Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Spectral Graph Theory and You:
Matrix Tree Theorem and Centrality Metrics
Jonathan Gootenberg
March 11, 2013
1 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Outline of Topics
1
Introduction
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
2
Matrix Tree Theorem
Preliminary concepts
Proof of Matrix Tree Theorem
3
PageRank and metrics of centrality
2 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Why use spectral graph theory?
The eigenvalues of a graph’s adjacency matrix (and others) can
reveal important information about the community structure.
The number of matrix trees
The most central nodes
3 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Easily calculate all spanning trees
4 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Find most central nodes in a network
5 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Some definitions: Baby’s first matrix
Definition (Adjacency Matrix A)
Given an undirected graph G = (V , E ), the adjacency matrix of G
is the n × n matrix A = A(G ) with entries aij such that
(
1, {vi , vj } ∈ E
aij =
0, otherwise
6 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Some definitions: Baby’s first matrix
G=

0
1
⇒ A(G ) = 
0
1
1
0
1
1
0
1
0
1

1
1

1
0
7 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Putting the spec in Spectral
Definition (Spectrum of a graph)
The spectrum of graph G is the set of eigenvalues of A(G ), along
with their multiplicities. If the eigenvalues of A(G ) are
λ0 > λ0 > . . . > λs−1 with multiplicites m(λ0 ), . . . , m(λs−1 ) then
λ0
λ1
···
λs−1
Spec G =
m(λ0 ) m(λ1 ) · · · m(λs−1 )
8 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Putting the spec in Spectral

0
1
A(G ) = 
0
1
√ 1
1 + 17
Spec G = 2
1
1
0
1
1
1
2

1
1

1
0
√ 1 − 17 −1 0
1
1 1
0
1
0
1
9 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
Putting the spec in Spectral

0
1
A(G ) = 
0
1
√ 1
1 + 17
Spec G = 2
1
1
0
1
1
1
2

1
1

1
0
√ 1 − 17 −1 0
1
1 1
0
1
0
1
9 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Definition (Characteristic Polynomial)
The characteristic polynomial of G , χ(G , λ) is
χ(G , λ) = det(λI − A)
=λn + c1 λn−1 + c2 λn−2 + . . . + cn
c1 =
X
λi
i
= Tr(A)
=0
10 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Definition (Characteristic Polynomial)
The characteristic polynomial of G , χ(G , λ) is
χ(G , λ) = det(λI − A)
=λn + c1 λn−1 + c2 λn−2 + . . . + cn
c1 =
X
λi
i
= Tr(A)
=0
10 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Definition (Characteristic Polynomial)
The characteristic polynomial of G , χ(G , λ) is
χ(G , λ) = det(λI − A)
=λn + c1 λn−1 + c2 λn−2 + . . . + cn
c1 =
X
λi
i
= Tr(A)
=0
10 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Definition (Characteristic Polynomial)
The characteristic polynomial of G , χ(G , λ) is
χ(G , λ) = det(λI − A)
=λn + c1 λn−1 + c2 λn−2 + . . . + cn
c1 =
X
λi
i
= Tr(A)
=0
10 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Characteristic Polynomial
χ(G , λ) = λn + c1 λn−1 + c2 λn−2 + . . . + cn
You can express ci in terms of the principal minors of A
Principal Minor
The principal minor det(A)J,J , J ⊂ {1, . . . , n} is the determiniant
of the subset of A the same subset J of columns and rows
11 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Characteristic Polynomial
χ(G , λ) = λn + c1 λn−1 + c2 λn−2 + . . . + cn
You can express ci in terms of the principal minors of A
ci (−1)i is the
P sum of the principal minors of size i × i
ci (−1)i = |J|=i det(A)J,J
11 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c2
All non-zero principal minors are of the form
0 1
1 0
Therefore, −c2 = |E |
11 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c3
The non-trivial principal
0 1
1 0
0 0
matrices of size 3
0 0 1 1 0
0 , 1 0 0 , 1
0 1 0 0 1
are
1 1
0 1
1 0
11 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Motivation
Basics of Spectral Graph Theory
Understanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c3
0 1 0 0 1 1 0 1 1
1 0 0 = 0, 1 0 0 = 0, 1 0 1 = 2
0 0 0 1 0 0 1 1 0
Therefore, −c3 is twice the number of triangles in G
11 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix
Definition (Incidence Matrix S)
Given an undirected graph G = (V , E ) with
V = {1, . . . , n}, E = {e1 , . . . , em } the incidence matrix of G is the
n × m matrix S = S(G ) with entries Sij such that


ej ends at i
1,
Sij = −1, ej starts at i


0,
otherwise
12 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix

G=

1
0
0
1
0
−1 −1 0
0
1

⇒ S(G ) = 
0
1 −1 0
0
0
0
1 −1 −1
13 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
14 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
Reorder the graph such that

S1


S2
S =
 ..
.
0 ···
···
..
.

0
.. 
.



Sc
where c is the number of connected components of G We wish to
show the rank of each component is ni − 1
14 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
For a given connected component Si , suppose we have a
summation over the rows sj
ni
X
αj sj = 0
j=1


1
0
0
1
0
−1 −1 0
0
1


0
1 −1 0
0
0
0
1 −1 −1
15 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
Consider a row sk , αk 6= 0

1
0
0
1
0
 −1 −1 0
0
1

 0
1 −1 0
0
0
0
1 −1 −1




15 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
Each non-zero column in sk has
sl =⇒ αk = αl

1
0
 −1 −1

 0
1
0
0
a corresponding row
0
1
0
0
−1 0
1 −1

0
1 

0 
-1
15 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
Since the graph is connected, all αj = 1, and
ni
X
sj = 0
j=1
Therefore, the rank of Si is ni − 1
Remark-S̃
Since Si has rank ni − 1, we can remove an arbitrary row to create
S̃i without loss of information
15 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |
Remark-S̃
Since Si has rank ni − 1, we can remove an arbitrary row to create
S̃i without loss of information
15 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n − 1 edges and
use the determinant to see if the resulting subgraph is connected.
We use create the matrix that is the combination of the columns
of the incidence matrix S̃: Q = S̃ S̃ T


0
 −1 1



1
0
0 1 0 
0
−1
1
2
−1
0


−1 −1 0 0 1  0


0 −1

 = −1 3 −1
0
1 −1 0 0  1
0
0
0 −1 2
0
1
0

16 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n − 1 edges and
use the determinant to see if the resulting subgraph is connected.
We use create the matrix that is the combination of the columns
of the incidence matrix S̃: Q = S̃ S̃ T


0
 −1 1



1
0
0 1 0 
0
−1
1
2
−1
0


−1 −1 0 0 1  0


0 −1

 = −1 3 −1
0
1 −1 0 0  1
0
0
0 −1 2
0
1
0

16 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n − 1 edges and
use the determinant to see if the resulting subgraph is connected.
We use create the matrix that is the combination of the columns
of the incidence matrix S̃: Q = S̃ S̃ T


0
 −1 1



1
0
0 1 0 
0
−1
1
2
−1
0


−1 −1 0 0 1  0


0 −1

 = −1 3 −1
0
1 −1 0 0  1
0
0
0 −1 2
0
1
0

16 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Graph Laplacian Q
Q is referred to as the graph laplacian, and can also be expressed
as
Q =D −A
where D is the degree matrix
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Lemma
If T = (V , E ) is an directed tree rooted at n, we can order E such
that ei ends at i.
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Lemma
If T = (V , E ) is an directed tree rooted at n, we can order E such
that ei ends at i.
Proof.
Label edges such that ei := (p(i), i), where p(i) is the parent of
i
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Lemma
If G = (V , E ), |E | = n − 1 is a directed graph that is not a tree,
det(S̃) = 0
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Lemma
If G = (V , E ), |E | = n − 1 is a directed graph that is not a tree,
det(S̃) = 0
Proof.
If |E | = n − 1 and G is not a tree, then it is not connected, and
rank(S̃) = rank(S) ≤ n − 2
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Theorem
The number of spanning trees of a graph G is equal to
det(Q), where Q = S̃ S̃ T
Lemma
If T = (V , E ), |E | = n − 1 is a tree with ei ∈ E ending at i ∈ V ,
det(S̃) = 1
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Lemma
If T = (V , E ), |E | = n − 1 is a tree with ei ∈ E ending at i ∈ V ,
det(S̃) = 1
Proof.
By the results of the previous lemmas, we can order the vertices
such that p(i) > i. Then


1 ∗ ··· ∗

.
0 1 . . . .. 


S̃ =  . .

..
.
.
.
.
. ∗
0 ··· 0 1
which is upper diagonal.
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Preliminary concepts
Proof of Matrix Tree Theorem
Matrix Tree Theorem
Proof.
We will use the linearity of the determinant to break down det(Q)
into a sum of determinants of subgraphs:
X
det(Q) =
det(S̃j )
where all Sj are subgraphs with n − 1 edges (Tr(Sj ) = n − 1).
17 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a web
surfer.
Jumps with probability q:
q
n.
Like typing a URL
P
f (j)
Surfs with probability 1 − q: j:j→i kout
(j) . Like clicking a link
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
18 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a web
surfer.
Jumps with probability q:
q
n.
Like typing a URL
P
f (j)
Surfs with probability 1 − q: j:j→i kout
(j) . Like clicking a link
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
18 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a web
surfer.
Jumps with probability q:
q
n.
Like typing a URL
P
f (j)
Surfs with probability 1 − q: j:j→i kout
(j) . Like clicking a link
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
18 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a web
surfer.
Jumps with probability q:
q
n.
Like typing a URL
P
f (j)
Surfs with probability 1 − q: j:j→i kout
(j) . Like clicking a link
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
18 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Solving the PageRank reccurance
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
Definition (Transition Matrix P)
Given an undirected graph G = (V , E ), the transition matrix of G
is the n × n matrix P = P(G ) with entries pij such that
(
P 1 , {vi , vj } ∈ E
k aik
pij =
0,
otherwise
19 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Solving the PageRank reccurance
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
X
j:j→i
f (j)
= fP
kout (j)
19 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Solving the PageRank reccurance
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
X
j:j→i
We can use the property
P
i
f (j)
= fP
kout (j)
f (i) = 1 = f 1T
19 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Solving the PageRank reccurance
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
X
j:j→i
f (j)
= fP
kout (j)
f (i) = 1 = f 1T
q
q
f = f 1T 1 + (1 − q)fP = f
J + (1 − q)P
n
n
We can use the property
P
i
19 / 19
Outline
Introduction
Matrix Tree Theorem
PageRank and metrics of centrality
Solving the PageRank reccurance
PageRank reccurance
f (i) =
X f (j)
q
+ (1 − q)
n
kout (j)
j:j→i
X
j:j→i
f (j)
= fP
kout (j)
f (i) = 1 = f 1T
q
q
f = f 1T 1 + (1 − q)fP = f
J + (1 − q)P
n
n
We can use the property
P
i
We can compute PageRank with the eigenvector!
19 / 19