Laplacians is a package containing graph algorithms, with an emphasis on tasks related to spectral and algebraic graph theory. It contains (and will contain more) code for solving systems of linear equations in graph Laplacians, low stretch spanning trees, sparsifiation, clustering, local clustering, and optimization on graphs.
All graphs are represented by sparse adjacency matrices. This is both for speed, and because our main concerns are algebraic tasks. It does not handle dynamic graphs. It would be very slow to implement dynamic graphs this way.
The documentation may be found by clicking on one of the "docs" links above.
Current Development Version
To get the current version of the master branch, run
pkg> add Laplacians#master
This version is compatible with Julia 1.4 and 1.5, but not earlier versions.
- Added two graph generators:
- Added a function
line_graphthat computes the line graph of an input graph.
Change: minor bug fix for spectral graph drawing.
Verified compatibility with Julia 1.2.
- Updating to use Julia's new Registrator.
harmonic_interpto interpolate harmonic functions on graphs. This is the fundamental routine used in Label Propagation / Semi-Supervised Learning on Graphs.
- Added a function
readIJV. It is a little more robust.
- Cleaned up
maxflowso that it now returns a flow and cut as a matrix and set.
pcga little more general.
fiedlerfor computing Fiedler vectors and values. That is, the smallest nonzero eigenvalue of the Laplacian.
- Fixed a bug in
thickenthat could cause it to loop forever, and cause
chimerato do the same.
- Changed the graph drawing code to use Plots instead of PyPlot.
plot_graphto plot in 3D.
- Fixed performance bug due to lazy matrix transpose.
- Changed more function names to agree with Julia naming conventions.
- Update documentation and examples.
This version works with Julia version 1.0.0.
The major change in this version is to the chimera and wted_chimera graph generators. They are now faster, and incorporate two-lifts and thickening. The old versions, using the pseudorandom generator from Julia V0.6 and Versions 0.2 of Laplacians, may be accessed by using the flag
ver=Laplacians.V06, as in
a = chimera(2000, 1, ver=Laplacians.V06)
There do seem to be differences in the very low order bits of graphs generated by wted_chimera with the V06 option and those generated in Julia V0.6. Not sure why.
The old generator is obtained by using the
RandomV06package for Julia.
Changed the names of many functions to bring closer to the Julia standard naming scheme. New names are empty_graph, path_graph, ring_graph, complete_graph, generalized_ring, rand_gen_ring, product_graph, join_graphs, two_lift ... Set deprecation warnings for the old names.
lex.jlto the directory
buggy, as on further testing we found bugs in it.
dropped wGrid3, as it produced a 4d grid so probably wasn't being used anyway. Dropped wGrid2 also.
Version 0.3.0, July 18 (or so), 2017
This is the first version that is compatible with Julia 0.7. Other changes:
- Dropped support for samplingSDDM and samplingLap solvers.
- The behavior of rand in Julia 0.7 is different, and this has changed the behavior of chimera. So, the chimera graphs generated in Version 0.3.0 and beyond will be different from those before.
Version 0.2.2, December 28, 2017
Fixed two bugs: one in shortestPaths, and one that prevented passing some parameters to approxchol_sddm. Improved the documentation for solving linear equations.
Version 0.2.1, September 18, 2017
Fixed a bug in
approxchol_sddm that caused it to be slow.
Version 0.2.0, July 17, 2017
This version is compatible with Julia 0.6. It will not work with Julia 0.5.X.
approxchol_sddm, a wrapper of
approxchol_lapthat solves SDDM systems.
Version 0.1.4, June 6, 2017
This is the current version. It is what you retrieve when you run
sparsify, an implementation of sparsification by effective resistance sampling, following Spielman and Srivastava.
conditionNumberfor checking how well one graph approximates another.
- Fixed a bug in the solution of Laplacian systems in disconnected graphs.
Version 0.1.3, June 2, 2017
- Changed the name of the approximate Cholesky solver from
approxchol_lap. Made improvements in this solver.
- Improved PCG so that it can now detect stagnation. Made options to do this even better when using it with a good preconditioner, like
- Added in code for comparing the running times of solvers. The difficulty here is that we need to stop them if they run too long. Added code to do this with threads inside Julia, and with
gtimeoutwhen calling Matlab to use icc, CMG, or LAMG.
Version 0.1.2, April 2, 2017
edgeElimLap- a fast Laplacian solver.
- fixed a bug in the unweighted version of
Version 0.1.1, December 26, 2016
- All of the linear equation solvers now have the same interface, and the Laplacian solvers work for disconnected graphs.
- Some support for calling solvers from Matlab has been added.
- Documentation is now through Documenter.jl.
Version 0.0.3 / 0.1.0, November 20, 2016
Versions 0.0.3 and 0.1.0 are the same. These versions works with Julia 0.5.
Warning: the behavior of chimera and wtedChimera differs between Julia 0.4 and Julia 0.5 because randperm acts differently in these.
Version 0.0.2, November 19, 2016
This is the version that works with Julia 0.4. It was captured right before the upgrade to Julia 0.5
The development of this package has been supported in part by the National Science Foundation Award CCF-1562041 and by the Office of Naval Research Award N00014-16-1-2374.