Spectral Graph Theory and its Applications Lillian Dai 6.454 Oct. - PowerPoint PPT Presentation

Spectral Graph Theory and its Applications Lillian Dai 6.454 Oct. 20, 2004 1

Outline • Basic spectral graph theory • Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral Partitioning Works: Planar Graphs and Finite Element Meshes,” 1996 2

Graph and Associated Matrices 2 ( ) Laplacian matrix = G V E , 1 3 = − L D A = = V n 4 G G G T = B B 4 G G = = E m 5 Adjacency matrix Degree matrix Incidency matrix       0 1 1 1 3 0 0 0 1 1 1 0 0       − 1 0 0 1 0 2 0 0 1 0 0 1 0       =  =  =  A D B G  G  G  − 1 0 0 1 0 0 2 0 0 1 0 0 1       − − − 1 1 1 0 0 0 0 3 0 0 1 1 1       3

Properties of the Laplacian Matrix 2 1 3 { } λ = 0,2,4,4 4 − − −   3 1 1 1 − −         1 0 1 2           − − 1 2 0 1 − − 1 1 1 1           =  L G  − − 1 0 2 1         − 1 1 1 1           − − − 1 1 1 3 1 0 3 0           • Symmetric -> real eigenvalues; eigenspaces are mutually orthogonal • Orthogonally diagonalizable -> an eigenvalue with multiplicity k 4 has k-dimensional eigenspace

More Properties of the Laplacian Matrix • Positive semidefinite -> non- { } λ = 0,2,4,4 negative eigenvalues • Row sum = 0 -> singular -> at − −         1 0 1 2 least one eigenvalue = 0, unity         − − 1 1 1 1 eigenvector (since row sum = 1)                 − 1 1 1 1         • Orthogonal eigenspaces 1 0 3 0         u = eigenvector of non-zero n eigenvalue ∑ = u 0 i ( ) = i 1 = ∈ � n x x x , ,..., x 1 2 n ( )( ) ( ) ∑ T 2 T = T T = T T = − ≥ x L x x B B x x B x B x x 0 G G G G G i j ( ) 5 ∈ i j , E × m × 1 m 1

Spectrum of Some Graphs Which graphs are determined by their Eigenvalues spectrum? , , { } ( ) Complete − n 1 0, n • Complete Graphs • Graphs with one edge ( ) − π 2 2cos k n Line • Graphs missing 1 edge = k 1,..., n • Regular graphs with ( ) − π 2 2cos 2 k n degree 2 Ring • Regular graphs of = k 1,..., n 2 degree n - 3 { } Star ( ) n − 2 0,1 ,2 6

Graph Connectedness λ ≤ λ ≤ ≤ λ ... 1 2 n { } λ = 0,2,4,4 λ > 0 For connected graphs, 2 � � ( ) ∑ 2 T = − Recall x L x x x − −         1 0 1 2 G i j         ( ) ∈ i j , E − − 1 1 1 1         �         − If is eigenvector for eigenvalue 0 1 1 1 1 x         = ( ) � x x ∈ i j , E 1 0 3 0 L x =         0 i j G Multiplicity of the 0 eigenvalue indicates # of connected components λ Fiedler Value 2 � v Fiedler Vector 7

Onto Graph Partitioning … 8

Graph Partitioning • Remove as little of the graph as possible to separate out a subset of vertices of some desired “size” • “Size” may mean the number of vertices, number of edges, etc. • Typical case is to remove as few edges as possible to disconnect the graph into two parts of almost equal size Diagram from Berkeley CS 267 lecture notes Isoperimetric problem One of the earliest problems in geometry – considered by the ancient Greeks: Find, among all closed curves of a given length, the one which Stein, 1841 encloses the maximum area 9

Applications • Load balancing while minimizing communication • Sparse matrix-vector multiplication • Optimizing VLSI layout • Communication network design 10

Bisection and Ratio-Partition • Divide vertices into two disjoint subsets and S S ( ) • Cut Size E S S , ( ) E S S , ( ) • Cut Ratio φ = S ( ) G min S , S ( ) φ = φ • Isoperimetric Number min S G G ⊂ S V ( ) E S S , Bisection Minimize subject to # of nodes in each partition differ by at most 1. NP-Complete ( ) φ G S Ratio-Partition Minimize 11

Spectral Partitioning • Find Fiedler vector of the Laplacian matrix – map to vertices • Choose some real number s { } • Partition vertices given by = ≤ V i v : s L i { } = > V i v : s L i { } v 1 ,... n v • Bisection , s = median of • Ratio partition , s is chosen to give the best cut ratio 12

Example 1 4 2 5 3 6 Fiedler vector [-1 -2 -1 1 2 1] 13

Spectral Partitioning For Planar Graphs • Guattery and Miller – Performance of Spectral Graph Partitioning, 1995 • Spielman and Teng, Spectral Partitioning Works on Planar Graphs, 1996 • Kelner, Spectral Partitioning Works on Graphs with Bounded Genus, 2004 14

Simple Spectral Bisection May Fail (Guattery & Miller) ( ) Θ n The simple spectral bisection method produces cut size of G , for any k for k 15

Optimal Bisector for Graphs with Bounded Genus (Kelner) Genus g of a graph G: smallest integer such that G can be embedded on a surface of genus g without any of its edges crossing one another. Eg. Planar graphs have genus 0 Sphere, disc, and annulus has genus 0 Torus has genus 1 There is a spectral algorithm that produces bisector of size ( ) O gn For every g , there is a class of bounded degree graphs that have no bisectors smaller than ( ) O gn 16

Improved Bisection Algorithm on Planar Graphs (Spielman and Teng) Bisector of size ( ) O n Why does the spectral method work? Why does it work well on planar graphs? Why does simple bisection fail even on planar graphs? 17

Another Look at Fiedler Value � ( ) ( ) ∑ 2 = ∈ n � x x x , ,..., x T = − Recall where x L x x x 1 2 n G i j ( ) ∈ i j , E ( ) = ∑ 2 � � − x x T x L x ( ) i j ∈ i j , E Rayleigh quotient : φ = G � � ∑ x T 2 x x x i λ = � φ min Fiedler value satisfies with the minimum 2 x ( ) ⊥ x 1,...,1 � occurring only when is a Fiedler vector. x � � � � T T λ x L x x x φ = = = λ G 2 � � � � x 2 T T x x x x 18

Connection Between Fiedler Value and Isoperimetric Number ( ) E S S , Recall Isoperimetric Number φ = min ( ) G ⊂ S V min S , S is the best ratio-partition possible Theorem 1 (Mihail ‘89) Let be a graph on nodes of G n n ∑ � = maximum degree . For any vector such that x 0 x ∈ n ∆ � i = i 1 � � T φ 2 φ 2 Good ratio-partition x L x ≥ λ ≥ G G G � � can be achieved if 2 T ∆ ∆ 2 2 x x Fiedler value is small { } { } ≤ > i v : s i v : s s Moreover, there is an so that the cut has i i ( ) ratio at most 2 φ ∆ 2 G 19

Upperbound on the Fiedler Value for Planar Graphs Theorem 2 (Spielman & Teng ‘96) For all planar graphs G with vertices and maximum degree ∆ n   1 ∆ 8 O n λ ≤   2   n   φ 2 ∆ ∆ 1 8 4 ≤ λ ≤ φ ≤ G O   2 G ∆ 2 n  n  n By bounding Fiedler value of planar graphs, ratio-partitioning method is shown to work well What about bisection? 20

Relationship Between Ratio-Partitioning and Bisection Lemma 3 Given an algorithm that will find a cut ratio of at most ( ) φ k in every k-node subgraph of , for some monotonically G φ decreasing function . Then repeated application of this algorithm can be used to find a bisection of of size at most G n ( ) ∫ φ x dx = x 1 1 ( ) ( ) ( ) n ( ) ∫ φ = x φ = − x dx 2 n 1 O n x = x 1 Bisection can be obtained by repeated application of ratio- partitioning 21

� � T φ 2 n x L x ∑ = ≥ G G x 0 Theorem 1 � � i T ∆ 2 x x = i 1 2 ( ) E S S , 1 3 Map graph vertices to a line φ = min ( ) G ⊂ S V min S , S 4 x x x ≤ ≤ ≤ x x ... x 1 2 n 1 2 n ( ) ∑ 2 � � − ( ) x x 2 T sum length of edge x L x ( ) i j ∈ i j , E = = G � � ∑ ( ) T 2 2 x x x sum length away from 0 i φ G i x ≤ If At least edges must cross over i n 2 i 22

∆ 8 λ ≤ Proof of Theorem 2 2 n Theorem 4 (Koebe-Andreev-Thurston) . Let G be a planar graph. Then, there exist a set of disks { } D 1 ,..., D n in the plane with disjoint interiors such that D touches D j i iff ( ) ∈ i j , E Kissing disks . 23

Proof of Theorem 2 cont. Stereographic Projection { } ( ) ( ) π π D 1 ,..., D n Circles in the plane -> circular caps on the sphere 24