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Laplacian Matrices of Graphs: Spectral and Electrical Theory
Daniel A. Spielman Dept. of Computer Science
Program in Applied Mathematics Yale University
Toronto, Sep. 28, 2011
Outline
Introduction to graphs
Physical metaphors
Laplacian matrices
Spectral graph theory
A very fast survey
Trailer for lectures 2 and 3
Graphs and Networks V: a set of vertices (nodes) E: a set of edges an edge is a pair of vertices
Dan
Donna Allan
Gary Maria
Nikhil
ShangHua
Difficult to draw when big
Examples of Graphs
Examples of Graphs
8
5
1
7 3 2
9 4 6
10
Examples of Graphs
8
5
1
7 3 2
9 4 6
10
How to understand a graph
Use physical metaphors Edges as rubber bands Edges as resistors
Examine processes Diffusion of gas Spilling paint
Identify structures Communities
How to understand a graph
Use physical metaphors Edges as rubber bands Edges as resistors
Examine processes Diffusion of gas Spilling paint
Identify structures Communities
Nail down some ver8ces, let rest se
Nail down some ver8ces, let rest se
Nail down some ver8ces, let rest se
Nail down some ver8ces, let rest se
If nail down a face of a planar 3connected graph, get a planar embedding!
Tuttes Theorem 63
3connected: cannot break graph by cuRng 2 edges
Tuttes Theorem 63
3connected: cannot break graph by cuRng 2 edges
Tuttes Theorem 63
Tuttes Theorem 63 3connected: cannot break graph by cuRng 2 edges
Tuttes Theorem 63 3connected: cannot break graph by cuRng 2 edges
Graphs as Resistor Networks
View edges as resistors connecting vertices
Apply voltages at some vertices. Measure induced voltages and current flow.
1V
0V
Graphs as Resistor Networks
View edges as resistors connecting vertices
Apply voltages at some vertices. Measure induced voltages and current flow.
Current flow measures strength of connection between endpoints.
More short disjoint paths lead to higher flow.
Graphs as Resistor Networks
View edges as resistors connecting vertices
Apply voltages at some vertices. Measure induced voltages and current flow.
1V
0V
0.5V
0.5V
0.625V 0.375V
Graphs as Resistor Networks
View edges as resistors connecting vertices
Apply voltages at some vertices. Measure induced voltages and current flow.
Induced voltages minimize
(a,b)E
(v(a) v(b))2
Subject to fixed voltages (by battery)
Learning on Graphs [ZhuGhahramaniLafferty 03]
Infer values of a function at all vertices from known values at a few vertices.
Minimize
Subject to known values
(a,b)E
(x(a) x(b))2
0
1
Learning on Graphs [ZhuGhahramaniLafferty 03]
Infer values of a function at all vertices from known values at a few vertices.
Minimize
Subject to known values
(a,b)E
(x(a) x(b))2
0
1 0.5
0.5
0.625 0.375
The Laplacian quadratic form
(a,b)E
(x(a) x(b))2
xTLx =
(a,b)E
(x(a) x(b))2The Laplacian matrix of a graph
The Laplacian matrix of a graph
To minimize subject to boundary constraints, set derivative to zero.
Solve equation of form
xTLx =
(a,b)E
(x(a) x(b))2
Lx = b
Weighted Graphs
Edge assigned a nonnegative real weight measuring strength of connection spring constant 1/resistance
wa,b R
xTLx =
(a,b)E
wa,b(x(a) x(b))2
(a, b)
Weighted Graphs
Edge assigned a nonnegative real weight measuring strength of connection spring constant 1/resistance
wa,b R
xTLx =
(a,b)E
wa,b(x(a) x(b))2
(a, b)
Ill show the matrix entries tomorrow
Measuring boundaries of sets
Boundary: edges leaving a set
S S
Measuring boundaries of sets
Boundary: edges leaving a set
S S
Measuring boundaries of sets
Boundary: edges leaving a set
S
0 0
0 0
0 0
1
1 0
1 1
1 1 1
0
0 0
1
S
Characteristic Vector of S:
x(a) =
1 a in S
0 a not in S
Measuring boundaries of sets
Boundary: edges leaving a set
S
0 0
0 0
0 0
1
1 0
1 1
1 1 1
0
0 0
1
S
Characteristic Vector of S:
x(a) =
1 a in S
0 a not in S
xTLx =
(a,b)E
(x(a) x(b))2 = boundary(S)
Cluster Quality
S
0
0 0
0 0
1
1 0
1 1
1 1 1
0
0 0
1
S
Number of edges leaving S Size of S
=boundary(S)
Sdef= (S) (sparsity)
Cluster Quality
S
0
0 0
0 0
1
1 0
1 1
1 1 1
0
0 0
1
S
Number of edges leaving S Size of S
=boundary(S)
Sdef= (S)
The Rayleigh Quotient of with respect to x L
(sparsity)
=xTLx
xTx=
(a,b)E(x(a) x(b))2
a x(a)2
Spectral Graph Theory A nbyn symmetric matrix has n real eigenvalues and eigenvectors such that
1 2 nv1, ..., vn
Lvi = ivi
Spectral Graph Theory A nbyn symmetric matrix has n real eigenvalues and eigenvectors such that
1 2 nv1, ..., vn
These eigenvalues and eigenvectors tell us a lot about a graph!
Theorems Algorithms Heuristics
Lvi = ivi
The Rayleigh Quotient and Eigenvalues A nbyn symmetric matrix has n real eigenvalues and eigenvectors such that
1 2 nv1, ..., vn
CourantFischer Theorem:
1 = minx =0
xTLx
xTxv1 = argmin
x =0
xTLx
xTx
Lvi = ivi
The Courant Fischer Theorem
The Courant Fischer Theorem
k = minxv1,...,vk1
xTLx
xTx
The first eigenvalue
Setting for all x(a) = 1 a
We find and 1 = 0 v1 = 1
1 = minx =0
xTLx
xTx
= minx =0
(a,b)E(x(a) x(b))2
x2
The second eigenvalue
if and only if G is connected 2 > 0
Proof: if G is not connected, are two functions with Rayleigh quotient zero
1 1
0
0
0 0
1 1
The second eigenvalue
if and only if G is connected 2 > 0
Proof: if G is not connected, are two functions with Rayleigh quotient zero
0 0
1
1
1 1
0 0
The second eigenvalue
if and only if G is connected 2 > 0
Proof: if G is connected,

+
x 1
a x(a) = 0means
So must be an edge (a,b) for which x(a) < x(b) and so (x(a) x(b))2 > 0
The second eigenvalue
if and only if G is connected 2 > 0
Proof: if G is connected,
  +
+
x 1
a x(a) = 0means
So must be an edge (a,b) for which x(a) < x(b) and so (x(a) x(b))2 > 0
The second eigenvalue
if and only if G is connected 2 > 0
Fiedler (73) called the algebraic connectivity of a graph The further from 0, the more connected.
Cheegers Inequality [Cheeger 70] [AlonMilman 85, JerrumSinclair 89, DiaconisStroock 91]
1. is big if and only if G does not have good clusters.
2. If is small, can use to find a good cluster. v2
Cheegers Inequality [Cheeger 70] [AlonMilman 85, JerrumSinclair 89, DiaconisStroock 91]
1. is big if and only if G does not have good clusters.
2/2 minSn/2
(S)
2d2
When every vertex has d edges,
(S) =boundary(S)
S
Cheegers Inequality [Cheeger 70] [AlonMilman 85, JerrumSinclair 89, DiaconisStroock 91]
1. is big if and only if G does not have good clusters.
2. If is small, can use to find a good cluster. v2
In a moment
Spectral Graph Drawing [Hall 70]
31 2
4
56 7
8 9
Arbitrary Drawing
Spectral Graph Drawing Plot vertex at draw edges as straight lines
[Hall 70]
(v2(a), v3(a))a
31 2
4
56 7
8 9
12
4
5
6
9
3
8
7Arbitrary Drawing
Spectral Drawing
A Graph
Drawing of the graph using v2, v3
Plot vertex at a (v2(a), v3(a))
The Airfoil Graph, original coordinates
The Airfoil Graph, spectral coordinates
The Airfoil Graph, spectral coordinates
Spectral drawing of Streets in Rome
Spectral drawing of Erdos graph: edge between coauthors of papers
Dodecahedron
Best embedded by first three eigenvectors
Condition for eigenvector
Spectral graph