From Graph Centrality to Data Depth

Eddie Aamari (1, 2, 3, 4) , Ery Arias-Castro (5) , Clément Berenfeld (6, 2, 7)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 CNRS - Centre National de la Recherche Scientifique
3 UFR Mathématiques UPCité - UFR Mathématiques [Sciences] - Université Paris Cité
4 SU - Sorbonne Université
5 UC San Diego - University of California [San Diego]
6 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
7 Université Paris Dauphine-PSL

Given a sample of points in a Euclidean space, we can define a notion of depth by forming a neighborhood graph and applying a notion of centrality. In the present paper, we focus on the degree, iterates of the H-index, and the coreness, which are all well-known measures of centrality. We study their behaviors when applied to a sample of points drawn i.i.d. from an underlying density and with a connectivity radius properly chosen. Equivalently, we study these notions of centrality in the context of random neighborhood graphs. We show that, in the large-sample limit and under some standard condition on the connectivity radius, the degree converges to the likelihood depth (unsurprisingly), while iterates of the H-index and the coreness converge to new notions of depth.

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Eddie Aamari, Ery Arias-Castro, Clément Berenfeld. From Graph Centrality to Data Depth. 2021. ⟨hal-03220416⟩