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Exchangeable graph-valued Feller processes

Abstract

The transition law of every exchangeable Feller process on the space of countable graphs is determined by a \(\sigma \)-finite measure on the space of \(\{0,1\}\times \{0,1\}\)-valued arrays. In discrete time, this characterization gives rise to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous time, the behavior is enriched by a Lévy–Itô–Khintchine-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Every such process almost surely projects to a Feller process in the space of graph limits.

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Correspondence to Harry Crane.

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Crane, H. Exchangeable graph-valued Feller processes. Probab. Theory Relat. Fields 168, 849–899 (2017). https://doi.org/10.1007/s00440-016-0726-0

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  • DOI: https://doi.org/10.1007/s00440-016-0726-0

Keywords

  • Exchangeability
  • Random graph
  • Graph limit
  • Lévy–Itô–Khintchine decomposition
  • Aldous–Hoover theorem
  • Erdős–Rényi random graph

Mathematics Subject Classification

  • 05C80
  • 60G09
  • 60J05
  • 60J25