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

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Spectral Graph Theory and its Applications

Lillian Dai 6.454

Oct. 20, 2004

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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

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Graph and Associated Matrices

Adjacency matrix

1 1 1 1 1 1 1 1 1 1

G

A       =      

Degree matrix

3 2 2 3

G

D       =      

Incidency matrix

1 1 1 1 1 1 1 1 1 1

G

B     −   =   −   − − −  

( )

, G V E = 4 V n = = 5 E m = =

Laplacian matrix

G G G T G G

L D A B B = − =

1 2 3 4

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Properties of the Laplacian Matrix

3 1 1 1 1 2 1 1 2 1 1 1 1 3

G

L − − −     − −   =   − −   − − −  

1 2 3 4

Symmetric -> real eigenvalues; eigenspaces are mutually
rthogonal
Orthogonally diagonalizable -> an eigenvalue with multiplicity k

has k-dimensional eigenspace

{ }

0,2,4,4 λ =

1 1 1 1             1 1     −         1 1 1 3 −     −     −     2 1 1 −            

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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 eigenspaces

u = eigenvector of non-zero eigenvalue

{ }

0,2,4,4 λ =

1 1 1 1             1 1     −         1 1 1 3 −     −     −     2 1 1 −            

( )( )

( )

2 , T T T T T T G G G G G i j i j E

x L x x B B x x B x B x x

∈

= = = − ≥

∑

( )

1 2

, ,...,

n n

x x x x = ∈

1 n i i

u

=

∑

1 m × 1 m×

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Star Ring Line Complete Eigenvalues

Spectrum of Some Graphs

, ,

( )

{ }

1

0,

n

−

( )

2 2cos k n π −

( )

2 2cos 2 k n π −

1,..., k n =

( )

{ }

2

0,1 ,2

n−

1,..., 2 k 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

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Graph Connectedness

For connected graphs,

{ }

0,2,4,4 λ =

1 1 1 1             1 1     −         1 1 1 3 −     −     −     2 1 1 −            

2

λ

1 2

...

n

λ λ λ ≤ ≤ ≤

Fiedler Value

v

Fiedler Vector

2

λ >

Multiplicity of the 0 eigenvalue indicates # of connected components

( )

2 , T G i j i j E

x L x x x

∈

= −

∑

Recall

If is eigenvector for eigenvalue 0

G

L x =

x
i

j

x x =

( )

, i j E ∈

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Onto Graph Partitioning …

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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

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Applications

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

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Bisection and Ratio-Partition

Divide vertices into two disjoint subsets and
Cut Size
Cut Ratio
Isoperimetric Number

S S

( )

, E S S

( )

, min ,

G

E S S S S S φ =

( )

min

G G S V

S φ φ

⊂

=

( )

, E S S

Bisection Minimize subject to # of nodes in each partition differ by at most 1. Ratio-Partition Minimize

( )

G S

φ

NP-Complete

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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 i

V i v s = ≤

{ }

:

L i

V i v s = >

{ }

1,... n

v v

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Example

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

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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

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Simple Spectral Bisection May Fail

(Guattery & Miller)

The simple spectral bisection method produces cut size of

( )

n Θ

for

k

G , for any k

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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 0 Torus has genus 1 For every g, there is a class of bounded degree graphs that have no bisectors smaller than (

)

O gn

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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?

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Another Look at Fiedler Value

Recall where Rayleigh quotient: Fiedler value satisfies with the minimum

ccurring only when is a Fiedler vector.

( )

2 , 2 T i j i j E G x T i

x x x L x x x x φ

∈

− = = ∑

∑

(

)

( )

2 , T G i j i j E

x L x x x

∈

= −

∑

( )

1 2

, ,...,

n n

x x x x = ∈

(

)

2 1,...,1

min

x x

λ φ

⊥

=

x

2

2 T T G x T T

x L x x x x x x x λ φ λ = = =

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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 S φ

⊂

=

Recall Isoperimetric Number is the best ratio-partition possible

G n ∆

n

x ∈

1

n i i

x

=

∑

2

T G G T

x L x x x φ ≥ ∆

s

2 2

2

G

φ λ ≥ ∆

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

{ }

:

i

i v s ≤

{ }

:

i

i v s >

( )

2

G

φ ∆

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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

G n ∆

2

8 n λ ∆ ≤ 1 O n      

2 2

8 2

G

n φ λ ∆ ≤ ≤ ∆ 4

G

n φ ∆ ≤ 1 O n      

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

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Relationship Between Ratio-Partitioning and Bisection

Lemma 3 Given an algorithm that will find a cut ratio of at most in 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 x

x dx φ

=

∫

( )

k φ φ

G

( )

1 x x φ =

( )

1

2 1

n x

x dx n φ

=

= −

∫

( )

O n

Bisection can be obtained by repeated application of ratio- partitioning

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Theorem 1

Map graph vertices to a line

2

x

2

T G G T

x L x x x φ ≥ ∆

1

n i i

x

=

∑

1 2

...

n

x x x ≤ ≤ ≤

( )

, min min ,

G S V

E S S S S φ

⊂

=

1

x

n

x

If

2 i n ≤

At least

Gi

φ

edges must cross over

i

x

( )

( ) ( )

2 2 , 2 2

sum length of edge sum length away from 0

T i j i j E G T i

x x x L x x x x

∈

− = =

∑ ∑

1

2 3 4

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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,..., n

D D

in the plane with disjoint interiors such that

i

D touches

j

D

iff (

)

, i j E ∈

.

Kissing disks

2

8 n λ ∆ ≤

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Proof of Theorem 2 cont.

Stereographic Projection

( ) ( )

{ }

1 ,..., n

D D π π

Circles in the plane -> circular caps on the sphere

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Proof of Theorem 2 cont.

Let

i

x

be the center of

( )

i

D π

n the sphere.

1

i

x =

2

1 n i i

x n

=

∑

Let

i

r

be the radius of the cap

( )

i

D π

( )

2 2 2 2

2

i j i j i j

x x r r r r − ≤ + ≤ +

(

)

, i j E ∈

2

4

i

r π π ≤

∑

( )

2 2 2 2 , ,

2 2 8

i j i j i i i j E i j E i

x x r r d r

∈ ∈

− ≤ + ≤ ≤ ∆

∑ ∑ ∑

(

)

2 , 2 2

8

i j i j E i

x x n x λ

∈

− ∆ ≤ ≤

∑ ∑

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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

2

G

φ λ ≥ ∆

2

8 n λ ∆ ≤