Sampler laws of the special digraphon families (directed umbrella #84, D3c / #87)

One headline identification per special family of Graphon.DigraphonConstructors (source crosswalk: Cai–Ackerman–Freer arXiv:1510.08440 — digraphons/sampling §2.3–2.4, the asymmetric-function model §3.1, undirected graphs and tournaments §3.2.1–3.2.2):

• Digraphon.ofTournament_sample_isTournament — the sample of the tournament digraphon is almost surely a tournament: no loops, and exactly one direction between any two distinct vertices;
• Digraphon.sampleEventIntegrand_ofKernel_ae — for the asymmetric-kernel digraphon the exact-event integrand factorizes over ordered pairs: the two directions of each pair are independent one-directional Bernoulli draws;
• Digraphon.map_sampleFinite_ofGraphon — the sample of the embedded ordinary graphon is the undirected W-random graph: its law is the pushforward of Graphon.samplePMF W k under the symmetric loopless embedding SimpleGraph.toFiniteDigraph.

Everything is driven by the labeling-free exact-event product formula (samplerSource_forall_sampleAdj) together with the a.e. simplexRep identifications of the families; a.e. facts about the latents transport to the sample space through the coordinate projections of the i.i.d. sources.

Small-event specializations of the exact-event formula

The tournament digraphon samples tournaments

theorem MeasureTheory.Digraphon.ofTournament_sample_isTournament {α : Type u_1} [MeasurableSpace α] {μ : Measure α} [IsProbabilityMeasure μ] (A : α × α →ₘ[μ.prod μ] ℝ) (hnn : ∀ᵐ (p : α × α) ∂μ.prod μ, 0 ≤ ↑A p) (hsum : ∀ᵐ (p : α × α) ∂μ.prod μ, ↑A p + ↑A p.swap = 1) : ∀ᵐ (ω : (ℕ → α) × (OffDiagPairIndex ℕ → ℝ)) ∂samplerSource μ, (∀ (i : ℕ), ((ofTournament A hnn hsum).sampleInfinite ω).adjBit i i = false) ∧ ∀ (i j : ℕ), i ≠ j → ((ofTournament A hnn hsum).sampleInfinite ω).adjBit j i = !((ofTournament A hnn hsum).sampleInfinite ω).adjBit i j

The tournament digraphon samples tournaments (D3c headline): almost surely, the sampled infinite digraph has no loops and exactly one direction between any two distinct vertices.

The asymmetric-kernel digraphon: independent directions

noncomputable def MeasureTheory.Digraphon.kernelDirMass {α : Type u_1} [MeasurableSpace α] {μ : Measure α} (A : α × α →ₘ[μ.prod μ] ℝ) {n : ℕ} (D : FiniteDigraph n) (y : Fin n → α) (q : Fin n × Fin n) : ℝ

The one-directional Bernoulli mass of a single ordered pair of the asymmetric-kernel digraphon: A at the ordered positions if the directed edge is prescribed present, 1 − A if absent.

Equations

• MeasureTheory.Digraphon.kernelDirMass A D y q = if D (digraphCoord q.1 q.2) = true then ↑A (y q.1, y q.2) else 1 - ↑A (y q.1, y q.2)

theorem MeasureTheory.Digraphon.sampleEventIntegrand_ofKernel_ae {α : Type u_1} [MeasurableSpace α] {μ : Measure α} [IsProbabilityMeasure μ] (A : α × α →ₘ[μ.prod μ] ℝ) (hmem : ∀ᵐ (p : α × α) ∂μ.prod μ, ↑A p ∈ Set.Icc 0 1) (L : α → Bool) (hL : Measurable L) {n : ℕ} (D : FiniteDigraph n) : (ofKernel A hmem L hL).sampleEventIntegrand D =ᵐ[Measure.pi fun (x : Fin n) => μ] fun (y : Fin n → α) => (∏ i : Fin n, if L (y i) = D (digraphCoord i i) then 1 else 0) * ∏ q : Fin n × Fin n with q.1 ≠ q.2, ENNReal.ofReal (kernelDirMass A D y q)

The asymmetric-kernel sample draws its two directions independently (D3c headline): the exact-event integrand factorizes over ordered off-diagonal pairs into one-directional Bernoulli masses, with the loop indicators reading the loop kernel L.

The graphon embedding samples the undirected W-random graph

theorem MeasureTheory.Digraphon.map_sampleFinite_ofGraphon {α : Type u_1} [MeasurableSpace α] {μ : Measure α} [IsProbabilityMeasure μ] (W : Graphon α μ) (k : ℕ) : Measure.map ((ofGraphon W).sampleFinite k) (samplerSource μ) = Measure.map SimpleGraph.toFiniteDigraph (W.samplePMF k).toMeasure

The embedded ordinary graphon samples the undirected W-random graph (D3c headline): the law of the k-vertex digraph sample of ofGraphon W is the pushforward of the undirected sample law Graphon.samplePMF W k under the symmetric loopless embedding.