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Fan Chung
University of California, San Diego
The PageRank of a graph
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What is PageRank?
•Ranking the vertices of a graph
•Partially ordered sets + graph theory
-Applications --web search,ITdata sets, xxxxxxxxxxxxxpartitioning algorithms,…
-Theory --- correlations among vertices
A new graph invariant for dealing with:
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fan
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Outline of the talk
• Motivation
• Local cuts and the Cheeger constant
using: eigenvectors random walks PageRank heat kernel
• Define PageRank
• Four versions of the Cheeger inequality
• Four partitioning algorithms
• Green’s functions and hitting time
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What is PageRank?
What is Rank?
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What is PageRank?
PageRank is defined on any graph.
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An induced subgraph of the collaboration graph with authors of Erdös number ≤ 2.
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A subgraph of the Hollywood graph.
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A subgraph of a BGP graph
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Yahoo IM graph Reid Andersen 2005
The Octopus graph
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Graph Theory has 250 years of history.
Leonhard Euler 1707-
1783
The Bridges of Königsburg
Is it possible to walk over every bridge once and only once?
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Geometric graphs
Algebraic graphs
General graphs
Topological graphs
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Massive dataMassive graphs
• WWW-graphs
• Call graphs
• Acquaintance graphs
• Graphs from any data a.base
protein interaction network Jawoong Jeong
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Big and bigger graphs
New directions in graph theory
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Many basic questions:
• Correlation among vertices?
• The `geometry’ of a network ?
• Quantitative analysis?
distance, flow, cut, …
eigenvalues, rapid mixing, …
• Local versus global?
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Google’s answer:
The definition for PageRank?
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A measure for the “importance” of a website
1 14 79 785( )x R x x x
2 1002 3225 9883 30027( )x R x x x x
The “importance” of a website is proportional
to the sum of the importance of all the sites
that link to it.
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A solution for the “importance” of a website
1 14 79 785( )x x x x
2 1002 3225 9883 30027( )x x x x x
Solve 1 2( , , , )nx x x for1
n
i ij jj
x a x
x
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A solution for the “importance” of a website
1 14 79 785( )x x x x
2 1002 3225 9883 30027( )x x x x x
Solve 1 2( , , , )nx x x for1
n
i ij jj
x a x
x
x = ρ A x ij n nA a
Adjacency matrix
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A solution for the “importance” of a website
1 14 79 785( )x x x x
2 1002 3225 9883 30027( )x x x x x
Solve 1 2( , , , )nx x x for1
n
i ij jj
x a x
x
x = ρ A x
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Graph models
directed graphs
weighted graphs
(undirected) graphs
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Graph models
directed graphs
weighted graphs
(undirected) graphs
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Graph models
directed graphs
weighted graphs
(undirected) graphs
1.5
1.2
3.3\
1.5
2
2.3
1
2.81.1
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In a directed graph,
authority
there are two types of “importance”:
hub Jon Kleinberg 1998
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Two types of the “importance” of a website
Solve
1 2( , , , )nx x x
and
x
x = r A y
Importance as Authorities :
1 2( , , , )my y y yImportance as Hubs :
y = s A xT
x = rs A A xT
y = rs A A yT
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Eigenvalue problem for n x n matrix:.
n ≈ 30 billion websites
Hard to compute eigenvalues
Even harder to compute eigenvectors
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In the old days,
compute for a given (whole) graph.
In reality,
can only afford to compute “locally”.
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A traditional algorithm
Input: a given graph on n vertices.
New algorithmic paradigm
Input: access to a (huge) graph
Efficient algorithm means polynomial algorithms
n3, n2, n log n, n
(e.g., for a vertex v, find its neighbors)Bounded number of access.
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A traditional algorithm
Input: a given graph on n vertices.
New algorithmic paradigm
Input: access to a (huge) graph
Efficient algorithm means polynomial algorithms
n3, n2, n log n, n
(e.g., for a vertex v, find its neighbors)Bounded number of access.
Exponential polynomial
Infinity finite
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The definition of PageRank given by
Brin and Page is based on
random walks.
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Random walks in a graph.
1,
( , )
0 .u
if u vdP u v
otherwise
G : a graph
P : transition probability matrix
2
I PW
A lazy walk:
ud := the degree of u.
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A (bored) surfer
• either surf a random webpage
with probability 1- α
Original definition of PageRank
α : the jumping constant
• or surf a linked webpage
with probability α
1 1 1( , ,...., ) (1 )n n np pW
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Two equivalent ways to define PageRank pr(α,s)
(1) (1 )p s pW
s: the seed as a row vector
Definition of personalized PageRank
α : the jumping constant
s
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Two equivalent ways to define PageRank p=pr(α,s)
(1) (1 )p s pW
0
(1 ) ( )t t
t
p sW
(2)
s = 1 1 1( , ,...., )n n n
Definition of PageRank
the (original) PageRank
some “seed”, e.g.,s = (1,0,....,0)
personalized PageRank
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Depends on the applications?
How good is PageRank as a measure of correlationship?
Isoperimetric properties
How “good” is the cut?
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Isoperimetric properties
“What is the shortest curve enclosing a unit area?”
In a graph G and an integer m,
what is the minimum cut disconnecting a subgraph of ≥ m vertices?In a graph G, what is the minimum cut e(S,V-S)
so that e(S,V-S) is the smallest?_____Vol S
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Two types of cuts:
• Vertex cut
• edge cut
How “good” is the cut?
S
E(S,V-S)
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e(S,V-S)_____Vol S
e(S,V-S)_____
|S|
Vol S = Σ deg(v)
v ε S|S| = Σ 1
v ε S
S
V-S
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The Cheeger constant
( , )min
min( , )GS
e S Sh
vol S vol S
The Cheeger constant for graphs
hG and its variations are sometimes called
“conductance”, “isoperimetric number”, …
Gh
Gh
The volume of S is ( ) xx S
vol S d
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The Cheeger constant
( , )min ( ) min
min( , )GS S
e S Sh h S
vol S vol S
The Cheeger inequality
2
22G
G
hh
The Cheeger inequality
: the first nontrivial eigenvalue of the xx(normalized) Laplacian of a connected graph.
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The spectrum of a graph
Many ways to define the spectrum of a graph.
How are the eigenvalues related to How are the eigenvalues related to
properties of graphs?properties of graphs?
•Adjacency matrix
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•Combinatorial Laplacian
L D A diagonal degree matrix
adjacency matrix
Gustav Robert Kirchhoff
1824-1887
The spectrum of a graph
•Adjacency matrix
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•Combinatorial Laplacian
L D A diagonal degree matrix
adjacency matrix
Gustav Robert Kirchhoff
1824-1887
The spectrum of a graph
•Adjacency matrix
Matrix tree theorem
0i
i
#spanning . trees
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•Combinatorial Laplacian
L D A diagonal degree matrix
adjacency matrix
Gustav Robert Kirchhoff
1824-1887
The spectrum of a graph
•Adjacency matrix
•Normalized Laplacian
Random walks
Rate of convergence
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The spectrum of a graph
not symmetric in general
•Normalized Laplacian
symmetric normalized
1( ) ( ( ) ( ))
y xx
f x f x f yd
L( , )x y 1 if x y
{ 1
x y
if x y and x yd d
Discrete Laplace operator
with eigenvalues 0 1 10 2n
( , )L x y 1 if x y
{ 1
x
if x y and x yd
loopless, simple
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The spectrum of a graph
•Normalized Laplacian
symmetric normalized
1( ) ( ( ) ( ))
y xx
f x f x f yd
Discrete Laplace operator
with eigenvalues 0 1 10 2n
=
( , )L x y 1 if x y
{ 1
x
if x y and x yd
L( , )x y 1 if x y
{ 1
x y
if x y and x yd d
not symmetric in general
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dictates many properties
of a graph.• expander
• diameter
• discrepancy
• subgraph containment
• ….
Spectral implications for finding good cuts?
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Finding a cut by a sweep
For : ( ) ,f V G R
1 2
1 2 ( )( ) ( ).
n
n
v v v
f vf v f v
d d d
1{ , , }, 1,..., ,fj jS v v j n
order the vertices
Consider sets
and the Cheeger constant of
( ) min ( )fj
jh f h S
.fjS
Define
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Using a sweep by the eigenvector,
can reduce the exponential number of
choices of subsets to a linear number.
Finding a cut by a sweep
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Using a sweep by the eigenvector,
can reduce the exponential number of
choices of subsets to a linear number.
Finding a cut by a sweep
Still, there is a lower bound guarantee
by using the Cheeger inequality.
2
22
hh
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Four types of Cheeger inequalities.
eigenvectors random walks PageRank heat kernel
Four proofs using:
Leading to four different
one-sweep partitioning algorithms.
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• graph spectral method
• random walks
• PageRank
• heat kernel
spectral partition algorithm
local partition algorithms
Four proofs of Cheeger inequalities
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Graph partitioning Local graph partitioning
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What is a local graph partitioning algorithm?
A local graph partitioning algorithm finds a small
cut near the given seed(s) with running time
depending only on the size of the output.
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Examples of local partitioning
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Examples of local partitioning
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Examples of local partitioning
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Examples of local partitioning
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Examples of local partitioning
h
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h
h
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• graph spectral method
• random walks
• PageRank
• heat kernel
spectral partition algorithm
local partition algorithms
Four proofs of Cheeger inequalities
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• graph spectral method
• random walks
• PageRank
• heat kernel
Lovasz, Simonovits, 90, 93 Spielman, Teng, 04
Andersen, Chung, Lang, 06
Chung, PNAS , 08.
Four proofs of Cheeger inequalities
Cheeger 60’s, Fiedler ’73Alon 86’, Jerrum+Sinclair
89’
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where is the first non-trivial eigenvalue of the Laplacian and is the minimum Cheeger ratio in a sweep using the eigenvector .
f
2 2
22 2
hh
Using eigenvector
the Cheeger inequality can be stated as
The Cheeger inequality
Partition algorithm
,
f
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Proof of the Cheeger inequality:
from definition
by Cauchy-Schwarz ineq.
summation by parts.
from the definition.
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2 2
28 8G h
h
where is the minimum Cheeger ratio over sweeps by using a lazy walk of k steps from every vertex for an appropriate range of k .
A Cheeger inequality using random walks
G
Lovász, Simonovits, 90, 93
2( )( , ) ( ) 1
8
k
k k
u
vol SW u S S
d
Leads to a Cheeger inequality:
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Using the PageRank vector.
A Cheeger inequality using PageRank
Recall the definition of PageRank p=pr(α,s):
(1) (1 )p s pW
0
(1 ) ( )t t
t
p sW
(2)
Organize the random walks by a scalar α.
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Random walks versus PageRank
How fast is the convergence to the
stationary distribution? Choose α to satisfy the
required property. For what k, can one
have
?
kf W
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with seed as a subset S
2 2
8log 8logS S
S S
hh
s s
Using the PageRank vector
and a Cheeger
inequality can be obtained :
where S is the Dirichlet eigenvalue of the Laplacian, and is the minimum Cheeger ratio over sweeps by using the appropriate personalized PageRank with seeds S.
A Cheeger inequality using PageRank
S
( ) ( ) / 4,vol S vol G
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2
2
( ( ) ( ))inf
( )u v
Sf
ww
f u f v
f w d
over all f satisfying the Dirichlet boundary
condition:
Dirichlet eigenvalues for a subset
( ) 0f v
S V
S
V
for all .v S
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min ( )
ST S
h h T
Local Cheeger constant for a subset
S V
S
V
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with seed as a subset S
2 2
8log 8logS S
S S
hh
s s
Using the PageRank vector
and a Cheeger
inequality can be obtained :
where S is the Dirichlet eigenvalue of the Laplacian, and is the minimum Cheeger ratio over sweeps by using personalized PageRank with seed S.
A Cheeger inequality using PageRank
S
( ) ( ) / 4,vol S vol G
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Algorithmic aspects of PageRank
• Fast approximation algorithm for x personalized PageRank
Can use the jumping constant to approximate PageRank with a support of the desired size.
greedy type algorithm, almost linear complexity
•
• Errors can be effectively bounded.
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A graph partition algorithm using PageRank
Given a set S with ,Sh
randomly choose a vertex v in S.
With probability at least1
,2
the one-sweep algorithm using
( , )f pr v
has an initial segment with the Cheeger
constant at most ( log | |).fh O S
1
2( ) ( ),vol S vol G
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Graph partitioning using PageRank vector.
198,430 nodes and 1,133,512 edges
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Kevin Lang 2007
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• graph spectral method
• random walks
• PageRank
• heat kernel
Lovasz, Simonovits, 90, 93 Spielman, Teng, 04
Andersen, Chung, Lang, 06
Chung, PNAS , 08.
Four proofs of Cheeger inequalities
Fiedler ’73, Cheeger, 60’sAlon 86’
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PageRank versus heat kernel
,0
(1 ) ( )k ks
k
p sW
,0
( )
!
kt
t sk
tWe s
k
Geometric sum
Exponential sum
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PageRank versus heat kernel
,0
(1 ) ( )k ks
k
p sW
,0
( )
!
kt
t sk
tWe s
k
Geometric sum
Exponential sum
(1 )p pW
recurrence
( )I Wt
Heat equation
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A Cheeger inequality using the heat kernel
2, / 4
,
( )( ) ( ) t ut
t uu
vol SS S e
d
Theorem:
where is the minimum Cheeger ratio over sweeps by using heat kernel pagerank over all u in S.
,t u
Theorem: For
, ( ) ( ) .2
St
t S
eS S
2/3( ) ( ) ,vol S vol G
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22( ... ...)
2 !
kt k
t
t tH e I tW W W
k
( )t I We
Definition of heat kernel
tLe2
2 ... ( 1) ...2 !
kk kt t
I tL L Lk
( )t tH I W Ht
,t s tsH
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2 2
8 8S S
S S
hh
Using the upper and lower bounds,
a Cheeger inequality can be
obtained :
where S is the Dirichlet eigenvalue of the Laplacian, and is the minimum Cheeger ratio over sweeps by using heat kernel with seeds S for appropriate t.
A Cheeger inequality using the heat kernel
S
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• Covering and packing using PageRank
Many applications of PageRank for problems in Graph Theory
• Pebbing and routing using PageRank
• Graph embedding using PageRank
• Relating graph invariants of subgraphs
to the host graph using PageRank
• Your favorite old problem using PageRank?
• Graph drawing using PageRank
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• Random graphs with general degrees
• pageranks
• Algorithmic game theory, graphical
games
New Directions in Graph Theory for information networks
Topics:
Using: • Spectral methods
• Probabilistic methods
• Quasirandom …
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