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Refinement and the index of a graph partition

Lax18.PartitionEnergy · concepts/Lax18/PartitionEnergy.lean · lax-18

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    Natural Language Statement

    Definition

    A partition Q refines a partition P when every class of Q is contained in a class of P.

    The index, also called the mean-square density or energy, of a partition is the weighted average of the squared densities between all ordered pairs of its classes. The weight of the pair (Vᵢ,Vⱼ) is |Vᵢ||Vⱼ|/|V|². Including diagonal pairs makes this an exact conditional mean-square and is convenient for the refinement argument.

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    1import Mathlib.Algebra.BigOperators.Group.Finset.Basic
    2import Lax18.RegularPartitions
    3
    4/-!
    5---
    6title: Refinement and the index of a graph partition
    7type: definition
    8---
    9A partition Q refines a partition P when every class of Q is contained in a
    10class of P.
    11
    12The index, also called the mean-square density or energy, of a partition is
    13the weighted average of the squared densities between all ordered pairs of
    14its classes. The weight of the pair (Vᵢ,Vⱼ) is |Vᵢ||Vⱼ|/|V|².
    15Including diagonal pairs makes this an exact conditional mean-square and is
    16convenient for the refinement argument.
    17-/
    18
    19namespace Lax18.PartitionEnergy
    20
    21open Lax18.EdgeDensity
    22open Lax18.FiniteGraphPartitions
    23open scoped BigOperators
    24
    25universe u
    26
    27variable {V : Type u} [Fintype V] [DecidableEq V]
    28
    29/-- `Q.Refines P` means that every class of `Q` lies in a class of `P`. -/
    30def Refines (Q P : VertexPartition V) : Prop :=
    31 ∀ j : Fin Q.partCount,
    32 ∃ i : Fin P.partCount, Q.part j ⊆ P.part i
    33
    34/-- The normalized weight of the ordered pair of classes `(i,j)`. -/
    35noncomputable def pairWeight (P : VertexPartition V)
    36 (i j : Fin P.partCount) : ℝ :=
    37 ((P.partSize i : ℝ) * (P.partSize j : ℝ)) /
    38 (Fintype.card V : ℝ) ^ 2
    39
    40/-- The weighted mean-square density (or index) of a graph partition. -/
    41noncomputable def partitionEnergy (G : SimpleGraph V)
    42 (P : VertexPartition V) : ℝ :=
    43 ∑ i : Fin P.partCount,
    44 ∑ j : Fin P.partCount,
    45 pairWeight P i j * (density G (P.part i) (P.part j)) ^ 2
    46
    47end Lax18.PartitionEnergy
    48
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