pyregg.md — Maximum Degree

Maximum Degree (MD)

Rare-event estimation for the probability that the maximum degree of a Gilbert random geometric graph G(X) does not exceed a threshold ℓ:

P(MD(G(X)) ≤ ℓ)

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

Estimate P(MD(G(X)) ≤ level) using Naïve Monte Carlo.

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.

  • level (int) – Threshold ℓ. The rare event is {MD(G(X)) ≤ level}.

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

  • warm_up (int, optional) – Minimum 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(MD(G(X)) ≤ level).

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

  • n_samples (int) – Number of samples used.

Examples

>>> import pyregg.md as md
>>> Z, RV, n = md.naive_mc(wind_len=10, kappa=0.65, int_range=1.0, level=2)
pyregg.md.conditional_mc(wind_len, kappa, int_range, level, max_iter=100000000, warm_up=1000, tol=0.001, *, seed=None)[source]

Estimate P(MD(G(X)) ≤ level) using Conditional Monte Carlo.

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.

  • level (int) – Threshold ℓ. The rare event is {MD(G(X)) ≤ level}.

  • 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(MD(G(X)) ≤ level).

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

  • n_samples (int) – Number of samples used.

Examples

>>> import pyregg.md as md
>>> Z, RV, n = md.conditional_mc(wind_len=10, kappa=0.65, int_range=1.0, level=2)
pyregg.md.importance_sampling(wind_len, kappa, int_range, level, grid_res=10, max_iter=100000000, warm_up=100, tol=0.001, *, seed=None)[source]

Estimate P(MD(G(X)) ≤ level) using Importance Sampling Monte Carlo.

Cells adjacent to any existing node with degree ≥ level, or with more than level existing neighbours, are blocked. 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.

  • level (int) – Threshold ℓ. The rare event is {MD(G(X)) ≤ level}.

  • 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(MD(G(X)) ≤ level).

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

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

Examples

>>> import pyregg.md as md
>>> Z, RV, n = md.importance_sampling(wind_len=10, kappa=0.65, int_range=1.0,
...                                   level=2, grid_res=10)