# 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} }

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

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

- Mathematics, Computer ScienceThe 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.

### Random networks, graphical models and exchangeability

- Mathematics
- 2017

We study conditional independence relationships for random networks and their interplay with exchangeability. We show that, for finitely exchangeable network models, the empirical subgraph densities…

### Edge exchangeable models for network data

- Computer ScienceArXiv
- 2016

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.

### Network representation using graph root distributions

- Mathematics, Computer ScienceThe Annals of Statistics
- 2018

This work develops a new parameterization for general exchangeable random graphs, where the nodes are independent random vectors in a linear space equipped with an indefinite inner product, and the edge probability between two nodes equals the inner product of the corresponding node vectors.

### Edge-exchangeable graphs and sparsity

- Computer Science, MathematicsNIPS
- 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 ScienceArXiv
- 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

- Computer Science, MathematicsNIPS
- 2015

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.

### Sparse Exchangeable Graphs and Their Limits via Graphon Processes

- MathematicsJ. Mach. Learn. Res.
- 2017

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.

### Completely random measures for modelling block-structured networks

- Computer Science
- 2015

This work re-introduce the use of block-structure for network models obeying Kallenberg's notion of exchangeability and thereby obtain a model which admits the inference of block -structure and edge inhomogeneity and derives a simple expression for the likelihood and an efficient sampling method.

### Exchangeable random measures for sparse and modular graphs with overlapping communities

- Computer ScienceJournal of the Royal Statistical Society: Series B (Statistical Methodology)
- 2020

It is shown that the approach proposed can recover interpretable structure of real world networks and can handle graphs with thousands of nodes and tens of thousands of edges.

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

- Mathematics, Computer ScienceThe 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 ScienceArXiv
- 2016

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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- 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 ScienceArXiv
- 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

- Computer Science, MathematicsNIPS
- 2015

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.

### Sparse Exchangeable Graphs and Their Limits via Graphon Processes

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- 2017

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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This work re-introduce the use of block-structure for network models obeying Kallenberg's notion of exchangeability and thereby obtain a model which admits the inference of block -structure and edge inhomogeneity and derives a simple expression for the likelihood and an efficient sampling method.

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