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1 Spectral Graph Theory and its Applications Lillian Dai 6.454 Oct. 20, 2004
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Page 1: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

1

Spectral Graph Theory and its Applications

Lillian Dai6.454 Oct. 20, 2004

Page 2: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

2

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

Page 3: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Graph and Associated Matrices

Adjacency matrix

0 1 1 11 0 0 11 0 0 11 1 1 0

GA

=

Degree matrix

3 0 0 00 2 0 00 0 2 00 0 0 3

GD

=

Incidency matrix

1 1 1 0 01 0 0 1 0

0 1 0 0 10 0 1 1 1

GB

− = −

− − −

( ),G V E=

4V n= =

5E m= =

Laplacian matrix

G G GT

G G

L D A

B B

= −

=

1

2

3

4

Page 4: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Properties of the Laplacian Matrix

3 1 1 11 2 0 11 0 2 11 1 1 3

GL

− − − − − = − − − − −

1

2

3

4

• Symmetric -> real eigenvalues; eigenspaces are mutually orthogonal

• Orthogonally diagonalizable -> an eigenvalue with multiplicity k has k-dimensional eigenspace

{ }0,2,4,4λ =

1111

01

10

111

3

− − −

2110

Page 5: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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More Properties of the Laplacian Matrix

• Positive semidefinite -> non-negative eigenvalues

• Row sum = 0 -> singular -> at least one eigenvalue = 0, unity eigenvector (since row sum = 1)

• Orthogonal eigenspacesu = eigenvector of non-zero eigenvalue

{ }0,2,4,4λ =

1111

01

10

111

3

− − −

2110

( )( ) ( )( )

2

,

0TT T T T T

G G G G G i ji j E

x L x x B B x x B x B x x∈

= = = − ≥∑

( )1 2, ,..., nnx x x x= ∈

1

0n

ii

u=

=∑

1 m× 1m×

Page 6: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Star

Ring

Line

Complete

Eigenvalues

Spectrum of Some Graphs

, , ( ){ }10, nn −

( )2 2cos k nπ−

( )2 2cos 2 k nπ−1,...,k n=

( ){ }20,1 ,2n−

1,..., 2k n=

Which graphs are determined by their spectrum?

• Complete Graphs• Graphs with one edge• Graphs missing 1 edge• Regular graphs with degree 2• Regular graphs of degree n - 3

Page 7: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Graph Connectedness

For connected graphs,{ }0,2,4,4λ =

1111

01

10

111

3

− − −

2110

1 2 ... nλ λ λ≤ ≤ ≤

Fiedler Value

v Fiedler Vector

2 0λ >

Multiplicity of the 0 eigenvalueindicates # of connected components

( )( )

2

,

TG i j

i j E

x L x x x∈

= −∑Recall

If is eigenvector for eigenvalue 00GL x =

x

i jx x= ( ),i j E∈

Page 8: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Onto Graph Partitioning …

Page 9: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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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

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 encloses the maximum area Stein, 1841

Diagram from Berkeley CS 267 lecture notes

Page 10: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Applications

• Load balancing while minimizing communication

• Sparse matrix-vector multiplication

• Optimizing VLSI layout

• Communication network design

Page 11: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Bisection and Ratio-Partition

• Divide vertices into two disjoint subsets and

• Cut Size

• Cut Ratio

• Isoperimetric Number

SS

( ),E S S

( )( )( )

,

min ,G

E S SS

S Sφ =

( )minG GS VSφ φ

⊂=

( ),E S SBisection Minimize subject to # of nodes in each partition differ by at most 1.

Ratio-Partition Minimize ( )G Sφ

NP-Complete

Page 12: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Spectral Partitioning

• Find Fiedler vector of the Laplacian matrix – map to vertices• Choose some real number s• Partition vertices given by

• Bisection, s = median of • Ratio partition, s is chosen to give the best cut ratio

{ }:L iV i v s= ≤

{ }:L iV i v s= >

{ }1,... nv v

Page 13: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Example

1

2

3

4

5

6

Fiedler vector [-1 -2 -1 1 2 1]

Page 14: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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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

Page 15: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Simple Spectral Bisection May Fail(Guattery & Miller)

The simple spectral bisection method produces cut size of ( )nΘ

for kG , for any k

Page 16: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Optimal Bisector for Graphs with Bounded Genus(Kelner)

There is a spectral algorithm that produces bisector of size ( )O gn

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 0Torus has genus 1

For every g, there is a class of bounded degree graphs that have no bisectors smaller than ( )O gn

Page 17: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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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?

Page 18: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Another Look at Fiedler Value

Recall where

Rayleigh quotient:

Fiedler value satisfies with the minimum occurring only when is a Fiedler vector.

( )( )2

,2

T i ji j EGx T

i

x xx L xx x x

φ ∈−

= =∑

( )( )

2

,

TG i j

i j E

x L x x x∈

= −∑ ( )1 2, ,..., nnx x x x= ∈

( )2 1,...,1min xx

λ φ⊥

=

x

22

T TG

x T Tx L x x xx x x x

λφ λ= = =

Page 19: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Connection Between Fiedler Value and Isoperimetric Number

Theorem 1 (Mihail ‘89) Let be a graph on nodes of

maximum degree . For any vector such that

Moreover, there is an so that the cut has ratio at most

( )( )

,min

min ,G S V

E S S

S Sφ

⊂=Recall Isoperimetric Number

is the best ratio-partition possible

G n

∆ nx∈1

0n

ii

x=

=∑2

2

TG GT

x L xx x

φ≥

s

2

2 2Gφλ ≥∆

Good ratio-partition can be achieved if Fiedler value is small

{ }: ii v s≤ { }: ii v s>

( )2 2Gφ ∆

Page 20: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Upperbound on the Fiedler Value for Planar Graphs

Theorem 2 (Spielman & Teng ‘96) For all planar graphs with vertices and maximum degree

Gn ∆

28n

λ ∆≤

1On

2

28

2G

λ ∆≤ ≤

∆4

G nφ ∆

≤1On

By bounding Fiedler value of planar graphs, ratio-partitioning method is shown to work well

What about bisection?

Page 21: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Relationship Between Ratio-Partitioning and Bisection

Lemma 3 Given an algorithm that will find a cut ratio of at mostin every k-node subgraph of , for some monotonically

decreasing function . Then repeated application of this algorithm can be used to find a bisection of of size at most

G

( )1

n

xx dxφ

=∫

( )kφφ

G

( ) 1xx

φ = ( ) ( )1

2 1n

xx dx nφ

== −∫ ( )O n

Bisection can be obtained by repeated application of ratio-partitioning

Page 22: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Theorem 1

Map graph vertices to a line

2x

2

2

TG GT

x L xx x

φ≥

∆ 1

0n

ii

x=

=∑

1 2 ... nx x x≤ ≤ ≤

( )( )

,min

min ,G S V

E S S

S Sφ

⊂=

1x nx

If 2i n≤ At least Giφ edges must cross over ix

( )( ) ( )( )

22

,2 2

sum length of edge

sum length away from 0

T i ji j EGT

i

x xx L xx x x

∈−

= =∑

12

3

4

Page 23: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Proof of Theorem 2

Theorem 4 (Koebe-Andreev-Thurston). Let G

be a planar graph. Then, there exist a set of disks { }1,..., nD D

in the plane with disjoint interiors such that iD touches jD

iff ( ),i j E∈

. Kissing disks

28n

λ ∆≤

Page 24: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Proof of Theorem 2 cont.

Stereographic Projection( ) ( ){ }1 ,..., nD Dπ π

Circles in the plane -> circular caps on the sphere

Page 25: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Proof of Theorem 2 cont.

Let ix be the center of ( )iDπ on the sphere.

1ix =2

1

n

ii

x n=

=∑

Let ir be the radius of the cap ( )iDπ

( ) ( )22 2 22i j i j i jx x r r r r− ≤ + ≤ + ( ),i j E∈

2 4irπ π≤∑

( )( )

( )

2 2 2 2

, ,

2 2 8i j i j i ii j E i j E i

x x r r d r∈ ∈

− ≤ + ≤ ≤ ∆∑ ∑ ∑

( )2

,2 2

8i ji j E

i

x x

nxλ ∈

− ∆≤ ≤∑

Page 26: Spectral Graph Theory and its Applicationsweb.mit.edu/6.454/www/2 Outline •Basic spectral graph theory •Graph partitioning using spectral methods D. Spielman and S. Teng, “Spectral

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Conclusion

Why does the spectral method work?- close relationship between Fiedler value and Isoperimetric number

Why does it work well on planar graphs?- planar graphs have nice collection of spherical cap embeddings

Why does simple bisection fail even on planar graphs?- even though good ratio-partitions can be found, the result may be unbalanced in the size of the partitions

2

2 2Gφλ ≥∆

28n

λ ∆≤


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