Exchangeable interval hypergraphs and limits of ordered discrete structures

Abstract

A hypergraph (V, E) is called an interval hypergraph if there exists a linear order l on V such that every edge e ∈ E is an interval w.r.t. l; we also assume that {j} ∈ E for every j ∈ V . Our main result is a de Finetti-type representation of random exchangeable interval hypergraphs on N (EIHs): the law of every EIH can be obtained by sampling from some random compact subset K of the triangle {(x, y) : 0 ≤ x ≤ y ≤ 1} at iid uniform positions U1, U2, . . . , in the sense that, restricted to the node set [n] := {1, . . . , n} every non-singleton edge is of the form e = {i ∈ [n] : x < Ui < y} for some (x, y) ∈ K. We obtain this result via the study of a related class of stochastic objects: erased-interval processes (EIPs). These are certain transient Markov chains (In, ηn) n∈N such that In is an interval hypergraph on V = [n] w.r.t. the usual linear order (called interval system). We present an almost sure representation result for EIPs. Attached to each transient Markov chain is the notion of Martin boundary. The points in the boundary attached to EIPs can be seen as limits of growing interval systems. We obtain a one-to-one correspondence between these limits and compact subsets K of the triangle with (x, x) ∈ K for all x ∈ [0, 1].Interval hypergraphs are a generalizations of hierarchies and as a consequence we obtain a representation result for exchangeable hierarchies, which is close to a result of Forman, Haulk and Pitman in [FHP]. Several ordered discrete structures can be seen as interval systems with additional properties, i.e. Schröder trees (rooted, ordered, no node has outdegree one) or even more special: binary trees. We describe limits of Schröder trees as certain tree-like compact sets. These can be seen as an ordered counterpart to real trees, which are widely used to describe limits of discrete unordered trees. Considering binary trees we thus obtain a homeomorphic description of the Martin boundary of Rémy's tree growth chain, which has been analyzed by Evans, Grübel and Wakolbinger in [EGW].

General erased-word processes: Product-type filtrations, ergodic laws and Martin boundaries

“…Note that RSS-chains over countable alphabets are of this form. We refer the reader to [Ve,EGW,EW,Ge17] for introductory literature.…”

“…One can introduce (ordered) embedding densities for all kind of combinatorial structures, define a notation of convergence similar to Corollary 4 (i) and ask for a nice description of the occurring limit density functions. [HKMRS] considered permutations and identified limits with 2-dimensional copulas, [EGW] studied ordered binary trees and [Ge17] generalized this to non-binary ordered trees and beyond. In all situations there is a one-to-one correspondence between limits of convergent combinatorial structures and certain ergodic exchangeable laws involving joinings with exchangeable linear order (see Section 3).…”

“…Remark 7. [EGW,Ge17] studied erased-type processes int the context of binary trees, Schröder trees and so-called interval-systems. [Ge17] shows that ergodic erased-type processes generated poly-adic filtrations of product-type in this situation.…”

PATRICIA Bridges

“…Finally, we note that an alternative concrete representation of extremal infinite Rémy bridges has recently been given in [11] as part of a program that extends the study of Markov chains with Rémy-like transition probabilities to classes of discrete structures other than binary trees; for example, [11] also considers the infinite bridges investigated in [9] that are similar to Rémy or PATRICIA infinite bridges but vertices of the successive trees may have more than two offspring and there is no left-to-right ordering of offspring. Steve Evans & Anton Wakolbinger…”