Efficient Gillespie algorithms for spreading phenomena in large and heterogeneous higher-order networks

[!NOTE] Code implemented using the Fortran Package Manager.

Main paper: Efficient Gillespie algorithms for spreading phenomena in large and heterogeneous higher-order networks, by Hugo P. Maia, Wesley Cota, Yamir Moreno, and Silvio C. Ferreira.

Reference: arxiv:2509.20174 DOI:10.48550/arXiv.2509.20174

Hyper-SIS Dynamical Model

This code simulates SIS dynamics on hypergraphs (Hyper-SIS). Each of the $N$ agents can be either susceptible ($\sigma_i = 0$) or infected ($\sigma_i = 1$). Infections occur via hyperedges, which are active if a critical mass of members is infected, while infected nodes recover spontaneously.

Key points:

• Node recovery rate: $\alpha = 1$.
• Hyperedge activation threshold: $\theta(m) = 1 + (m-1)\theta_0$, where $m$ is the hyperedge order.
• Infection rate as a function of hyperedge order: $\beta(m) = \beta[1 + b(m-1)]$.
• Pairwise infection rate: $\beta(1) = \beta$.
• Parameters par_b and par_theta correspond to $b$ and $\theta_0$.

See the main paper for full details.

Using it as a Fortran dependency

Add this package as a dependency using the Fortran Package Manager (fpm):

[dependencies]
hyperSIS.git = "https://github.com/gisc-ufv/hyperSIS"

See the documentation and main program for details.

Python package

[!TIP] 💡 A Google Colab notebook demonstrating all installation and usage steps is available here.

The easiest way to use this project is through its Python interface.

Via PyPI (pip)

Use pip install hyperSIS to install it.

Build python package

In this case, you need to clone the repository manually.

Before installing, make sure that at least one Fortran compiler is available. By default, the package assumes GNU Fortran (gfortran) installed and available in your PATH. See Installing GFortran for help.

Steps:

1. Clone the repository and enter it:

   git clone https://github.com/gisc-ufv/hyperSIS.git
   cd hyperSIS
   

2. Activate your preferred Python environment (e.g., venv, conda, etc.):

   # Example with venv
   python -m venv venv
   source venv/bin/activate
   
   # Example with conda
   conda create -n hyperSIS python=3.11
   conda activate hyperSIS
   

3. Install the Python package:

   pip install ./python
   

   • If you want to use another compiler and/or Fortran flags, set the FC and FFLAGS.

   # Optional: customize the Fortran compiler and flags
   export FC=gfortran # default is gfortran
   export FFLAGS="-O3 -march=native -funroll-loops" # adjust optimization flags
   pip install ./python
   

Usage (Python)

[!TIP] 💡 See examples.ipynb and Google Colab for examples.

Import the package with

import hyperSIS as hs

The simulation interface revolves around two main objects:

1. SimulationArgs A dataclass containing all parameters required to configure a hyperSIS simulation, including network specification, algorithm choices, temporal settings, initial conditions, and epidemic parameters.

2. run_simulation(beta1: float, args: SimulationArgs) The function that executes the simulation with the given arguments. Returns a SimulationResult object containing the processed results, including network mapping, temporal evolution, and statistics of infected nodes.

Simulation arguments

The SimulationArgs dataclass contains all configurable parameters for running a hyperSIS simulation.

• verbose: bool

  • Enable verbose output.
  • Default: True

• verbose_level: str

  • Logging level: 'info', 'warning', 'error', 'debug'.
  • Default: warning

• seed: int

  • Random seed for reproducibility.
  • Default: 42

• remove_files: bool

  • Remove temporary files after execution.
  • Default: False

• network: NetworkFormat

  • Network specification as a tuple. Optional parameters are in brackets:
    • ("edgelist", path, [delimiter], [comment], [cache])
    • ("fortran-edgelist", path, [cache])
    • ("bipartite", path, [delimiter], [comment], [cache])
    • ("xgi", name_or_object, [cache])
    • ("xgi_json", path, [cache])
    • ("hif", path, [cache])
    • ("PL", gamma, N, [sample])
    • ("networkx", nx.Graph, [cache])
  • Default: ("PL", 3.0, 100, 1)

• output_dir: Optional[str]

  • Directory to store simulation output. If None, a temporary folder is used.
  • Default: None

• algorithm: str

  • Simulation algorithm: 'HB_OGA' or 'NB_OGA'.
  • Default: HB_OGA

• sampler: str

  • Sampling method: 'rejection_maxheap' or 'btree'.
  • Default: btree

• tmax: int

  • Maximum simulation time.
  • Default: 100

• use_qs: bool

  • Whether to use the quasi-stationary method.
  • Default: False

• n_samples: int

  • Number of samples per simulation.
  • Default: 10

• time_scale: str

  • Temporal scale for output: 'uniform' or 'powerlaw'.
  • Default: uniform

• initial_condition: tuple

  • Initial state specification:
    • ('fraction', float) → fraction of infected nodes
    • ('number', int) → exact number of initially infected nodes
  • Default: ("fraction", 1.0)

• export_states: bool

  • Whether to export the full state trajectory.
  • Default: False

• build_xgi_hypergraph: bool

  • Whether to build and return the xgi hypergraph representation of the network.
  • Default: False

• par_b: float

  • Epidemic infection rate scale $b$ in $\beta(m) = \beta[1 + b(m-1)]$.
  • Default: 0.5

• par_theta: float

  • Epidemic critical mass threshold $\theta_0$ in $\theta(m) = 1 + (m-1)\theta_0$.
  • Default: 0.5

Function

run_simulation(beta1: float, args: SimulationArgs)

Runs a Hyper-SIS simulation on the specified network.

Parameters:

• beta1: float Base infection rate $\beta(1)$ for pairwise interactions.
• args: SimulationArgs Simulation parameters, including network specification, algorithm choice, number of samples, initial condition, and epidemic parameters par_b and par_theta.

Returns:

• SimulationResult Object containing:

  • network: NetworkFormat – the network specification used.
  • node_map: dict – mapping from original node IDs to Fortran node IDs.
  • temporal: TemporalResult – temporal dynamics with:
    • t: np.ndarray – mean time per Gillespie tick.
    • rho_avg: np.ndarray – mean number of infected nodes over all runs.
    • rho_var: np.ndarray – variance of infected nodes.
    • n_samples: int – number of runs where infection is non-zero.
    • active_states: Optional[dict] – detailed active states per sample and time (if requested), formatted as {sample_id: {time: {"nodes": [...], "edges": [...]}}}.
  • xgi_hypergraph: Optional[xgi.core.hypergraph.Hypergraph] – representation of the structure as an xgi hypergraph, if generated.

Fortran executable hyperSIS_sampling

Read hyperSIS_sampling.md for instructions.

How to Cite

[!IMPORTANT] When using this package, please cite the following paper:

Efficient Gillespie algorithms for spreading phenomena in large and heterogeneous higher-order networks, by Hugo P. Maia, Wesley Cota, Yamir Moreno, and Silvio C. Ferreira (2026)

Reference: Nature Communications DOI:10.1038/s41467-026-75402-0 Preprint: arxiv:2509.20174 DOI:10.48550/arXiv.2509.20174

The BibTeX entry is:

@article{Maia2026_hyperSIS,
   title={Efficient Gillespie algorithms for spreading phenomena in large and heterogeneous higher-order networks},
   ISSN={2041-1723},
   url={http://dx.doi.org/10.1038/s41467-026-75402-0},
   DOI={10.1038/s41467-026-75402-0},
   journal={Nature Communications},
   publisher={Springer Science and Business Media LLC},
   author={Maia, Hugo P. and Cota, Wesley and Moreno, Yamir and Ferreira, Silvio C.},
   year={2026},
   month=July }