Sampling perspectives on sparse exchangeable graphs

Abstract

Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the statistical modeling of network data, the theory of large graph limits, exchangeability, and network sampling. The purpose of the present paper is to clarify the relationships between these subjects by explaining each in terms of a certain natural sampling scheme associated with the graphex model. The first main technical contribution is the introduction of sampling convergence, a new notion of graph limit that generalizes left convergence so that it becomes meaningful for the sparse graph regime. The second main technical contribution is the demonstration that the (somewhat cryptic) notion of exchangeability underpinning the graphex framework is equivalent to a more natural probabilistic invariance expressed in terms of the sampling scheme.

45 pages, 1 figure

References in corpus (3)

• Graph limits and exchangeable random graphs
• On exchangeable random variables and the statistics of large graphs and hypergraphs
• Graphons and cut metric on sigma-finite measure spaces

Cited by in corpus (11)

• Exponential-Family Models of Random Graphs: Inference in Finite-, Super-, and Infinite Population Scenarios
• On sparsity, power-law and clustering properties of graphex processes
• Nonparametric Modeling of Higher-Order Interactions via Hypergraphons
• Next Waves in Veridical Network Embedding
• Empirical Risk Minimization and Stochastic Gradient Descent for Relational Data
• A Nonparametric Bayesian Model for Sparse Dynamic Multigraphs
• Asymptotics of Network Embeddings Learned via Subsampling
• Exchangeable modelling of relational data: checking sparsity, train-test splitting, and sparse exchangeable Poisson matrix factorization
• Asymptotic Behavior of Common Connections in Sparse Random Networks
• Robustness on Networks
• Tractably Modelling Dependence in Networks Beyond Exchangeability