Logit Graph¶

Fit, simulate, and compare the Logistic Random Graph (LG) model against ER / WS / BA using spectral GIC.

logit-graph is the Python package for the Logistic Random Graph (LG) model — a logistic random-graph model with neighborhood degree effects — with maximum-likelihood estimation, AIC-based model selection, and a scikit-learn-style API. It lets you:


Estimate	Pick the neighborhood radius `d̂` by AIC and the parameters `σ̂` (baseline) and `α̂` (degree effect) from a real graph
Generate	Run the LG iterative generation algorithm so the normalized-Laplacian spectrum matches the data
Compare	Rank LG against Erdős–Rényi, Watts–Strogatz, and Barabási–Albert on the same spectral GIC

Reference

This package accompanies Ottoni, Takahashi & Fujita, A Logistic Random Graph Model with Neighborhood Degree Effects (IMA Journal of Complex Networks, 2026), and the corresponding master's thesis. Notation and model names below follow the paper.

Installation¶

pip install "logit-graph>=0.1.3"

Quickstart¶

The paper-consistent pipeline is simulate_graph → select_d_ensemble → estimate_sigma_from_graph → GraphModelComparator:

from logit_graph import simulate_graph, select_d_ensemble, estimate_sigma_from_graph

# Generate a graph at a known (n, d, sigma)
adj = simulate_graph(n=200, d=1, sigma=-4.0, n_iter=30_000,
                     feature_mode="incremental", target_density=0.10, seed=42)

# Recover the neighborhood radius d by AIC, then the baseline sigma
d_hat, aic_stats = select_d_ensemble(graphs=[adj], d_candidates=[0, 1, 2, 3])
sigma_hat = estimate_sigma_from_graph(adj, d=d_hat, feature_mode="incremental")

Where to next¶

API Guide — task-oriented walkthrough of the main entry points with examples.
API Reference — full reference generated from the source docstrings.
Connectomes case study — applying LG to real brain networks.

The model (paper notation)¶

Edge probability via the logistic link (Eq. 3.1):

p_ij = logistic(η_ij), η_ij = σ + α·[g(S_i) + g(S_j)], g(s) = log(1 + s)

where S_i = Σ_{l ∈ N_d(i)} k_l is the neighborhood degree sum over the d-hop ball N_d(i) = {l : dist(i,l) ≤ d}, σ is the baseline connection propensity (baseline log-odds, typically negative), and α ≥ 0 is the strength of the neighborhood degree effect.

The radius d controls how far the degree feature reaches: d=0 uses each node's own degree only (N_0(i)={i}, so S_i = k_i); d=1 adds the neighbors' degrees; larger d aggregates over a wider ball. d is selected from the data by AIC (select_d_ensemble). The iterative generation algorithm draws each pair from its predetermined value on the lagged graph G(t-1) (Eq. 3.5), which is also what makes maximum-likelihood estimation of (σ, α) consistent.

Extension (§3.7). Any pairwise features can be appended: logit p_ij = σ + Σₖ θₖ·φₖ(i,j). The unified five-feature LG model of the thesis adds coarse/fine community indicators (γc, γf) and a latent-proximity feature (λ) to the degree term.

Models are ranked by GIC = 2·KL + 2·|θ| (lower is better), where KL is the Kullback–Leibler divergence between the normalized-Laplacian spectral densities of the observed and generated graphs and |θ| is the number of free parameters; the un-penalized KL is reported as the goodness-of-fit.