Tractably modelling dependence in networks beyond exchangeability

Abstract

We propose a general framework for modelling network data that is designed to describe aspects of non-exchangeability, with an explicit parameter describing the degree of non-exchangeability. Conditional on latent (unobserved) variables, the edges of the network are generated by their finite growth history (via a latent order) while the marginal probabilities of the adjacency matrix are modeled by a generalization of a graph limit function (or a graphon). In particular, we study the estimation, clustering and degree behavior of the network in this setting. We determine (i) the least squares estimator of a composite graphon attaining the minimax rate under weak dependence with respect to squared error loss; (ii) that spectral clustering is able to consistently detect the latent membership when the block-wise constant composite graphon is considered under additional conditions; and (iii) we are able to construct models with heavy-tailed empirical degrees under specific scenarios and parameter choices. We find conditions under which the spectral clustering is consistent under non-exchangeability, revealing that the application scope of classification can be broader than classic $i.i.d$. or exchangeable assumptions. In aggregate, we explore why and under which general conditions non-exchangeable network data can be described by a stochastic block model. The new modelling framework is able to capture empirically important characteristics of network data such as sparsity combined with heavy tailed degree distribution, and add understanding as to what generative mechanisms will make them arise.