pyregg.forest — Forest (Acyclicity)

Forest (Acyclicity)

Rare-event estimation for the probability that a Gilbert random geometric graph G(X) is a forest (acyclic, contains no cycle):

P(G(X) is a forest)

Three estimators are provided: Naïve Monte Carlo, Conditional Monte Carlo, and Importance Sampling Monte Carlo.

References

Hirsch, C., Moka, S. B., Taimre, T., & Kroese, D. P. (2022).

Rare events in random geometric graphs. Methodology and Computing in Applied Probability, 24, 1367–1383.

Moka, S., Hirsch, C., Schmidt, V., & Kroese, D. P. (2025).

Efficient rare-event simulation for random geometric graphs via importance sampling. arXiv:2504.10530.

pyregg.forest.naive_mc(wind_len, kappa, int_range, max_iter=100000000, warm_up=100000, tol=0.001, *, seed=None)[source]

Estimate P(G(X) is a forest) using Naïve Monte Carlo.

Generates independent realisations of the Gilbert graph and estimates the probability as the fraction that are acyclic (contain no cycle).

Parameters:

  • wind_len (float) – Side length of the square observation window [0, wind_len]².

  • kappa (float) – Intensity of the Poisson point process (expected points per unit area).

  • int_range (float) – Interaction range (connection radius) of the Gilbert graph. Two points are connected if their distance is at most int_range.

  • max_iter (int, optional) – Maximum number of samples. Default is 10**8.

  • warm_up (int, optional) – Minimum number of samples before checking convergence. Default is 100,000.

  • tol (float, optional) – Stop when estimated RV / n < tol. Default is 0.001.

  • seed (int, optional) – Integer seed for reproducibility (keyword-only). By default no seed is set and results are non-deterministic.

Returns:

  • probability (float) – Estimated rare-event probability P(G(X) is a forest).

  • rel_variance (float) – Estimated relative variance of the estimator.

  • n_samples (int) – Number of samples used.

Examples

>>> import pyregg.forest as forest
>>> Z, RV, n = forest.naive_mc(wind_len=10, kappa=0.3, int_range=1.0)
pyregg.forest.conditional_mc(wind_len, kappa, int_range, max_iter=100000000, warm_up=1000, tol=0.001, *, seed=None)[source]

Estimate P(G(X) is a forest) using Conditional Monte Carlo.

Points are added sequentially until the first cycle is created (step n). At each step the probability contribution is computed analytically via poisson.cdf(n-1, mu), yielding substantially lower variance than NMC.

Parameters:

  • wind_len (float) – Side length of the square observation window [0, wind_len]².

  • kappa (float) – Intensity of the Poisson point process (expected points per unit area).

  • int_range (float) – Interaction range (connection radius) of the Gilbert graph.

  • max_iter (int, optional) – Maximum number of samples. Default is 10**8.

  • warm_up (int, optional) – Minimum samples before checking convergence. Default is 1,000.

  • tol (float, optional) – Stop when estimated RV / n < tol. Default is 0.001.

  • seed (int, optional) – Integer seed for reproducibility (keyword-only). By default no seed is set and results are non-deterministic.

Returns:

  • probability (float) – Estimated rare-event probability P(G(X) is a forest).

  • rel_variance (float) – Estimated relative variance of the estimator.

  • n_samples (int) – Number of samples used.

Examples

>>> import pyregg.forest as forest
>>> Z, RV, n = forest.conditional_mc(wind_len=10, kappa=0.3, int_range=1.0)
pyregg.forest.importance_sampling(wind_len, kappa, int_range, grid_res=10, max_iter=100000000, warm_up=100, tol=0.001, *, seed=None)[source]

Estimate P(G(X) is a forest) using Importance Sampling Monte Carlo.

The window is partitioned into a uniform grid. A cell is blocked if its conservative grid-cell neighbourhood contains two existing nodes that already belong to the same connected component (i.e., placing a point there would create a cycle). Cycle detection uses a union-find data structure. A likelihood-ratio correction yields an unbiased estimate with substantially lower variance than CMC.

Parameters:

  • wind_len (float) – Side length of the square observation window [0, wind_len]².

  • kappa (float) – Intensity of the Poisson point process (expected points per unit area).

  • int_range (float) – Interaction range (connection radius) of the Gilbert graph.

  • grid_res (int, optional) – Number of grid cells per interaction-range interval. The window is divided into (wind_len / int_range × grid_res)² cells total. Default is 10.

  • max_iter (int, optional) – Maximum number of IS samples. Default is 10**8.

  • warm_up (int, optional) – Minimum samples before checking convergence. Default is 100.

  • tol (float, optional) – Stop when estimated RV / n < tol. Default is 0.001.

  • seed (int, optional) – Integer seed for reproducibility (keyword-only). By default no seed is set and results are non-deterministic.

Returns:

  • probability (float) – Estimated rare-event probability P(G(X) is a forest).

  • rel_variance (float) – Estimated relative variance of the IS estimator.

  • n_samples (int) – Number of IS samples used.

Examples

>>> import pyregg.forest as forest
>>> Z, RV, n = forest.importance_sampling(wind_len=10, kappa=0.3, int_range=1.0,
...                                       grid_res=10)