Q-Distance Matrix Compression

GraphicalDistanceMatrix{T<:Real}

A structure representing a graphical distance matrix computed from a graph. This specialized matrix type allows efficient storage and computation of resistance distances, similar to how ToeplitzMatrices.jl defines specialized matrices.

Fields

• graph::SimpleGraph: The underlying graph
• matrix::Matrix{T}: The precomputed resistance distance matrix (can be lazily evaluated)
• algorithm::Symbol: The algorithm used to compute resistance distances (:naive, :vectorized, :backsolve, :sparse, :compact)
• computed::Bool: Whether the matrix has been computed

Examples

using GraphicalDistance, Graphs
g = barabasi_albert(100, 5)
R = GraphicalDistanceMatrix(g)
# Access as a standard matrix
R[1, 2]
# Or compute on demand using a specific algorithm
R = GraphicalDistanceMatrix(g, :compact)

_subidentity_csc(n, cols)

Build the nxm sparse matrix whose columns are e{cols[1]},…,e{cols[m]} by directly constructing the internal CSC arrays.

graph(R::GraphicalDistanceMatrix)

Get the underlying graph.

resistance_distance(R::GraphicalDistanceMatrix, i::Integer, j::Integer)

Get the resistance distance between vertices i and j.

spsolve(A::SparseMatrixCSC, B::SparseMatrixCSC) -> SparseMatrixCSC

Solve the system AX = B where A is a lower triangular sparse matrix (SparseMatrixCSC) and B is a sparse matrix (SparseMatrixCSC) representing the right-hand side.

Arguments

• A::SparseMatrixCSC: A lower triangular sparse matrix.
• B::SparseMatrixCSC: A sparse matrix representing the right-hand side.

Returns

• X::SparseMatrixCSC: The solution matrix in sparse format.

Notes

• Assumes that A is lower triangular and non-singular.