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1. Spectral Graph Sparsification: overview of theory and practical methods Yiannis Koutis University of Puerto Rico - Rio Piedras

2. Graph Sparsification or ‘Sketching’ Compute a smaller graph that preserves some crucial property of the input Motivation: Computational efficiency with approximation guarantees • BFS: Breadth First Spanning Tree • Spanner: Spanning subgraph that approximately preserves distances

3. Spectral Graph Sparsification Compute a smaller graph that preserves some crucial property of the input We want to approximately preserve the eigenvalues and eigenvectors of the graph Laplacian Motivation: Speed-up many clustering and partitioning algorithms based on computing Laplacian eigenvectors

4. Spectral Graph Sparsification Compute a smaller graph that preserves some crucial property of the input We want to approximately preserve the quadratic form xTLx of the Laplacian L Implies spectral approximations for both the Laplacian and the normalized Laplacian

5. The Graph Laplacian 30 2 1 1 20 15

6. Spectral Sparsification by Picture • H is a reweighted subgraph of G • H is obtained using randomness (sampling)

7. Combinatorial sketching Spectral sketching Linear system solvers Better combinatorial sketching

8. Outline • Sums of random positive matrices • Combinatorial sketching to incremental sparsification • Incremental sparsification for solving • Parallel and distributed sparsification • Deep sparsification by effective resistances • A heuristic for better clustering

9. Matrix Ordering • Whenever for all vectors x we have • We write G¹H • In this notation, a spectral sparsifier H satisfies

10. Sums of random matrices [Tropp ‘12, adapted]: 1. Let S be a nxn PSD matrix and Y1, … , Yk be independent random PSD matrices also of size nxn. 2. Let Y = S+i Yi 3. Let Z = E[Y] S : Combinatorial Sketch 4. Suppose Yi ¹ R ¢ Z Yi : Edges R: should be O(C/log n)

11. Outline • Sums of random positive matrices • Combinatorial sketching to incremental sparsification • Incremental sparsification for solving • Parallel and distributed sparsification • Deep sparsification by effective resistances • A heuristic for better clustering

12. A simple algorithm 1. Compute a spanner S’: 2. Let H:= S’ 3. For every edge e not in H: H:=H + k*e, with probability 1/k H has O(nlog n) + m/k edges n: number of vertices m: number of edges

13. and a simple proof • Suppose the graph G has n vertices and m edges • For simplicity we let G be unweighted (generalization is easy) • By definition of spanner, for every edge e of G, there is a path pe in S ‘ that joins the two endpoints of e and has length logn. • Algebraically, if Ge is the Laplacian of edge e:

14. and a simple proof • Apply Tropp’s Theorem on: • Combinatorial Sketch: • Samples:

15. Incremental Sparsification for Solving H has O(nlog n) + m/k edges [Spielman and Teng] If we can construct H with same guarantees but only n+m/k edges then we can solve linear systems on Laplacians in O(mlog2n) time [K, Miller Peng 10] Use low-stretch tree instead of spanner. It preserves distances on average. Use sampling with skewed probability distribution.

16. Incremental Sparsification for Solving [K, Miller Peng 10,11] If A is a symmetric diagonally dominant matrix then an approximate solution to Ax= b can be computed in O(m log n log log n log (1/²)) time, where ² is the required precision Fact: Approximations to the j first eigenvectors can be computed via solving O(j log n) linear systems

17. Outline • Sums of random positive matrices • Combinatorial sketching to incremental sparsification • Incremental sparsification for solving • Parallel and distributed sparsification • Deep sparsification by effective resistances • A heuristic for better clustering

18. Parallel and Distributed Sparsification 1. Can we do better than incremental sparsification ? 2. Is there a parallel sparsification algorithm? 3. Is there a distributed sparsification algorithm ? • Spanners hold the key to questions #2, #3 • There are very efficient parallel and distributed algorithms for computing spanners. From this we get parallel and distributed incremental sparsification. This doesn’t itself imply parallel solvers. • The main problem is question #1.

19. Parallel and Distributed Sparsification • A better combinatorial sketch: • t-bundle spanner: A collection of graphs S1….St such that Si is a spanner for G – (S1+….+ Si-1) • A t-bundle spanner can be computed with t sequential calls to a spanner computation algorithm. • If t is small then the algorithm remains efficient (polylogarithmic parallel time and distributed rounds)

20. A simple algorithm 1. Compute O(log4 n)-bundle spanner S 2. Let H:= S 3. For every edge e not in H: H:=H + 2*e, with probability 1/2 H has O(nlog5 n) + m/2 edges n: number of vertices m: number of edges

21. A simple algorithm • H has O(nlog5 n) + m/2 edges • Small size reduction (factor of 2) • Very tight spectral approximation • Repeat recursively on H. In O(log n) rounds we get O(nlog6 n) edges and a constant spectral approximation.

22. A parallel solver • This is the best known parallel sparsification routine. • Improves the total work guarantees of a parallel solver recently described by Peng and Spielman. • Parallel solver that works in polylogarithmic time and does O(mlog3 n) work.

23. Outline • Sums of random positive matrices • Combinatorial sketching to incremental sparsification • Incremental sparsification for solving • Parallel and distributed sparsification • Deep sparsification by effective resistances • A heuristic for better clustering

24. Spielman-Srivastava: Deep sparsification • Spielman and Srivastava proved: There is a graph H with edges such that • The algorithm is based on sampling edges with probabilities proportional to the effective resistances of the edges in the graph • Proof uses similarly Tropp’s theorem

25. Combinatorial sketching Spectral sketching Linear system solvers Better combinatorial sketching

26. The Incidence Matrix 30 2 1 1 20 15 W is the diagonal matrix containing the square roots of edge weights

27. Spielman-Srivastava: Deep sparsification • Effective resistances can be approximated closely by solving O(log n) linear Laplacian systems as follows: 1. Let L be the Laplacian of the graph 2. Let B be the incident matrix of the graph 3. Let Q be a random Johnson-Lindenstrauss projection of size m x O(log n) 4. Solve the systems L X = BT Q 5. Effective resistance between vertices i and j is equal to the ||Xi – Xi||2 where Xi is the ith row of X • The solution X is a n x O(log n) matrix. • Each row can be interpreted as an embedding of corresponding vertex to the O(log n)-dimensional Euclidean space. • Let’s call this the effective resistance embedding.

28.• http://ccom.uprrp.edu/~ikoutis/SpectralAlgorithms.htm

29. Heuristic: clustering based on the effective resistance embedding • Small effective resistance for an edge e means that there are a lot of short connections between the two endpoints . • Points that are close in the geometric embedding should be close in this connectivity sense in the graph. • Idea: Produce a k-clustering of the graph by running k-means on the effective resistance embedding • Produced clusterings appear to have better properties than clusterings based on geometric embeddings using the k-first eigenvectors. It is also much faster for most values of k.

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