Information is localized in growing network models

TL;DR

Mechanistic growing network models often yield intractable likelihoods; this work proves that, for a broad class of such models, the likelihood factors through local subgraphs, i.e., the informative content about parameters is localized within $2k+1$-hop receptive fields around each growth step. Building on this, the authors develop neural posterior density estimators (NDEs) based on graph neural networks with limited receptive fields to approximate $p(\theta|G)$ from observed graphs, enabling Bayesian inference without observing full histories. They analyze nine growing-network models, showing that localization predictions match NDE-based posteriors on simulated data; for non-localized models, NDEs still recover high-fidelity posteriors at a fraction of the cost. The results justify analyzing local subgraphs embedded in larger networks and offer a practical, fast, largely model-agnostic inference framework for mechanistic network growth.

Abstract

Mechanistic network models can capture salient characteristics of empirical networks using a small set of domain-specific, interpretable mechanisms. Yet inference remains challenging because the likelihood is often intractable. We show that, for a broad class of growing network models, information about model parameters is localized in the network, i.e., the likelihood can be expressed in terms of small subgraphs. We take a Bayesian perspective to inference and develop neural density estimators (NDEs) to approximate the posterior distribution of model parameters using graph neural networks (GNNs) with limited receptive size, i.e., the GNN can only "see" small subgraphs. We characterize nine growing network models in terms of their localization and demonstrate that localization predictions agree with NDEs on simulated data. Even for non-localized models, NDEs can infer high-fidelity posteriors matching model-specific inference methods at a fraction of the cost. Our findings establish information localization as a fundamental property of network growth, theoretically justifying the analysis of local subgraphs embedded in larger, unobserved networks and the use of GNNs with limited receptive field for likelihood-free inference.

Figures (2)

• Figure 1: Illustration of network growth for a probabilistic redirection model Krapivsky2001. At each step $t$, a new node $\sigma_t$ is added, and a single seed node $s$ is sampled uniformly at random. The new node is connected to $s$ with probability $1-\theta$ and to one of its uniformly chosen neighbors otherwise. New connections only depend on the direct neighborhood $B_{s}^{(1)}$ of the seed $s$, and the model is 1-localized. The receptive field of $\sigma_t$ is its three-hop subgraph $B^{(3)}_{\sigma_t}$. Nodes outside the receptive field cannot affect the new node and vice-versa.
• Figure 2: Neural density estimators can learn high fidelity posteriors. Each panel shows an estimate of the variational mutual information $\mathcal{I}$ as a function of GIN depth $\ell$, based on a test set of 1,000 graphs for each of the models in \ref{['tbl:models']}. The larger $\mathcal{I}$, the better the inference. Five of the nine models are localized, and dotted vertical lines indicate the expected receptive field by which we expect $\mathcal{I}$ to saturate. Dashed horizontal lines correspond to posterior inference for models with tractable or approximated likelihood. Error bars are bootstrapped 95% confidence intervals and are smaller than markers.