| import OAI.Combinatorics.CliqueFree.WalkEntropy |
| import OAI.Combinatorics.CliqueFree.TriangleState |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| namespace CliqueFreeIndependence.WeightedGraph |
|
|
| universe u v |
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| variable {V : Type u} [Fintype V] |
|
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| attribute [local instance 10000] edgeStateDecEq |
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|
| namespace LocalWalk |
| open FiniteEntropy FiniteKernel |
| variable {G : SimpleGraph V} {w : V → ℝ} |
| lemma triangle_pair_sum [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) |
| (f : EdgeState G → EdgeState G → ℝ) : |
| (∑ t : TriangleState G, TriangleState.weight w t * f t.edge12 t.edge13) = |
| normalizer G w * ∑ e, law G w e * ∑ z, transition G w e z * f e z := by |
| rw [TriangleState.corner_sum] |
| simp only [mul_sum] |
| apply sum_congr rfl |
| intro e _ |
| apply sum_congr rfl |
| intro z _ |
| have he := (hc _ _ e.adj).ne' |
| have hZ := (normalizer_pos hw hc).ne' |
| by_cases hs : e.src = z.src |
| · by_cases ha : G.Adj e.dst z.dst |
| · simp only [hs, ha, true_and, ↓reduceIte, transition, law] |
| rw [hs] at he |
| field_simp |
| · simp [hs, ha, transition] |
| · simp [hs, transition] |
|
|
| lemma triangle_edge_sum [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) (f : EdgeState G → ℝ) : |
| (∑ t : TriangleState G, TriangleState.weight w t * f t.edge12) = |
| normalizer G w * ∑ e, law G w e * f e := by |
| rw [triangle_pair_sum hw hc (fun e _ ↦ f e)] |
| simp only [← sum_mul, (transition_stochastic hw hc _).2, one_mul] |
| end LocalWalk |
|
|
| end CliqueFreeIndependence.WeightedGraph |
|
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| end |
|
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| end OAI |
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|
