Post on 19-Aug-2020
Network Analysis
Graphs as Matrices
Michele Coscia & Luca RossiITU København
Background:
Background:
Looking at a problem from a different
perspective
Sergey Brin & Larry Page
(1973-)
55 Million!
55 Million!
PageRank
PageRank● p of visiting v…
PageRank● p of visiting v…● … = p of visiting a neighbor u of v *
p of going from u to v
PageRank● p of visiting v…● … = p of visiting a neighbor u of v *
p of going from u to v● for v in V…
PageRank● p of visiting v…● … = p of visiting a neighbor u of v *
p of going from u to v● for v in V…● 55 million times!
PageRankG = (V, E)
PageRankG = (V, E) A = 1 if uv in E
PageRankG = (V, E) A = 1 if uv in E
PageRankG = (V, E) A = 1 if uv in E
PageRank
for v in V...
G = (V, E) A = 1 if uv in E
PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
Row-normalized A
PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
Row-normalized A
Eigenvectorof A
Random Walks
Basics
A B C D E FA
B
C
D
E
F
A B
Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
Yes! Yes!
Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
Yes! Yes!
X
X
Basics
Basics
1 2 3 4 5 6 7 8 9
Basics
1 2 3 4 5 6 7 8 9123456789
Basics
1 2 3 4 5 6 7 8 9123456789
Basics
1 2 3 4 5 6 7 8 9123456789
Basics
1 2 3 4 5 6 7 8 9123456789
Basics
1 2 3 4 5 6 7 8 9123456789
Basics
1 2 3 4 5 6 7 8 9123456789
Adjacency Matrix
Properties of Adjacency Matrices
Properties of Adjacency Matrices
3
Properties of Adjacency Matrices
3
3
2
1
Properties of Adjacency Matrices
Stochastic Adjacency
A B C D E FA
B
C
D
E
F
AX X X
Stochastic Adjacency
A B C D E FA
B
C
D
E
F
Pick oneof your friends
at random
AX X X
Stochastic Adjacency
A B C D E FA
B
C
D
E
F
Pick oneof your friends
at random
A
Mmm…E!
X X X
Stochastic Adjacency
Stochastic Adjacency● p of going from
v to u
Stochastic Adjacency● p of going from
v to u
● A / A.sum(axis = 1)
Stochastic Adjacency● p of going from
v to u
● A / A.sum(axis = 1)
Stochastic Adjacency● p of going from v
to u
● A / A.sum(axis = 1)
● Useful eigen-properties
Walk Probabilities
Walk Probabilities
Walk Probabilities
Walk Probabilities
Walk Probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
Walk probabilities
p(1→2) in two steps?
Walk probabilities
p(1→2) in two steps?
(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))
Walk probabilities
p(1→2) in two steps?
(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))
p(u→v)?
Powers of Stochastic
A
Powers of Stochastic
A
Powers of Stochastic
A
Powers of Stochastic
A A2
Powers of Stochastic
A A2
Powers of Stochastic
A A2
Powers of Stochastic
A A2
Powers of Stochastic
A A2
Powers of Stochastic
Powers of Stochastic
Powers of Stochastic
Powers of Stochastic
Powers of Stochastic
1
Powers of Stochastic
Powers of Stochastic
Powers of Stochastic
A2
A2
A2 A3
A2 A30
Stationary Distribution
Stationary Distribution● A∞ = π
Stationary Distribution● A∞ = π
Stationary Distribution● A∞ = π
● πA = π
Stationary Distribution● A∞ = π
● πA = π
● ~ PageRank
Disconnected Mazes
Disconnected Mazes
Disconnected Mazes
Disconnected Mazes
?
Disconnected Mazes
?
Multiple Components
A
Multiple Components
A30
Multiple Components
A30
Multiple Components
A30
Multiple Components
A30
Multiple Components
A30
Google’s Solution
Google’s Solution
Google’s Solution
Beamme up,Scotty!
Google’s Solution
Google’s Solution
A
Google’s Solution
A
Google’s Solution
A+ε
Stationary Distribution vs Degree
Stationary Distribution vs Degree
Stationary Distribution vs Degree
Corr = 1
Stationary Distribution vs Degree
Corr = 1π = k / k.sum()
Stationary Distribution vs PageRank
Stationary Distribution vs PageRank
Stationary Distribution vs PageRank
Stationary Distribution vs PageRank
Stationary Distribution vs PageRank
Graph Spectrum
A Recap on Eigenvectors
v
(x, y)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
(x, y)
A Recap on Eigenvectors
v
w = Av
(x’, y’)(λx, λy)
(x, y)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
(x, y)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
Eigenvector
(x, y)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
Eigenvector
Eigenvalue
(x, y)
Right & Left
Av = λv
Right & Left
Av = λv Right Eigenvector
Right & Left
Av = λv Right Eigenvector
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
λ2
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvectors
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvectors
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1 v
2 v
3 … v
n-2 v
n-1 v
n
Eigenvectors
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1 v
2 v
3 … v
n-2 v
n-1 v
n
“Largest” Eigenvector
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
Eigenvectors & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
λ2
Eigenvectors & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
λ2
v1 v
2
Eigenvectors & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
λ2
v1 v
2
a
a
a
a
a
a
0
0
0
0
0
0
0
0
0
b
b
b
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1
λ1
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1
a
a
a
a
a
a
a
a
a
λ1
Left Largest Eigenvector
Left Largest Eigenvector
Left Largest Eigenvector
Left Largest Eigenvector
vL1 = π
Left Largest Eigenvector
vL1 = π
πA = π
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
λ1
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
λ1
Left & Right
Left & RightvL
1
Left & RightvL
1
Left & RightvL
1
v1
a
a
a
a
a
a
a
a
a
Left & RightvL
1
vR1
v1
a
a
a
a
a
a
a
a
a
Left & RightvL
1
vR1 A
Left & RightvL
1
vR1 AT
Left & RightvL
1
vR1 AT
A / A.sum(axis = 0)
Left & Right
AT
A / A.sum(axis = 0)
Left & Right
AT
A / A.sum(axis = 0)
vL1
vR1
Graph Laplacian
Degree Matrix
Degree MatrixZeros everywhere
Degree MatrixZeros everywhere
i-th diagonal element...
Degree MatrixZeros everywhere
i-th diagonal element... … is the degree of node i
Degree MatrixZeros everywhere
i-th diagonal element... … is the degree of node i
D
Laplacian Matrix
D A
Laplacian Matrix
D AL = -
Laplacian Matrix
L = D - A
Laplacian Matrix
L = D - A
Laplacian Matrix
L = D - A
Laplacian Matrix
L = D - A
0
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
A
L
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
A
L λ1 ≤ λ
2 ≤ λ
3 ≤ … ≤ λ
n-2 ≤ λ
n-1 ≤ λ
n
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
A
L λ1 ≤ λ
2 ≤ λ
3 ≤ … ≤ λ
n-2 ≤ λ
n-1 ≤ λ
n
0
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
A
L λ1 ≤ λ
2 ≤ λ
3 ≤ … ≤ λ
n-2 ≤ λ
n-1 ≤ λ
n
0
# of 0s = # of components
Eigenvectors
Eigenvectors
v1
Eigenvectors
v1
a
a
a
a
a
a
a
a
a
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
Eigenvectors
Eigenvectors
+
Eigenvectors
+-
Eigenvectors
Eigenvectors
2nd Eigenvector
Eigenvectors
2nd Eigenvector
3rd Eigenvector
Eigenvectors
Eigenvectors
Eigenvectors
Textbook Chapters● Adjacency Matrix
– Section 2.4● Stochastic, Graph Spectrum &
Laplacian– http://www.michelecoscia.com/?
page_id=1488● Done with Chapter 2!
– Section 2.11