| import OAI.Combinatorics.CliqueFree.Reach |
| import OAI.Combinatorics.CliqueFree.FiniteKernel |
|
|
| namespace OAI |
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| noncomputable section |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| attribute [local instance] Classical.propDecidable |
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| namespace CliqueFreeIndependence.WeightedGraph |
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| 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) |
|
|
| /-- The disjoint union of the neighborhood transition kernels. -/ |
| noncomputable def transition : Kernel (EdgeState G) := fun e f ↦ |
| if e.src = f.src then |
| if G.Adj e.dst f.dst then w f.dst / commonMass G w e.src e.dst else 0 |
| else 0 |
|
|
| noncomputable def normalizer : ℝ := |
| ∑ e : EdgeState G, w e.src * w e.dst * commonMass G w e.src e.dst |
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|
| noncomputable def law : EdgeState G → ℝ := fun e ↦ |
| (w e.src * w e.dst * commonMass G w e.src e.dst) / normalizer G w |
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| noncomputable def vertexWeight : EdgeState G → ℝ := fun e ↦ w e.dst |
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| noncomputable def fiber (u : V) : Finset (EdgeState G) := by |
| classical |
| exact univ.filter (fun e ↦ e.src = u) |
|
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| @[simp] lemma mem_fiber (e : EdgeState G) (u : V) : e ∈ fiber G u ↔ e.src = u := by |
| classical |
| simp [fiber] |
|
|
| lemma fiber_mass (u : V) : (∑ e ∈ fiber G u, vertexWeight G w e) = neighborMass G w u := by |
| classical |
| simp only [fiber, sum_filter, vertexWeight, EdgeState.sum_fiber, neighborMass, mass] |
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|
| lemma normalizer_eq : normalizer G w = 6 * triangleMass G w := by |
| rw [normalizer, EdgeState.sum_eq (fun u v ↦ w u * w v * commonMass G w u v), |
| triangleMass_eq_sum] |
| simp only [triangleAt, mul_sum, mul_assoc] |
|
|
| variable {G w} |
| lemma transition_stochastic (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) : Stochastic (transition G w) := by |
| classical |
| intro e |
| constructor |
| · intro f |
| dsimp [transition] |
| split_ifs |
| · exact (div_pos (hw _) (hc _ _ e.adj)).le |
| all_goals exact le_rfl |
| · change (∑ f : EdgeState G, if e.src = f.src then |
| (if G.Adj e.dst f.dst then w f.dst / commonMass G w e.src e.dst else 0) else 0) = 1 |
| rw [EdgeState.sum_fiber' e.src (fun z ↦ |
| if G.Adj e.dst z then w z / commonMass G w e.src e.dst else 0)] |
| have hid (z : V) : (if G.Adj e.dst z then w z / commonMass G w e.src e.dst else 0) = |
| (if G.Adj e.dst z then w z else 0) / commonMass G w e.src e.dst := by |
| split_ifs <;> simp |
| simp_rw [hid] |
| rw [← sum_div, ← commonMass_eq_sum, div_self (hc _ _ e.adj).ne'] |
|
|
| lemma normalizer_pos [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) : 0 < normalizer G w := by |
| exact sum_pos (fun e _ ↦ mul_pos (mul_pos (hw _) (hw _)) (hc _ _ e.adj)) univ_nonempty |
|
|
| lemma law_isProb [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) : IsProb (law G w) := by |
| have hZ := normalizer_pos hw hc |
| constructor |
| · intro e |
| exact (div_pos (mul_pos (mul_pos (hw _) (hw _)) (hc _ _ e.adj)) hZ).le |
| · unfold law |
| rw [← sum_div] |
| exact div_self hZ.ne' |
| |
| lemma transition_reversible [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) |
| (hc : ∀ u v, G.Adj u v → 0 < commonMass G w u v) : |
| Reversible (law G w) (transition G w) := by |
| classical |
| intro e f |
| by_cases hsrc : e.src = f.src |
| · by_cases hadj : G.Adj e.dst f.dst |
| · simp only [transition, hsrc, ite_true, hadj, hadj.symm, law] |
| have he := (hc _ _ e.adj).ne' |
| have hf := (hc _ _ f.adj).ne' |
| rw [hsrc] at he |
| field_simp [he, hf, (normalizer_pos hw hc).ne'] |
| · simp [transition, hsrc, hadj, G.adj_comm f.dst] |
| · simp [transition, hsrc, Ne.symm hsrc] |
|
|
| lemma transition_fixed_src (e f : EdgeState G) (h : e.src ≠ f.src) : |
| transition G w e f = 0 := by simp [transition, h] |
|
|
| lemma transition_density (hw : ∀ u, 0 < w u) {x : ℝ} (hx : 0 < x) |
| (hc : ∀ u v, G.Adj u v → 1 / x ≤ commonMass G w u v) : |
| ∀ e f, transition G w e f ≤ x * vertexWeight G w f := by |
| classical |
| intro e f |
| have hc0 : 0 < commonMass G w e.src e.dst := (one_div_pos.2 hx).trans_le (hc _ _ e.adj) |
| dsimp [transition, vertexWeight] |
| split_ifs |
| · apply (div_le_iff₀ hc0).2 |
| have h := (div_le_iff₀ hx).1 (hc _ _ e.adj) |
| nlinarith [mul_le_mul_of_nonneg_left h (hw f.dst).le] |
| all_goals exact (mul_pos hx (hw _)).le |
|
|
| lemma smoothed_fixed_src (n : ℕ) (e f : EdgeState G) (h : e.src ≠ f.src) : |
| smoothed (transition G w) n e f = 0 := |
| preserves_smoothed (label := EdgeState.src) transition_fixed_src n e f h |
|
|
| lemma smoothed_supported (n : ℕ) (e f : EdgeState G) (h : f ∉ fiber G e.src) : |
| smoothed (transition G w) n e f = 0 := by |
| apply smoothed_fixed_src |
| intro he |
| apply h |
| exact (mem_fiber G f e.src).2 he.symm |
|
|
| lemma smoothed_density (hw : ∀ u, 0 < w u) {x : ℝ} (hx : 0 < x) |
| (hc : ∀ u v, G.Adj u v → 1 / x ≤ commonMass G w u v) (n : ℕ) : |
| ∀ e f, smoothed (transition G w) n e f ≤ x * vertexWeight G w f := by |
| have hc0 : ∀ u v, G.Adj u v → 0 < commonMass G w u v := |
| fun u v huv ↦ (one_div_pos.2 hx).trans_le (hc u v huv) |
| exact FiniteKernel.smoothed_density (transition_stochastic hw hc0) (transition_density hw hx hc) n |
|
|
| lemma entropy_bounds [Nonempty (EdgeState G)] (hw : ∀ u, 0 < w u) {x : ℝ} (hx : 0 < x) |
| (hc : ∀ u v, G.Adj u v → 1 / x ≤ commonMass G w u v) |
| (hL : ∀ u, neighborMass G w u ≤ Real.exp x) (n : ℕ) (e : EdgeState G) : |
| -Real.log x ≤ entropy (vertexWeight G w) (smoothed (transition G w) n e) ∧ |
| entropy (vertexWeight G w) (smoothed (transition G w) n e) ≤ x := by |
| have hc0 : ∀ u v, G.Adj u v → 0 < commonMass G w u v := |
| fun u v huv ↦ (one_div_pos.2 hx).trans_le (hc u v huv) |
| have hp := stochastic_smoothed (transition_stochastic hw hc0) n e |
| constructor |
| · exact entropy_ge_neg_log_of_density (w := vertexWeight G w) (fun f ↦ hw f.dst) hp hx |
| (smoothed_density hw hx hc n e) |
| · have h := entropy_supported_le_log (w := vertexWeight G w) (fiber G e.src) (fun f _ ↦ hw f.dst) hp |
| (smoothed_supported n e) (Real.exp_pos x) (by rw [fiber_mass]; exact hL e.src) |
| simpa only [Real.log_exp] using h |
|
|
| end LocalWalk |
|
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| end CliqueFreeIndependence.WeightedGraph |
|
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| end |
|
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| end OAI |
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