Approximate Bayesian Inference on Mechanisms of Network Growth and Evolution

TL;DR

This work tackles inference for mixtures of network growth mechanisms by proposing an edgewise, event-level mixture model and an approximate Bayesian framework based on a Graph Neural Network–Mixture Density Network (GNN-MDN). By mapping observed network structure to a conditional density over growth/evolution parameters, the method circumvents intractable likelihoods and delivers posterior estimates for mechanism weights and Poisson rates. Validation on simulated data demonstrates good posterior recovery and coverage, while application to real social networks reveals the importance of triangle-formation mechanisms and highlights limitations in capturing highly clustered communities. The approach offers a scalable, flexible pathway for mechanistic network inference and can be extended to incorporate additional mechanisms, covariates, or temporal dynamics.

Abstract

Mechanistic models can provide an intuitive and interpretable explanation of network growth by specifying a set of generative rules. These rules can be defined by domain knowledge about real-world mechanisms governing network growth or may be designed to facilitate the appearance of certain network motifs. In the formation of real-world networks, multiple mechanisms may be simultaneously involved; it is then important to understand the relative contribution of each of these mechanisms. In this paper, we propose the use of a conditional density estimator, augmented with a graph neural network, to perform inference on a flexible mixture of network-forming mechanisms. This event-wise mixture-of-mechanisms model assigns mechanisms to each edge formation event rather than stipulating node-level mechanisms, thus allowing for an explanation of the network generation process, as well as the dynamic evolution of the network over time. We demonstrate that our approximate Bayesian approach yields valid inferences for the relative weights of the mechanisms in our model, and we utilize this method to investigate the mechanisms behind the formation of a variety of real-world networks.

Figures (8)

• Figure 1: In a nodewise mixture-of-mechanisms model (a), mechanisms are assigned to each incoming node. In contrast, in an edgewise mixture-of-mechanisms model (b), each edge can be assigned to a different mechanism. In (c), a new node with 2 edges is to be added to the network, such that the number of triangles in the network is to be maximized. Under most growth models, edges can only be added between the new node and the existing network (top). By allowing for new edges between existing nodes, we can create one additional triangle (bottom), for a total of four triangles in this network
• Figure 2: A Graph Neural Network (GNN) and a Multilayer Perceptron (MLP), which serves as our Mixture Density Network (MDN), are trained in series to minimize Expected Posterior Entropy (EPE). The resultant output is a conditional density estimate, which approximates the posterior distribution of parameters $\phi_g$, $\phi_e$, $\tilde{\alpha}$, and $\tilde{\beta}$.
• Figure 3:
• Figure 4: Simulation results for three scenarios. For each scenario, we generated 10 realizations (replications) of the true network and utilized GNN-MDN to estimate the 95% credible intervals and medians. True values for each parameter are shown as dotted horizontal lines. Due to the stochasticity of the data generation process behind the true networks, the credible intervals associated with each realization are expected to be different.
• Figure 5: Coverage properties for parameters of interest, using posterior samples of GNN-MDN. If coverage properties are fulfilled, all points should lie on the identity line.