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Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More*

Explore faster algorithms for computing the stationary distribution, simulating random walks, and more in Directed Spectral Graph Theory. Learn about solving linear systems, extensions to non-Eulerian Laplacians, and improved methods for solving Eulerian Laplacian systems. Discover the use of undirected structure, symmetric symmetrization, and special eigenvalue structures. Uncover techniques like preconditioning, CG, sparsification, and low-rank updates along with the Sherman-Morrison Formula. Find out how to handle non-Eulerian Laplacian systems efficiently with iterative approaches and reduce access to solver computations. Gain insights into applications such as Personalized PageRank, hitting times, commute times, and escape probabilities in graph theory. Begin your journey into Directed Spectral Graph Theory and stay updated on the latest advancements in running time improvements and algorithm simplifications.

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Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More*

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  1. Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More* * Directed Spectral Graph Theory 101 Michael Cohen Jonathan Kelner John Peebles Richard Peng Aaron Sidford Adrian Vladu

  2. Shortest Paths • Spanning Trees • Max Flow • Min Cost Flow • … • Solve Ux = b • max flow • approximate: [CKMST 11, LRS 12], [KLOS, S 14], [P 15] • exact: [M 13, M 16] • min cost flow: [DS 08], [LS 14], [CMSV 16] • balanced partitioning: [OSV 12] • sampling random spanning trees: [KM 09], [MST 15] • sparsification [SS 08, BSS 09, AZLO 15, LS 15] No Directed Structure Spectral Graph Theory Can solve Ux = b in nearly linear time [ST 04] Improvements: [KMP 10 11, KOSZ 13, LS 13, CKMPPRX 14, PS 14, KLPSS 16, KS 16, …] Combinatorial Object Algebraic Object 1 1 2 subroutine 2 1 5 U = D- A = 1 4 4 3 1

  3. Directed Spectral Graph Theory ? Random walk matrix: W = ATDout-1 5 1 • (I-W) Dout 1 1 2 /1 2 1 /5 /2 5 • AT = • - 1 4 /5 /1 2 4 3 1 /1 1 1

  4. Directed Spectral Graph Theory ? Random walk matrix: W = ATDout-1 To solve we should at least know solution to . is stationary: 5 1 • (I-W) Dout 1 1 2 2 1 5 ℒ = Dout- AT = 1 4 2 4 3 1 1 1

  5. Directed Spectral Graph Theory 1 2 • There is a large class of directed graphs for which we know the kernel of (aka know ‘s stationary distribution) • So and • Goal #1: Solve linear systems on Eulerian Laplacians • Goal #2: Extend to non-Eulerian Laplacians of strongly-connected graphs via calls to Eulerian solver 2 2 3 2 calls to EulerianLaplacian solver improved to in follow up paper, soon on arXiv

  6. Solving EulerianLaplacian Systems • We can employ undirected structure • The symmetrization is an undirected Laplacian Not true for non-Eulerian graphs 2 1 2 1

  7. Solving EulerianLaplacian Systems • We can employ undirected structure • The symmetrization is an undirected Laplacian Not true for non-Eulerian graphs • harmonic symmetrization is symmetric PSD • when is symmetric, same as harmonic symmetrization because

  8. Solving EulerianLaplacian Systems • Classic approach: preconditioning • want to solve • have access to solver for , which is a decent approximation to , i.e. • off-the-shelf method (Chebyshev, Conjugate Gradient) solves in iterations • Theorem ( as preconditioner):and • So iterations, each involves applying in time.

  9. Solving EulerianLaplacian Systems • Classic approach: preconditioning • want to solve • have access to solver for , which is a decent approximation to , i.e. • off-the-shelf method (Chebyshev, Conjugate Gradient) solves in iterations • Theorem ( as preconditioner):and • So iterations, each involves applying in time. Special eigenvalue structure: at most eigenvalues . Preconditioned Conjugate Gradient converges in iterations: .

  10. Solving EulerianLaplacian Systems • Do we have a time algorithm? • Not really, CG may require a polynomially large number of bits • Instead, use sparsification + low rank updates running time • Soon on arXiv: [CKPPSV + Anup Rao ]

  11. Solving Non-EulerianLaplacian Systems 2/7 2/7 • Every strongly connected graph is Eulerian up to a diagonal scaling • Consider the rescaled Laplacian • Let be the stationary probability • Rescale all arcs leaving vertex by is Eulerian • But we did not know to begin with… x 2/7 1 1 2 x 2/7 x 1/7 1/7 1 .5 x 1/7 x 2/7 5 1 .5 x 1/7 1/7 1/7 4 3 1

  12. Solving Non-EulerianLaplacian Systems • Extension of Eulerian solver: can solve where is Laplacian and is nonnegative diagonal matrix such that all column sums (call RCDD matrix)

  13. Solving Non-EulerianLaplacian Systems Find diagonal scaling which makes Eulerian Find diagonal scaling and small diagonal which make RCDD relax , , ,

  14. Solving Non-EulerianLaplacian Systems Sherman-Morrison Formula scaling of Laplacian rank-1 update RCDD requires one solve in and one solve in access to solver access to solver + + can find kernel

  15. Solving Non-EulerianLaplacian Systems sum of entries in gets reduced by half scaling of Laplacian strongly RCDD Eulerian scaling of Laplacian RCDD tightly RCDD reduce access to solver access to solver + + can find kernel

  16. Solving Non-EulerianLaplacian Systems RCDD , , , Done a iterations!

  17. Strongly-Connected Solving EulerianLaplacian Systems • Do we have a time algorithm? • Not really, CG may require a polynomially large number of bits • Instead, use sparsification + low rank updates running time • Soon on arXiv: [CKPPRSV]

  18. Applications • Personalized PageRank vector with restart probability , and vector of teleport probabilities : • Hitting times • Commute times • Escape probabilities: probability a random walk reaches u before v • with such that , • Sketching all-pairs commute times • Can write , where • Sketch quadratic form, just like in the undirected case [SS 09]

  19. Conclusions • First dent into Directed Spectral Graph Theory that can be used algorithmically • Running time improvements, simplifications? Currently have algorithm with running time , to be posted on arXiv. • More applications?

  20. Thank You! Michael Cohen Jonathan Kelner John Peebles Richard Peng Aaron Sidford Adrian Vladu

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