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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
Spectral Graph Theory and You:Matrix Tree Theorem and Centrality Metrics
Jonathan Gootenberg
March 11, 2013
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
Outline of Topics
1 IntroductionMotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
2 Matrix Tree TheoremPreliminary concepts
Proof of Matrix Tree Theorem
3 PageRank and metrics of centrality
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Why use spectral graph theory?
The eigenvalues of a graphs adjacency matrix (and others) canreveal important information about the community structure.The number of matrix treesThe most central nodes
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Easily calculate all spanning trees
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Find most central nodes in a network
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Some denitions: Babys rst matrix
Denition (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
aij =1, {v i , v j } E 0, otherwise
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Some denitions: Babys rst matrix
G = A(G ) =0 1 0 11 0 1 10 1 0 1
1 1 1 0
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Putting the spec in Spectral
Denition (Spectrum of a graph)
The spectrum of graph G is the set of eigenvalues of A(G ), alongwith their multiplicities. If the eigenvalues of A(G ) are 0 > 0 > . . . > s 1 with multiplicites m( 0), . . . , m( s 1) then
Spec G = 0 1 s 1
m( 0) m( 1) m( s 1)
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Outline
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Putting the spec in Spectral
A(G ) =
0 1 0 1
1 0 1 10 1 0 11 1 1 0
Spec G =12 1 + 17 12 1 17 1 0
1 1 1 1
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
Putting the spec in Spectral
A(G ) =
0 1 0 1
1 0 1 10 1 0 11 1 1 0
Spec G =12 1 + 17 12 1 17 1 0
1 1 1 1
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Outline M i i
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OutlineIntroduction
Matrix Tree TheoremPageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Denition (Characteristic Polynomial)The characteristic polynomial of G , (G , ) is
(G , ) = det( I A)= n + c 1 n 1 + c 2 n 2 + . . . + c n
c 1 =i
i
= Tr( A)
= 0
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Outline M ti ti
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Denition (Characteristic Polynomial)The characteristic polynomial of G , (G , ) is
(G , ) = det( I A)= n + c 1 n 1 + c 2 n 2 + . . . + c n
c 1 =i
i
= Tr( A)
= 0
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Outline Motivation
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Denition (Characteristic Polynomial)The characteristic polynomial of G , (G , ) is
(G , ) = det( I A)= n + c 1 n 1 + c 2 n 2 + . . . + c n
c 1 =i
i
= Tr( A)
= 0
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Outline Motivation
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Denition (Characteristic Polynomial)The characteristic polynomial of G , (G , ) is
(G , ) = det( I A)= n + c 1 n 1 + c 2 n 2 + . . . + c n
c 1 =i
i
= Tr( A)
= 0
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Outline Motivation
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Characteristic Polynomial (G , ) = n + c 1 n 1 + c 2 n 2 + . . . + c n
You can express c i in terms of the principal minors of A
Principal MinorThe 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
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
MotivationBasics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Characteristic Polynomial
(G , ) = n
+ c 1n 1
+ c 2n 2
+ . . . + c n
You can express c i in terms of the principal minors of A
c i (1) i is the sum of the principal minors of size i i c i (1)
i
= |J |= i det(A)J , J
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
Basics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c 2
All non-zero principal minors are of the form0 11 0
Therefore,
c
2 =
|E
|
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
Basics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c 3
The non-trivial principal matrices of size 3 are
0 1 01 0 00 0 0
,0 1 11 0 01 0 0
,0 1 11 0 11 1 0
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
Basics of Spectral Graph TheoryUnderstanding the characteristic polynomial
What does the spectrum of a graph tell us?
Consider c 3
0 1 01 0 00 0 0
= 0 ,0 1 11 0 01 0 0
= 0 ,0 1 11 0 11 1 0
= 2
Therefore, c 3 is twice the number of triangles in G
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OutlineIntroduction Preliminary concepts
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
Preliminary conceptsProof of Matrix Tree Theorem
The incidence matrix
Denition (Incidence Matrix S )Given an undirected graph G = ( V , E ) with
V = {1, . . . , n}, E = {e 1, . . . , e m } the incidence matrix of G is then m matrix S = S (G ) with entries S ij such that
S ij =1, e j ends at i
1, e j starts at i
0, otherwise
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IntroductionMatrix Tree Theorem
PageRank and metrics of centrality
Preliminary conceptsProof of Matrix Tree Theorem
The incidence matrix
G = S (G ) =1 0 0 1 0
1 1 0 0 10 1 1 0 00 0 1
1
1
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Matrix Tree TheoremPageRank and metrics of centrality
y pProof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |
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Matrix Tree TheoremPageRank and metrics of centrality
y pProof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |Reorder the graph such that
S =
S 1 0S 2
...... . . .0 S c
where c is the number of connected components of G We wish toshow the rank of each component is ni 1
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |For a given connected component S i , suppose we have a
summation over the rows s j n i
j =1
j s j = 0
1 0 0 1 0
1 1 0 0 10 1 1 0 00 0 1 1 115/19
OutlineIntroduction Preliminary concepts
f f
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |Consider a row s k , k = 0
1 0 0 1 0
1 1 0 0 10 1
1 0 00 0 1 1 1
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OutlineIntroduction
M i T ThPreliminary conceptsP f f M i T Th
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |Each non-zero column in s k has a corresponding rows l = k = l
1 0 0 1 0
1
1 0 0 1
0 1 1 0 00 0 1 1 -1
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OutlineIntroduction
Matri Tree TheoremPreliminary conceptsProof of Matri Tree Theorem
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theoremrank(S) = n - |number of connected components of G |Since the graph is connected, all j = 1, and
n i
j =1s j = 0
Therefore, the rank of S i is ni
1
Remark-S Since S i has rank ni 1, we can remove an arbitrary row to createS i without loss of information
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Matrix Tree TheoremPreliminary conceptsProof of Matrix Tree Theorem
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
The incidence matrix implies connectivity of the graph
Theorem
rank(S) = n - |number of connected components of G |Remark-S Since S i has rank ni 1, we can remove an arbitrary row to createS i without loss of information
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Matrix Tree TheoremPreliminary conceptsProof of Matrix Tree Theorem
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n 1 edges anduse the determinant to see if the resulting subgraph is connected.We use create the matrix that is the combination of the columnsof the incidence matrix S : Q = S S T
1 0 0 1 0
1
1 0 0 1
0 1 1 0 0
1 1 00 1 10 0
11 0 00 1 0
=2 1 0
1 3
1
0 1 2
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OutlineIntroductionMatrix Tree Theorem
Preliminary conceptsProof of Matrix Tree Theorem
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n 1 edges anduse the determinant to see if the resulting subgraph is connected.We use create the matrix that is the combination of the columnsof the incidence matrix S : Q = S S T
1 0 0 1 0
1
1 0 0 1
0 1 1 0 0
1 1 00 1 10 0
11 0 00 1 0
=2 1 0
1 3
1
0 1 2
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OutlineIntroductionMatrix Tree Theorem
Preliminary conceptsProof of Matrix Tree Theorem
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Matrix Tree TheoremPageRank and metrics of centrality
Proof of Matrix Tree Theorem
Matrix Tree Theorem
In the following proof, we will try all selections of n 1 edges anduse the determinant to see if the resulting subgraph is connected.We use create the matrix that is the combination of the columnsof the incidence matrix S : Q = S S T
1 0 0 1 0
1
1 0 0 1
0 1 1 0 0
1 1 00 1 10 0
11 0 00 1 0
=2 1 0
1 3
1
0 1 2
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OutlineIntroductionMatrix Tree Theorem
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S
S
T
Graph Laplacian Q Q is referred to as the graph laplacian, and can also be expressedas
Q = D Awhere D is the degree matrix
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OutlineIntroductionMatrix Tree Theorem
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S S T
LemmaIf T = ( V , E ) is an directed tree rooted at n, we can order E suchthat e i ends at i.
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Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S S T
LemmaIf T = ( V , E ) is an directed tree rooted at n, we can order E suchthat e i ends at i.
Proof.Label edges such that e i := ( p (i ), i ), where p (i ) is the parent of i
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OutlineIntroductionMatrix Tree Theorem
k d i f li
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S S T
LemmaIf G = ( V , E ),
|E
| = n
1 is a directed graph that is not a tree,
det( S ) = 0
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OutlineIntroductionMatrix Tree Theorem
P R k d t i f t lit
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S S T
LemmaIf G = ( V , E ), |E | = n 1 is a directed graph that is not a tree,det( S ) = 0Proof.If |E | = n 1 and G is not a tree, then it is not connected, andrank(S ) = rank( S ) n 2
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PageRank and metrics of centrality
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
TheoremThe number of spanning trees of a graph G is equal to
det( Q ), where Q = S S T
LemmaIf T = ( V , E ),
|E
| = n
1 is a tree with e i
E ending at i
V ,
det( S ) = 1
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PageRank and metrics of centrality
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
LemmaIf T = ( V , E ), |E | = n 1 is a tree with e i E ending at i V ,det( S ) = 1Proof.By the results of the previous lemmas, we can order the verticessuch that p (i ) > i . Then
S =
1 0 1
.. .
...... . . . . . . 0 0 1
which is upper diagonal.17/19
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PageRank and metrics of centrality
Preliminary conceptsProof of Matrix Tree Theorem
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PageRank and metrics of centrality
Matrix Tree Theorem
Proof.We will use the linearity of the determinant to break down det(Q )
into a sum of determinants of subgraphs:
det( Q ) = det( S j )
where all S j are subgraphs with n
1 edges (Tr(S j ) = n
1).
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PageRank and metrics of centrality
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PageRank and metrics of centrality
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a websurfer.
Jumps with probability q : q n . Like typing a URL
Surfs with probability 1q : j : j i f ( j )k out ( j ) . Like clicking a linkPageRank reccurance
f (i ) = q n + (1 q ) j : j i
f ( j )k out ( j )
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g y
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a websurfer.
Jumps with probability q : q n . Like typing a URL
Surfs with probability 1q : j : j i f ( j )k out ( j ) . Like clicking a linkPageRank reccurance
f (i ) = q n + (1 q ) j : j i
f ( j )k out ( j )
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PageRank and metrics of centrality
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g y
Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a websurfer.
Jumps with probability q : q n . Like typing a URL
Surfs with probability 1q : j : j i f ( j )k out ( j ) . Like clicking a linkPageRank reccurance
f (i ) = q n + (1 q ) j : j i
f ( j )k out ( j )
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PageRank and metrics of centrality
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Formulation of PageRank
PageRank (PR) is a centrality metric which approximates a websurfer.
Jumps with probability q : q n . Like typing a URL
Surfs with probability 1q : j : j i f ( j )k out ( j ) . Like clicking a linkPageRank reccurance
f (i ) = q n + (1 q ) j : j i
f ( j )k out ( j )
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PageRank and metrics of centrality
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Solving the PageRank reccurance
PageRank reccurance
f (i ) = q n
+ (1 q ) j : j i
f ( j )k out ( j )
Denition (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 p ij such that
p ij = 1
k a ik , {v i , v j } E
0, otherwise
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PageRank and metrics of centrality
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Solving the PageRank reccurance
PageRank reccurance
f (i ) =
q
n + (1 q ) j : j i f ( j )
k out ( j )
j : j i
f ( j )k out ( j )
= fP
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Solving the PageRank reccurance
PageRank reccurance
f (i ) = q
n + (1
q )
j : j i
f ( j )
k out ( j )
j : j i
f ( j )k out ( j )
= fP
We can use the property i f (i ) = 1 = f 1T
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PageRank and metrics of centrality
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Solving the PageRank reccurance
PageRank reccurance
f (i ) = q n
+ (1 q ) j : j i
f ( j )k out ( j )
j : j i
f ( j )k out ( j )
= fP
We can use the propertyi f (i ) = 1 = f 1T
f = q n
f 1T 1 + (1 q )fP = f q n
J + (1 q )P
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Solving the PageRank reccurance
PageRank reccurance
f (i ) = q n
+ (1 q ) j : j i
f ( j )k out ( j )
j : j i
f ( j )k out ( j )
= fP
We can use the property i f (i ) = 1 = f 1T
f = q n
f 1T 1 + (1 q )fP = f q n
J + (1 q )P We can compute PageRank with the eigenvector!
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