# Sparse graphs using exchangeable random measures

@article{Caron2017SparseGU,
title={Sparse graphs using exchangeable random measures},
author={François Caron and Emily B. Fox},
journal={Journal of the Royal Statistical Society. Series B, Statistical Methodology},
year={2017},
volume={79},
pages={1295 - 1366}
}
• Published 6 January 2014
• Computer Science
• Journal of the Royal Statistical Society. Series B, Statistical Methodology
Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this array—which can aid in modelling, computations and theoretical analysis—the Aldous–Hoover theorem informs us that the graph is necessarily either dense or empty. We instead consider representing the graph as an exchangeable random measure and appeal to the Kallenberg representation theorem for this object. We explore using completely…
165 Citations

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### Sampling and Estimation for (Sparse) Exchangeable Graphs

• Mathematics, Computer Science
The Annals of Statistics
• 2019
The graphex framework is developed as a tool for statistical network analysis by identifying the sampling scheme that is naturally associated with the models of the framework, and by introducing a general consistent estimator for the parameter (the graphex) underlying these models.

### Edge exchangeable models for network data

• Computer Science
ArXiv
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It is observed that edges, not vertices, act as the statistical units in networks constructed from interaction data, making a theory of edge-labeled networks more natural for many applications, and gives rise to a class of nonparametric models, akin to graphon models in the vertex exchangeable setting.

### Edge-exchangeable graphs and sparsity

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NIPS
• 2016
It is shown how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering and other familiar combinatorial structures.

### The Class of Random Graphs Arising from Exchangeable Random Measures

• Mathematics, Computer Science
ArXiv
• 2015
A class of random graphs is introduced that meets many of the desiderata one would demand of a model to serve as the foundation for a statistical analysis of real-world networks, and is given a representation theorem via a straightforward specialization of Kallenberg's representation theorem.

### Private Graphon Estimation for Sparse Graphs

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NIPS
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This work designs algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members by outputting a node-differentially-private nonparametric block model approximation.

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J. Mach. Learn. Res.
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By generalizing the classical definition of graphons as functions over probability spaces to functions over $\sigma$-finite measure spaces, this work can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases.

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This work re-introduce the use of block-structure for network models obeying Kallenberg’s notion of exchangeability and thereby obtain a collapsed model which both admits the inference of block and edge inhomogeneity and performs well on real network datasets.

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