| import OAI.Combinatorics.GraphThreshold.GraphEncoding |
| import OAI.Combinatorics.GraphThreshold.FiniteProbability |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| universe u |
|
|
| variable {n : ℕ} {V : Type u} |
|
|
| /-- Regard an actual subgraph as a subgraph of the complete graph. Its vertex set, |
| including all isolated vertices, is unchanged. -/ |
| def subgraphToComplete {G : SimpleGraph (Fin n)} (K : G.Subgraph) : |
| (⊤ : SimpleGraph (Fin n)).Subgraph where |
| verts := K.verts |
| Adj := K.Adj |
| adj_sub h := h.ne |
| edge_vert := K.edge_vert |
| symm := K.symm |
|
|
| /-- A complete-graph subgraph whose edges occur in G is the same actual subgraph of G. -/ |
| def subgraphFromComplete (G : SimpleGraph (Fin n)) |
| (K : (⊤ : SimpleGraph (Fin n)).Subgraph) (hK : K.spanningCoe ≤ G) : |
| G.Subgraph where |
| verts := K.verts |
| Adj := K.Adj |
| adj_sub h := hK h |
| edge_vert := K.edge_vert |
| symm := K.symm |
|
|
| /-- Copies are put in bijection without choosing or counting graph embeddings. -/ |
| def completeCopyEquiv (G : SimpleGraph (Fin n)) (F : SimpleGraph V) : |
| {K : G.Subgraph // Nonempty (F ≃g K.coe)} ≃ |
| {K : (⊤ : SimpleGraph (Fin n)).Subgraph // |
| Nonempty (F ≃g K.coe) ∧ K.spanningCoe ≤ G} where |
| toFun K := ⟨subgraphToComplete K.val, K.property, fun _ _ h => K.val.adj_sub h⟩ |
| invFun K := ⟨subgraphFromComplete G K.val K.property.2, K.property.1⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| theorem copyCount_eq_complete_card (G : SimpleGraph (Fin n)) (F : SimpleGraph V) : |
| G.copyCount F = Nat.card |
| {K : (⊤ : SimpleGraph (Fin n)).Subgraph // |
| Nonempty (F ≃g K.coe) ∧ K.spanningCoe ≤ G} := by |
| classical |
| calc |
| G.copyCount F = Fintype.card {K : G.Subgraph // Nonempty (F ≃g K.coe)} := by |
| rw [SimpleGraph.copyCount, Fintype.card_subtype] |
| apply congrArg Finset.card |
| ext K |
| simp |
| _ = Fintype.card {K : (⊤ : SimpleGraph (Fin n)).Subgraph // |
| Nonempty (F ≃g K.coe) ∧ K.spanningCoe ≤ G} := |
| Fintype.card_congr (completeCopyEquiv G F) |
| _ = _ := Nat.card_eq_fintype_card.symm |
|
|
| theorem copyCount_eq_sum_complete (G : SimpleGraph (Fin n)) (F : SimpleGraph V) : |
| (G.copyCount F : ℝ) = |
| ∑ K : (⊤ : SimpleGraph (Fin n)).Subgraph, |
| if Nonempty (F ≃g K.coe) then (if K.spanningCoe ≤ G then (1 : ℝ) else 0) |
| else 0 := by |
| classical |
| rw [copyCount_eq_complete_card, Nat.card_eq_fintype_card, Fintype.card_subtype] |
| rw [Finset.card_eq_sum_ones] |
| simp only [Nat.cast_sum, Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro K _ |
| by_cases hF : Nonempty (F ≃g K.coe) <;> simp [hF] |
|
|
| /-- Adding ambient isolated vertices does not change the edge count. -/ |
| theorem edgeCount_spanningCoe {G : SimpleGraph (Fin n)} (K : G.Subgraph) : |
| edgeCount K.spanningCoe = edgeCount K.coe := by |
| unfold edgeCount |
| rw [SimpleGraph.Subgraph.edgeSet_spanningCoe, ← K.image_coe_edgeSet_coe] |
| exact Nat.card_congr |
| (Equiv.Set.image (Sym2.map (Subtype.val : K.verts → Fin n)) K.coe.edgeSet |
| (Sym2.map.injective Subtype.val_injective)).symm |
|
|
| theorem card_graphEdges_of_copy (F : SimpleGraph V) |
| (K : (⊤ : SimpleGraph (Fin n)).Subgraph) (hK : Nonempty (F ≃g K.coe)) : |
| (graphEdges K.spanningCoe).card = edgeCount F := by |
| obtain ⟨f⟩ := hK |
| rw [card_graphEdges, edgeCount_spanningCoe] |
| exact (Nat.card_congr f.mapEdgeSet).symm |
|
|
| theorem subgraph_survival_probability (p : ℝ) |
| (K : (⊤ : SimpleGraph (Fin n)).Subgraph) : |
| bernoulliProbability p (fun W : Finset (Edge n) => |
| K.spanningCoe ≤ graphOfEdges W) = p ^ edgeCount K.coe := by |
| have h : ∀ W : Finset (Edge n), |
| K.spanningCoe ≤ graphOfEdges W ↔ graphEdges K.spanningCoe ⊆ W := by |
| intro W |
| calc |
| K.spanningCoe ≤ graphOfEdges W ↔ |
| graphOfEdges (graphEdges K.spanningCoe) ≤ graphOfEdges W := by |
| rw [graphOfEdges_graphEdges] |
| _ ↔ graphEdges K.spanningCoe ⊆ W := graphOfEdges_le_iff |
| simp_rw [h] |
| rw [bernoulliProbability_subset, card_graphEdges, edgeCount_spanningCoe] |
|
|
| /-- The exact expected number of unlabelled, non-induced copies. The counted |
| objects include their vertex sets, so the identity retains isolated vertices. -/ |
| theorem expectedCopies_formula (n : ℕ) (p : ℝ) (F : SimpleGraph V) : |
| expectedCopies n p F = |
| ((⊤ : SimpleGraph (Fin n)).copyCount F : ℝ) * p ^ edgeCount F := by |
| classical |
| calc |
| expectedCopies n p F = |
| ∑ K : (⊤ : SimpleGraph (Fin n)).Subgraph, |
| if Nonempty (F ≃g K.coe) then p ^ edgeCount F else 0 := by |
| unfold expectedCopies bernoulliExpectation |
| simp_rw [copyCount_eq_sum_complete, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro K _ |
| by_cases hK : Nonempty (F ≃g K.coe) |
| · simp only [hK, ite_true] |
| have hsurv := subgraph_survival_probability p K |
| have hedge : edgeCount K.coe = edgeCount F := |
| (Nat.card_congr hK.some.mapEdgeSet).symm |
| rw [hedge] at hsurv |
| rw [← hsurv] |
| unfold bernoulliProbability |
| apply Finset.sum_congr rfl |
| intro W _ |
| by_cases hW : K.spanningCoe ≤ graphOfEdges W <;> simp [hW] |
| · simp [hK] |
| _ = ((⊤ : SimpleGraph (Fin n)).copyCount F : ℝ) * p ^ edgeCount F := by |
| rw [copyCount_eq_sum_complete, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro K _ |
| by_cases hK : Nonempty (F ≃g K.coe) <;> simp [hK] |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
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