Sample ensemble magnitudes

TL;DR

Fit a null model once, sample many null networks, compute the same scalar magnitude on every sample, and compare the observed value to the null ensemble.

How many samples?

Use 100 samples for a quick check and 1000+ samples for reported analyses. Increase the count when tail percentiles matter.

Python example

import numpy as np

from menobis.analysis import compute_all_stats
from menobis.models import Constraint, ModelFamily, fit_model, sample_model
from menobis.utilities.synthetic import (
    derive_synthetic_constraints,
    generate_pa_geographic_network,
)

network = generate_pa_geographic_network(30, average_degree=6.0, seed=12)
constraints = derive_synthetic_constraints(network)

fit = fit_model(
    family=ModelFamily.ME,
    constraint=Constraint.STRENGTH,
    strength_out=constraints.strength_out,
    strength_in=constraints.strength_in,
    self_loops=False,
)


def mean_y2(edges):
    stats = compute_all_stats(edges)
    return float(np.mean(stats.y2_out))

observed = mean_y2(network.edges)
null_values = np.array([
    mean_y2(sample_model(
        family=ModelFamily.ME,
        constraint=Constraint.STRENGTH,
        fit=fit,
        seed=seed,
    ))
    for seed in range(100)
])

print(observed, null_values.mean(), np.percentile(null_values, [2.5, 97.5]))

Magnitudes to start with

Magnitude	MENoBiS helper
Y₂ disparity	compute_all_stats(edges).y2_out or .y2_in
Average nearest-neighbour strength	compute_all_stats(edges).s_nn_out or .s_nn_in
Clustering	clustering_coefficient(edges) or weighted_clustering_coefficient(edges)
Occupation histogram	weight_distribution(edges)

Interpret the comparison

Observed value	Interpretation
inside central interval	magnitude is compatible with the chosen constraints
above upper percentile	magnitude is larger than expected under the null
below lower percentile	magnitude is smaller than expected under the null

Null-model dependence

A magnitude can look surprising under strength-only constraints and become typical after adding cost or degree constraints. Choose constraints before interpreting significance.