pyregg.ntg — Number of Triangles

Number of Triangles (NTG)

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

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

Estimate P(NTG(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 {NTG(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(NTG(G(X)) ≤ level).
rel_variance (float) – Estimated relative variance of the estimator.
n_samples (int) – Number of samples used.

Examples

>>> import pyregg.ntg as ntg
>>> Z, RV, n = ntg.naive_mc(wind_len=10, kappa=0.5, int_range=1.0, level=0)

pyregg.ntg.conditional_mc(wind_len, kappa, int_range, level, max_iter=100000000, warm_up=1000, tol=0.001, *, seed=None)[source]

Estimate P(NTG(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 {NTG(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(NTG(G(X)) ≤ level).
rel_variance (float) – Estimated relative variance of the estimator.
n_samples (int) – Number of samples used.

Examples

>>> import pyregg.ntg as ntg
>>> Z, RV, n = ntg.conditional_mc(wind_len=10, kappa=0.5, int_range=1.0, level=0)

pyregg.ntg.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(NTG(G(X)) ≤ level) using Importance Sampling Monte Carlo.

Uses a two-phase blocking strategy: no blocking while the triangle count is below the threshold; once the threshold is reached, cells that would create a new triangle 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 {NTG(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(NTG(G(X)) ≤ level).
rel_variance (float) – Estimated relative variance of the IS estimator.
n_samples (int) – Number of IS samples used.

Examples

>>> import pyregg.ntg as ntg
>>> Z, RV, n = ntg.importance_sampling(wind_len=10, kappa=0.5, int_range=1.0,
...                                    level=0, grid_res=10)