Emerging Activity Temporal Hypergraph (EATH)

• EATH is a generative model that replicates time-varying group interactions in complex systems by integrating node-level stochastic processes and memory in group formation.
• It simulates realistic temporal, topological, and higher-order dynamics using empirical group size distributions and tunable parameters based on real data.
• The framework enables controlled surrogate benchmark generation and the simulation of dynamical processes like contagion spread and collective phenomena.

The Emerging Activity Temporal Hypergraph (EATH) is a generative model and formalism designed to replicate and analyze the dynamics of time-varying group interactions in complex systems. Developed to address limitations in data availability and to enable controlled simulations for dynamical processes on higher-order networks, the EATH framework introduces a parameterizable model that reproduces the temporal, topological, and higher-order structure observed in empirical temporal hypergraph datasets. Its construction incorporates node-level stochastic activity processes, group formation mechanisms with memory, and tunable parameters drawn from real data, enabling the synthesis of surrogate time-varying hypergraphs suitable for benchmarking, theoretical analysis, and the paper of contagion and collective phenomena.

1. Formal Definition and Model Architecture

The EATH model generates a sequence of temporal hypergraph snapshots, each representing simultaneous group (hyperedge) interactions among a population of nodes. Let $\mathscr{H} = \{\mathcal{H}_t\}_{t=1}^{T/\delta t}$ denote the temporal hypergraph, with each snapshot $\mathcal{H}_t$ comprising active hyperedges formed at time $t$.

Components

• Nodes: Each node $i$ is assigned intrinsic properties (persistence activity, instantaneous activity, order propensity).
• Node Activity Dynamics: Each node independently alternates between low-activity and high-activity states, governed by parameters controlling activity persistence and external/system-level modulation.
• Emergence of System Activity: The global group interaction activity at each step emerges from the sum of node activities, normalized to match a specified average.
• Group (hyperedge) Formation: At each time, hyperedges are created by sampling group sizes from an empirical or specified distribution, assigning nodes probabilistically based on their activity, group size preference, and memory mechanisms that encode past co-participation.

Mathematical Representation

• Node activity at time $t$:

$$a_t(i) = \begin{cases} a_h(i) & i \in h(t) \ \gamma a_h(i) & i \in l(t) \end{cases}$$

where $h(t)$, $l(t)$ are the high- and low-activity node sets, $a_h(i)$ is the instantaneous active degree, and $\gamma \ll 1$.

• State transition probabilities:

$$r_{l \rightarrow h}(i, t) = \Lambda_t \varrho_l \frac{a_T(i)}{\langle a_T(i) \rangle}$$

$$r_{h \rightarrow l}(i, t) = (1 - \Lambda_t) \varrho_h$$

where $a_T(i)$ is the persistence activity, $\Lambda_t$ is an exogenous modulation (e.g., reflecting daily schedule), $\varrho_l$, $\varrho_h$ control typical durations of the active states.

• Number of active hyperedges at $t$:

$E_t = \lambda \frac{A_t}{\langle A_t \rangle}$

with $A_t = \sum_{i \in \mathcal{V}} a_t(i)$, and $\lambda$ the mean group activity parameter.

• Hyperedge (group) sizes: Sampled from a target or empirical distribution $\Psi(m)$. Nodes are drawn according to their current $a_t(i)$ and order propensity $\varphi_i(m)$.

2. Emergence Mechanisms and Memory

Emergent activity in the system is generated through mechanisms that reflect both randomness and memory in group formation:

• Random formation (probability $p$): New hyperedges are formed by sampling active nodes proportional to their instantaneous activity and group size preference.
• Short-term memory (probability $1-p$): Existing groups are continued or modified based on their recent existence, mimicking observed group stability and reactivation.
• Long-term memory: Node pairs with higher historical co-participation (encoded in a fixed memory matrix $\omega_0(i,j)$) are favored when forming new groups, reproducing realistic temporal recurrences in empirical datasets.

3. Fit to Empirical Data and Validation

Parameters of the EATH model can be directly estimated from empirical face-to-face interaction or activity datasets:

• Node-level properties: Distributions of degree, hyperdegree, and participation frequencies across group sizes are closely reproduced by surrogates generated via EATH.
• Temporal patterns: Burstiness, cyclicity, and variations in activity timelines are mimicked by adjusting input parameters ($\Lambda_t$, $a_T(i)$, $a_h(i)$) to empirical analogs.
• Memory effects: Surrogates generated with memory mechanisms display realistic group lifetimes, recurrence statistics, and the correlation of group reactivation, matching measured properties from schools, conferences, and hospital data.

Statistical agreement is quantified with measures such as the burstiness parameter $\Delta B$, Pearson's correlations across node participation ranks, and distributional comparisons of topological observables.

4. Higher-Order Contagion and Dynamical Process Simulation

The EATH model is employed to generate synthetic temporal hypergraphs for simulations of dynamical processes sensitive to higher-order structure:

• Nonlinear contagion dynamics: Simulations (using discrete-time nonlinear SIR models) yield outbreak sizes and basic reproduction numbers ($R_0, R_\infty$) in close agreement with original datasets, provided memory mechanisms are included. Memoryless versions (lacking group/churn recurrence) systematically overestimate epidemic sizes, highlighting the structural importance captured by EATH.
• Temporal alignment of epidemic peaks: Synthetic activities preserve both the timing and collective surge properties of real data, suitable for scenario testing and theoretical exploration.

5. Model Flexibility and Hybridization

EATH supports flexible generation of surrogate datasets with tunable or hybrid properties:

• Parameterization: All stochastic process parameters (activity levels, group size distributions, memory matrices) can be fit from a single empirical context, merged from multiple sources, or set synthetically.
• Hybrid hypergraphs: By combining population-level properties from one dataset with context-level activity patterns from another (“hybrid substitution” or “hybrid union”), EATH can synthesize activity scenarios with controlled differences—allowing for hypothesis-driven comparisons of group interaction regimes.
• Scalability: Model structure ensures scalability to large populations and time windows due to local node processes and polynomial constraint on group sizes.

6. Implications and Applications

The EATH model is applicable to a range of domains where empirical collection of high-resolution group interaction data is difficult, incomplete, or costly:

• Synthetic benchmark generation: EATH enables the systematic testing of algorithms and hypotheses on realistic, tunable surrogates.
• Analysis of dynamical processes: Surrogates generated with EATH make it possible to separate the contributions of temporal patterns, group memory, and node propensities to phenomena such as epidemic spreading, opinion dynamics, or social reinforcement.
• Hybrid scenario construction: By decoupling context and agent-level parameters, EATH allows evaluation of intervention, population mixing, or behavioral change on macro-level emergent outcomes.
• Generalization and future developments: The EATH framework can be extended to include additional features such as explicit community structure, different group aggregation/disaggregation rules, or coupling with agent-based models for domain-specific analysis.

7. Summary Table: Key Properties and Mechanisms

Aspect	EATH Feature / Mechanism	Empirical Matching
Node-level diversity	Heterogeneous activity dynamics (persistence and instantaneous)	Preserves burstiness, degree/strength distribution
Group formation	Tunable size distribution, order propensity, memory-based construction	Matches group size and lifetimes
Temporal structure	Exogenous cycles, per-node state transitions	Replicates system-level periodicity
Short/long-term memory	Recent and historical co-participation matrices	Produces realistic group recurrence
Hybridization	Input flexibility for population/activity separation	Enables transfer/hybrid experiments

8. Future Directions

• Extension to explicit community structure may allow modeling domain-specific features such as persistent teams or latent classes.
• Incorporation of richer inter/intra-group correlations, e.g., explicit modeling of aggregation/disaggregation processes or stochastic block models, is plausible for further realism.
• Broader deployment for benchmarking, rare event simulation, and control-theoretic studies is anticipated where empirical data is unavailable or privacy-constrained.

EATH thus serves as a bridge between empirical high-resolution group interaction data and tractable synthetic temporal hypergraphs, capturing the interplay between temporal, topological, and memory-driven effects in the emergence of collective dynamical phenomena.