Optimized Gillespie Algorithms

• Optimized Gillespie algorithms are enhanced stochastic simulation methods that incorporate phantom processes to efficiently model complex, higher-order contagion phenomena.
• They leverage local sampling and structural decomposition to cut computational overhead, achieving near-linear scaling compared to traditional quadratic methods.
• These techniques preserve the statistical exactness of continuous-time dynamics, enabling accurate simulations of epidemic and contagion processes on large networks.

Optimized Gillespie algorithms encompass a class of stochastic simulation methods that extend, enhance, or generalize the original Gillespie algorithm for simulating Markovian and non-Markovian dynamics in complex systems. These algorithms address algorithmic bottlenecks, improve scaling with system size, and enable efficient simulation of processes such as epidemic and contagion dynamics on large, heterogeneous, and higher-order networks. A core theme is the reduction of computational complexity—often achieved by the introduction of auxiliary or "phantom" processes, structural decomposition, and sampling optimizations—while strictly preserving the statistical exactness of the underlying continuous-time stochastic models.

1. Gillespie Algorithm and the Challenge of Higher-Order Networks

The classic Gillespie algorithm provides an exact, event-driven simulation scheme for continuous-time Markov processes. In its canonical form, the algorithm samples the time to the next event from an exponential distribution and then selects which event occurs based on current propensities that encode all allowed transitions. Each event typically requires recomputation or updating of the list of possible future events—a process that scales poorly in large, heterogeneous, or structurally elaborate networks.

Higher-order networks generalize standard graphs (pairwise interactions) to hypergraphs, in which hyperedges connect arbitrary-size groups of nodes. Epidemic or contagion dynamics on hypergraphs (e.g., group contagion with critical mass activation) may involve order-heterogeneous processes and many-body thresholds. The naive application of the Gillespie method to such systems is computationally prohibitive, with complexity often scaling as $\mathcal{O}(N^2)$ for network size $N$, especially as the number and order of hyperedges grow or the network structure becomes highly heterogeneous (Maia et al., 24 Sep 2025).

2. Phantom Processes and Local Sampling

Phantom processes represent a central algorithmic innovation facilitating dramatic efficiency gains. In standard Gillespie-based epidemic simulations, the event list is updated after each infection, recovery, or susceptible event; this involves scanning (in the worst case) all combinations of nodes and hyperedges. Optimized Gillespie algorithms introduce phantom processes—events that occur stochastically, are sampled and counted for time propagation, but leave the system state unchanged (e.g., an infection attempt directed toward an already-infected node).

By allowing infection "attempts" to target any node irrespective of whether the resulting state changes, the need to maintain or update the full list of actual (i.e., state-changing) events at every step is circumvented. Time advances according to all attempted events, and the distribution of events remains statistically exact as per the master equation, provided that acceptance/rejection probabilities are implemented correctly.

In the context of higher-order (hypergraph) dynamics, phantom processes are included both at the level of hyperedges—as in the hyperedge-based optimized Gillespie algorithm (HB-OGA)—and at the level of nodes—as in the node-based optimized Gillespie algorithm (NB-OGA) (Maia et al., 24 Sep 2025). In both cases, all possible (including failed) transition attempts contribute to the time increment.

3. Algorithmic Structure: Hyperedge- and Node-Based Variants

The two principal optimized Gillespie variants for higher-order networks are:

Variant	Central Data Structure	Core Optimization
HB-OGA	List of potentially active hyperedges	Infection attempts considered over all hyperedges; stateless infection attempts via phantom processes
NB-OGA	List of susceptible (“quiescent”) nodes	Node-centric sampling; each node's hyperedge participation precomputed; local acceptance rules via phantom processes

Hyperedge-based OGA (HB-OGA) maintains, for each hyperedge $h$ of order $m_h$, the current number of infected nodes and evaluates infection attempts as

$$G = \sum_{h \in \Lambda^{(H)}} \beta(m_h)\, \omega_h,$$

where $\omega_h = m_h + 1 - \theta(m_h)$, and $\theta(m_h)$ is the activation threshold. Each infection attempt samples a hyperedge proportionally to its activation rate, and then a target node; if the node is already infected, the attempt is a phantom process.

Node-based OGA (NB-OGA) samples susceptible nodes (quiescent with respect to at least one active hyperedge) at rates proportional to an overestimate of their total infection attempt rate. Each sampled attempt is accepted with probability determined by the actual number of active infectious hyperedges connected to the node; rejected samples constitute phantom processes.

This structural shift ensures that, for large and heterogeneous networks, the number of per-event updates becomes dependent only on local connectivity and not the global set of hyperedges or nodes. Auxiliary data (e.g., incident hyperedges per node, infection status per hyperedge) are precomputed and updated locally, significantly reducing computational overhead.

4. Complexity, Scaling, and Empirical Performance

Optimized Gillespie algorithms with phantom processes achieve nearly linear scaling in CPU time with respect to system size $N$, as compared to the standard algorithm's quadratic scaling. The following comparative summary outlines the key complexity benefits:

Algorithm	Time Complexity (per event)	Limiting Bottleneck
Standard Gillespie	$\mathcal{O}(N^2)$	Complete event list update across high-order, dense hypergraphs
HB-OGA / NB-OGA	Near $\mathcal{O}(N)$	Local updates of node/hyperedge participation and state tracking

Empirical studies in the paper demonstrate that, for networks with millions of nodes and orders of heterogeneity (e.g., power-law distributed hyperedge sizes, presence of “hyperblob” outlier structures), the optimized methods reduce simulation times by several orders of magnitude (Maia et al., 24 Sep 2025). Furthermore, both stationary and transient epidemic observables are statistically identical between optimized and standard implementations, validating the statistical exactness of the optimization.

5. Applications to Contagion and Beyond

Optimized higher-order Gillespie algorithms are directly applicable to the simulation of SIS-type (susceptible-infected-susceptible) epidemic models with group thresholds on hypergraphs. This allows for the exploration of phenomena such as catastrophic activation, discontinuous transitions, and hysteresis—dynamical features absent in traditional pairwise models.

The general framework is also extensible to other types of spreading and contagion processes, including SIR, SEIR, rumor spreading, and opinion formation dynamics that intrinsically involve group or higher-order interactions. The methods are robust to variations in both network structure (homogeneous vs. heterogeneous hyperedge distributions) and contagion dynamics (critical mass thresholds, non-pairwise activation functions).

6. Algorithmic Exactness, Limitations, and Future Directions

The inclusion of phantom processes ensures that time increments and event statistics remain exactly distributed as per the continuous-time master equation. This is achieved without adversarial impact on the measured macroscopic observables and with substantial reductions in running time.

A limitation, as identified in the data, is that for networks with extremely fat-tailed degree or order distributions, the use of naive rejection sampling in node-based algorithms can incur inefficiencies due to high rejection rates. The paper suggests further optimizations such as integration of alias sampling or binary tree techniques to lower the cost of sampling in such regimes (Maia et al., 24 Sep 2025).

Further research opportunities include extensions to:

• More complex dynamical rules (e.g., SEIR models, co-spreading, competing contagion)
• Improved data structures for extremely broad degree or order distributions
• Systematic treatment of rare-event sampling in the context of higher-order interactions
• Theoretical study of the impact of hyperedge structure on dynamical critical phenomena

7. Significance in Computational Network Science

Optimized Gillespie algorithms for higher-order dynamics represent a foundational advance in the efficient simulation of collective phenomena on complex networks. By reducing computational bottlenecks from global to local, and achieving linear or near-linear scaling, these methods make feasible the study of contagion in structurally realistic settings, such as social platforms, modular infrastructures, and neural assemblies, at the million-node scale and beyond.

This innovation fundamentally extends the scope of stochastic simulation, supports investigations into non-trivial emergent behaviors in higher-order models, and underpins advances in both epidemic forecasting and the broader theory of dynamical systems on networks (Maia et al., 24 Sep 2025).