Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this arraywhich can aid in modelling, computations and theoretical analysisthe 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 random measures (CRMs) to define the exchangeable random measure, and we show how our CRM construction enables us to achieve sparse graphs while maintaining the attractive properties of exchangeability. We relate the sparsity of the graph to the Levy measure defining the CRM. For a specific choice of CRM, our graphs can be tuned from dense to sparse on the basis of a single parameter. We present a scalable Hamiltonian Monte Carlo algorithm for posterior inference, which we use to analyse network properties in a range of real data sets, including networks with hundreds of thousands of nodes and millions of edges.

Sparse graphs using exchangeable random measures

CASARIN, Roberto;IACOPINI, MATTEO;ROSSINI, LUCA
2017-01-01

Abstract

Statistical network modelling has focused on representing the graph as a discrete structure, namely the adjacency matrix. When assuming exchangeability of this arraywhich can aid in modelling, computations and theoretical analysisthe 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 random measures (CRMs) to define the exchangeable random measure, and we show how our CRM construction enables us to achieve sparse graphs while maintaining the attractive properties of exchangeability. We relate the sparsity of the graph to the Levy measure defining the CRM. For a specific choice of CRM, our graphs can be tuned from dense to sparse on the basis of a single parameter. We present a scalable Hamiltonian Monte Carlo algorithm for posterior inference, which we use to analyse network properties in a range of real data sets, including networks with hundreds of thousands of nodes and millions of edges.
File in questo prodotto:
File Dimensione Formato  
DiscussionFinal.pdf

non disponibili

Descrizione: jrssbrandgraph
Tipologia: Documento in Pre-print
Licenza: Accesso chiuso-personale
Dimensione 304.95 kB
Formato Adobe PDF
304.95 kB Adobe PDF   Visualizza/Apri

I documenti in ARCA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10278/3691785
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 108
  • ???jsp.display-item.citation.isi??? 93
social impact