order component" (HOC, e.g., a giant 2nd-order component where hyperedges overlap by at least two nodes) is necessary for global outbreak in pure higher-order contagion models (where infection requires multiple infecteds per group). In real data, empirically observed HOCs fundamentally determine the possibility of epidemic invasion from single seeds (Kim et al., 2022).
• Facet structure and group heterogeneity: Analytic and simulation results confirm that nested group structure, heterogeneous hyperdegree (node memberships), and hyperedge size/power laws all modulate the width and location of transition windows, bistable intervals, and localization of outbreaks on the largest groups (St-Onge et al., 2021, Kim et al., 2023).

5. Influence Maximization, Seeding, and Intervention Strategies

Hypergraph contagion models alter not only global outbreak thresholds but also local strategies for maximizing or suppressing spread:

• In nonlinear contagion models (e.g., with superlinear infection kernels or group thresholds), optimal influence maximization shifts from targeting high-degree nodes ("influential spreaders") to targeting influential groups—entire hyperedges of large size (St-Onge et al., 2021).
• The "collective influence" metric, evaluated via the spectrum of the weighted non-backtracking matrix on the hypergraph, provides a principled method for ranking seed sets; adaptive seeding algorithms that exploit group structure outperform traditional degree-based heuristics (Zhang et al., 2023).
• Intervention levers decompose across epidemiological and structural axes: modulate base infection rate ($\beta$), alter recovery ($\delta$), disrupt high-order connectivity ($\lambda(W)$), or adjust behavioral coefficients such as threshold levels ($c_f$) (Higham et al., 2021, Higham et al., 2021).

Adaptive hypergraph models, in which nodes can rewire their group memberships in response to local outbreak signals, display new dynamical regimes: rewiring can suppress, exacerbate, or non-monotonically affect prevalence, depending on its targeting accuracy, speed, and the characteristic group-activity scale (Burgio et al., 2023, Burgio et al., 20 May 2025).

6. Inferring Contagion Mechanisms and Effective Model Bias

Distinguishing between truly complex contagion (requiring multi-node exposure) and effective superlinear behavior arising from heterogeneous group structure is crucial:

• On realistic weighted hypergraphs, ignoring heterogeneity in group transmission risk induces a nonlinear infection kernel at the aggregate level—producing effective thresholds, early outbreak acceleration, and even discontinuous transitions (St-Onge et al., 2023).
• Bayesian inference procedures, applied to empirical event sequences, consistently overestimate the nonlinearity parameter ($\alpha > 1$) when group-specific rates are unobserved, falsely classifying simple linear contagion as complex. This underscores the importance of explicit group-level modeling and parameter estimation in real systems.
• Practically, activity normalization and dynamic parameter-fitting frameworks are robust to structural uncertainty when the effective per-hyperedge infection rate can be estimated or deduced (Tan et al., 28 Aug 2025).

7. Broader Implications and Applications

Contagion dynamics on hypergraphs expose theoretical and practical limitations of standard network-based models. The essential novelty is the emergence of genuinely collective critical phenomena—abrupt transitions, bistability, and intermittency—driven by group-structural features and infection nonlinearity. However, the macroscopic incidence curves, endemic levels, and even much of the early outbreak trajectory can, in many cases, be closely matched by an optimally parameterized, lower-order (pairwise) "replica" model, provided a suitable global network-activity normalization is performed (Tan et al., 28 Aug 2025).

Empirical studies confirm that genuinely group-driven transitions (beyond what can be explained by rescaled pairwise mixing) often manifest as strong anomalous transients or in regimes with extreme nonlinearity, large group overlaps, or pronounced mesoscopic community structure (Arruda et al., 2021).

As a result, for inference and control in real-world systems, a practical program is to estimate global or instantaneous network activity rates—via survey, contact tracing, or topological reconstruction—and parameterize effective pairwise models accordingly, with deviations indicating potential higher-order or emergent dynamics. Nonetheless, for situations involving explicit group super-spreader events, critical-mass phenomena, or adaptive (rewiring) populations, full hypergraph modeling remains indispensable.

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References

• (Tan et al., 28 Aug 2025) Do triangles matter? Replicating hypergraph disease dynamics with lower-order interactions
• (Higham et al., 2021) Epidemics on Hypergraphs: Spectral Thresholds for Extinction
• (Kim et al., 2023) Contagion dynamics on hypergraphs with nested hyperedges
• (St-Onge et al., 2021) Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs
• (Xu et al., 2022) Dynamics of the threshold model on hypergraphs
• (Landry et al., 2020) The effect of heterogeneity on hypergraph contagion models
• (Higham et al., 2021) Mean Field Analysis of Hypergraph Contagion Model
• (Arruda et al., 2020) Phase transitions and stability of dynamical processes on hypergraphs
• (Arruda et al., 2019) Social contagion models on hypergraphs
• (Cisneros-Velarde et al., 2020) Multi-group SIS Epidemics with Simplicial and Higher-Order Interactions
• (Zhang et al., 2023) Influence Maximization based on Simplicial Contagion Models in Hypergraphs
• (Liang et al., 2024) Discrete-time SIS Social Contagion Processes on Hypergraphs
• (Malizia et al., 15 Jan 2026) Nested hyperedges promote the onset of collective transitions but suppress explosive behavior
• (Jhun et al., 2019) Simplicial SIS model in scale-free uniform hypergraph
• (Burgio et al., 2023) Adaptive hypergraphs and the characteristic scale of higher-order contagions using generalized approximate master equations
• (Arruda et al., 2021) Multistability, intermittency and hybrid transitions in social contagion models on hypergraphs
• (Burgio et al., 2023) A triadic approximation reveals the role of interaction overlap on the spread of complex contagions on higher-order networks
• (St-Onge et al., 2023) Nonlinear bias toward complex contagion in uncertain transmission settings
• (Burgio et al., 20 May 2025) Characteristic scales and adaptation in higher-order contagions
• (Kim et al., 2022) Higher-Order Components Dictate Higher-Order Contagion Dynamics in Hypergraphs