Linear Algebra
Graphs.jl provides the following matrix operations on both directed and undirected graphs in the LinAlg submodule:
Graphs.LinAlg.AdjacencyGraphs.LinAlg.AveragingAdjacencyGraphs.LinAlg.AveragingLaplacianGraphs.LinAlg.CombinatorialAdjacencyGraphs.LinAlg.GraphMatrixGraphs.LinAlg.NoopGraphs.LinAlg.NormalizedAdjacencyGraphs.LinAlg.NormalizedLaplacianGraphs.LinAlg.StochasticAdjacencyGraphs.LinAlg.StochasticLaplacianGraphs.LinAlg.adjacency_matrixGraphs.LinAlg.adjacency_spectrumGraphs.LinAlg.degreesGraphs.LinAlg.degreesGraphs.LinAlg.incidence_matrixGraphs.LinAlg.laplacian_matrixGraphs.LinAlg.laplacian_spectrumGraphs.LinAlg.spectral_distanceGraphs.LinAlg.symmetrizeGraphs.LinAlg.symmetrize
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Graphs.LinAlg — ModuleLinAlgA package for using the type system to check types of graph matrices.
Graphs.LinAlg.Adjacency — TypeAdjacency{T}The core Adjacency matrix structure. Keeps the vertex degrees around. Subtypes are used to represent the different normalizations of the adjacency matrix. Laplacian and its subtypes are used for the different Laplacian matrices.
Adjacency(lapl::Laplacian) provides a generic function for getting the adjacency matrix of a Laplacian matrix. If your subtype of Laplacian does not provide a field A for the Adjacency instance, then attach another method to this function to provide an Adjacency{T} representation of the Laplacian. The Adjacency matrix here is the final subtype that corresponds to this type of Laplacian.
Graphs.LinAlg.AveragingAdjacency — TypeAveragingAdjacency{T}The matrix whose action is to average over each neighborhood.
Graphs.LinAlg.AveragingLaplacian — TypeAveragingLaplacian{T}Laplacian version of the AveragingAdjacency matrix.
Graphs.LinAlg.CombinatorialAdjacency — TypeCombinatorialAdjacency{T,S,V}The standard adjacency matrix.
Graphs.LinAlg.GraphMatrix — TypeGraphMatrix{T}An abstract type to allow opertions on any type of graph matrix
Graphs.LinAlg.Noop — TypeNoopA type that represents no action.
Implementation Notes
- The purpose of
Noopis to help write more general code for the
different scaled GraphMatrix types.
Graphs.LinAlg.NormalizedAdjacency — TypeNormalizedAdjacency{T}The normalized adjacency matrix is $\hat{A} = D^{-1/2} A D^{-1/2}$. If A is symmetric, then the normalized adjacency is also symmetric with real eigenvalues bounded by [-1, 1].
Graphs.LinAlg.NormalizedLaplacian — TypeNormalizedLaplacian{T}The normalized Laplacian is $\hat{L} = I - D^{-1/2} A D^{-1/2}$. If A is symmetric, then the normalized Laplacian is also symmetric with positive eigenvalues bounded by 2.
Graphs.LinAlg.StochasticAdjacency — TypeStochasticAdjacency{T}A transition matrix for the random walk.
Graphs.LinAlg.StochasticLaplacian — TypeStochasticLaplacian{T}Laplacian version of the StochasticAdjacency matrix.
Graphs.LinAlg.degrees — Methoddegrees(adjmat)Return the degrees of a graph represented by the CombinatorialAdjacencyadjmat.
Graphs.LinAlg.degrees — Methoddegrees(graphmx)Return the degrees of a graph represented by the graph matrix graphmx.
Graphs.LinAlg.symmetrize — Functionsymmetrize(adjmat, which=:or)Return a symmetric version of graph (represented by CombinatorialAdjacencyadjmat) as a CombinatorialAdjacency. which may be one of :triu, :tril, :sum, or :or. Use :sum for weighted graphs.
Implementation Notes
Only works on Adjacency because the normalizations don't commute with symmetrization.
Graphs.LinAlg.symmetrize — Functionsymmetrize(A::SparseMatrix, which=:or)Return a symmetric version of graph (represented by sparse matrix A) as a sparse matrix. which may be one of :triu, :tril, :sum, or :or. Use :sum for weighted graphs.
Graphs.LinAlg.adjacency_matrix — Functionadjacency_matrix(g[, T=Int; dir=:out])Return a sparse adjacency matrix for a graph, indexed by [u, v] vertices. Non-zero values indicate an edge from u to v. Users may override the default data type (Int) and specify an optional direction.
Optional Arguments
dir=:out: :in, :out, or :both are currently supported.
Implementation Notes
This function is optimized for speed and directly manipulates CSC sparse matrix fields.
Graphs.LinAlg.adjacency_spectrum — Functionadjacency_spectrum(g[, T=Int; dir=:unspec])Return the eigenvalues of the adjacency matrix for a graph g, indexed by vertex. Default values for T are the same as those in adjacency_matrix.
Optional Arguments
dir=:unspec: Options for dir are the same as those in laplacian_matrix.
Performance
Converts the matrix to dense with $nv^2$ memory usage.
Implementation Notes
Use eigs(adjacency_matrix(g); kwargs...) to compute some of the eigenvalues/eigenvectors.
Graphs.LinAlg.incidence_matrix — Functionincidence_matrix(g[, T=Int; oriented=false])Return a sparse node-arc incidence matrix for a graph, indexed by [v, i], where i is in 1:ne(g), indexing an edge e. For directed graphs, a value of -1 indicates that src(e) == v, while a value of 1 indicates that dst(e) == v. Otherwise, the value is 0. For undirected graphs, both entries are 1 by default (this behavior can be overridden by the oriented optional argument).
If oriented (default false) is true, for an undirected graph g, the matrix will contain arbitrary non-zero values representing connectivity between v and i.
Graphs.LinAlg.laplacian_matrix — Methodlaplacian_matrix(g[, T=Int; dir=:unspec])Return a sparse Laplacian matrix for a graph g, indexed by [u, v] vertices. T defaults to Int for both graph types.
Optional Arguments
dir=:unspec: :unspec, :both, :in, and:outare currently supported. For undirected graphs,dirdefaults to:out; for directed graphs,dirdefaults to:both`.
Graphs.LinAlg.laplacian_spectrum — Functionlaplacian_spectrum(g[, T=Int; dir=:unspec])Return the eigenvalues of the Laplacian matrix for a graph g, indexed by vertex. Default values for T are the same as those in laplacian_matrix.
Optional Arguments
dir=:unspec: Options for dir are the same as those in laplacian_matrix.
Performance
Converts the matrix to dense with $nv^2$ memory usage.
Implementation Notes
Use eigs(laplacian_matrix(g); kwargs...) to compute some of the eigenvalues/eigenvectors.
Graphs.LinAlg.spectral_distance — Functionspectral_distance(G₁, G₂ [, k])Compute the spectral distance between undirected n-vertex graphs G₁ and G₂ using the top k greatest eigenvalues. If k is ommitted, uses full spectrum.
References
- JOVANOVIC, I.; STANIC, Z., 2014. Spectral Distances of Graphs Based on their Different Matrix Representations