Exchangeable graph-valued Markov processes Harry Crane Department - - PowerPoint PPT Presentation

Exchangeable graph-valued Markov processes

Harry Crane

Department of Statistics Rutgers

June 20, 2014

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 1 / 7

Graph-valued Markov process

GN: graphs with vertex set N = {1, 2, . . .}. Gn: graphs with vertex set [n] := {1, . . . , n}. adjacency matrix/array: G = (Gij)i,j≥1 ∈ GN. relabeling: σ : N → N (permutation), G → Gσ := (Gσ(i)σ(j))i,j≥1. restriction: GN → Gn, (Gij)i,j≥1 → G|[n] := (Gij)1≤i,j≤n. Γ = (Γt)t≥0 is a Markov process on GN satisfying exchangeability: for all σ : N → N, Γσ := (Γσ

t )t≥0 is a version of Γ.

(Markovian) consistency: Γ[n] := (Γt|[n])t≥0 is a Markov chain on Gn, for every n = 1, 2, . . ..

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 2 / 7

Weakly exchangeable arrays

An exchangeable graph Γ is a weakly exchangeable {0, 1}-valued array Γ = (Γij)i,j≥1. Aldous–Hoover theorem: Γ =L Γ∗ = (Γ∗

ij )i,j≥1 with

Γ∗

ij = f(α, ξi, ξj, η{i,j}),

i, j ≥ 1, where f(·, b, c, ·) = f(·, c, b, ·) and α, (ξi)i≥1, (η{i,j})1≤i<j are i.i.d. Uniform[0,1]. (I) overall effect: α (II) vertex effect: {ξi}i≥1 (III) edge effect: {η{i,j}}1≤i<j

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 3 / 7

Characterization of discontinuities

Theorem

Γ = (Γt)t≥0 an exchangeable, consistent Markov process on GN. Then there are three types of discontinuity: (I) global jump: a positive fraction of all edges changes status; (II) single-vertex jump: a positive fraction of edges incident to a single vertex change, everything else stays the same; (III) single-edge flip: a single edge changes status, everything else stays the same.

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 4 / 7

Characterization of discontinuities

Theorem

Γ = (Γt)t≥0 an exchangeable, consistent Markov process on GN. Then there are three types of discontinuity: (I) global jump: a positive fraction of all edges changes status; (II) single-vertex jump: a positive fraction of edges incident to a single vertex change, everything else stays the same; (III) single-edge flip: a single edge changes status, everything else stays the same.

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 5 / 7

Characterization of discontinuities

Theorem

Γ = (Γt)t≥0 an exchangeable, consistent Markov process on GN. Then there are three types of discontinuity: (I) global jump: a positive fraction of all edges changes status; (II) single-vertex jump: a positive fraction of edges incident to a single vertex change, everything else stays the same; (III) single-edge flip: a single edge changes status, everything else stays the same.

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 6 / 7

Characterization of discontinuities

Theorem

Γ = (Γt)t≥0 an exchangeable, consistent Markov process on GN. The jump measure decomposes into three parts: (I) unique σ-finite measure on {0, 1} × {0, 1}-valued arrays: random function W : [0, 1]4 × {0, 1} → {0, 1} (weakly exchangeable array) so that Γt− → Γt with Γt(i, j) = W(α, ξi, ξj, η{i,j}, Γt−(i, j)), where {α; (ξi); (η{i,j})} are i.i.d. Uniform[0,1]. (II) unique σ-finite measure on 2 × 2 stochastic matrices: there is a unique i = 1, 2, . . . for which (Γt−(i, 1), Γt−(i, 2), . . .) jumps according to a 2 × 2 stochastic matrix S. (III) unique constants c01, c10 ≥ 0: determine jump rates of each edge. Comments: Compare to the Lévy-Itô characterization of exchangeable coalescent processes (binary coagulation, multiple collisions).

(I) binary coagulation: two blocks merge, everything else stays the same (“continuous jumps”); (II) multiple collisions: multiple blocks merge simultaneously (“discrete jumps”).

There is an associated projection of Γ into the space of graph limits.

Harry Crane (Rutgers) Graph-valued Markov processes Pitman: June 20-21, 2014 7 / 7