Tractably Modelling Dependence in Networks Beyond Exchangeability
This work addresses the challenge of capturing realistic network characteristics like sparsity and heavy-tailed degrees for researchers in statistical network analysis, though it is incremental in extending existing graphon-based methods.
The authors tackled the problem of modeling non-exchangeable network data by proposing a framework that generalizes graphons and incorporates latent growth histories, achieving minimax estimation of composite graphons, consistent spectral clustering under certain conditions, and the ability to model heavy-tailed degree distributions.
We propose a general framework for modelling network data that is designed to describe aspects of non-exchangeable networks. Conditional on latent (unobserved) variables, the edges of the network are generated by their finite growth history (with latent orders) 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 our setting. We determine (i) the minimax estimator of a composite graphon 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. This explores 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. Keywords: statistical network analysis, exchangeable arrays, stochastic block model, nonlinear stochastic processes.