Graphons and cut metric on sigma-finite measure spaces

2016/08/05 by Svante Janson, Janson, Svante
Computer Science · Mathematics · #05C80 #05C99 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1608.01833

openalex publication_date 2016/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Borgs, Chayes, Cohn and Holden (2016+) recently extended the definition of graphons from probability spaces to arbitrary σ-finite measure spaces, in order to study limits of sparse graphs. They also extended the definition of the cut metric, and proved various results on the resulting metric space. We continue this line of research and give various further results on graphons and the cut metric in this general setting, extending known results for the standard case of graphons on probability spaces. In particular, we characterize pairs of equivalent graphons, and we give new results on completeness and compactness.

Citations

• Completely random measures for modelling block-structured networks
• The Class of Random Graphs Arising from Exchangeable Random Measures

Related

• Graphons, cut norm and distance, couplings and rearrangements
• The cut metric for probability distributions
• Identifiability for graphexes and the weak kernel metric
• Probability-graphons: Limits of large dense weighted graphs
• Regularity partitions and the topology of graphons