network analysis graphs as matrices - michele coscia · graphs as network analysis matrices michele...
TRANSCRIPT
![Page 1: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/1.jpg)
Network Analysis
Graphs as Matrices
Michele Coscia & Luca RossiITU København
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Background:
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Background:
Looking at a problem from a different
perspective
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Sergey Brin & Larry Page
(1973-)
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55 Million!
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55 Million!
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PageRank
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PageRank● p of visiting v…
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PageRank● p of visiting v…● … = p of visiting a neighbor u of v *
p of going from u to v
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PageRank● p of visiting v…● … = p of visiting a neighbor u of v *
p of going from u to v● for v in V…
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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!
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PageRankG = (V, E)
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PageRankG = (V, E) A = 1 if uv in E
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PageRankG = (V, E) A = 1 if uv in E
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PageRankG = (V, E) A = 1 if uv in E
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PageRank
for v in V...
G = (V, E) A = 1 if uv in E
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PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
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PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
Row-normalized A
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PageRank
for v in V... R := AR
G = (V, E) A = 1 if uv in E
~
Row-normalized A
Eigenvectorof A
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Random Walks
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Basics
A B C D E FA
B
C
D
E
F
A B
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Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
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Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
Yes! Yes!
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Basics
A B C D E FA
B
C
D
E
F
Do youknow each
other?
A B
Yes! Yes!
X
X
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Basics
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Basics
1 2 3 4 5 6 7 8 9
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Basics
1 2 3 4 5 6 7 8 9123456789
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Basics
1 2 3 4 5 6 7 8 9123456789
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Basics
1 2 3 4 5 6 7 8 9123456789
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Basics
1 2 3 4 5 6 7 8 9123456789
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Basics
1 2 3 4 5 6 7 8 9123456789
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Basics
1 2 3 4 5 6 7 8 9123456789
Adjacency Matrix
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Properties of Adjacency Matrices
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Properties of Adjacency Matrices
3
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Properties of Adjacency Matrices
3
3
2
1
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Properties of Adjacency Matrices
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Stochastic Adjacency
A B C D E FA
B
C
D
E
F
AX X X
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Stochastic Adjacency
A B C D E FA
B
C
D
E
F
Pick oneof your friends
at random
AX X X
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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
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Stochastic Adjacency
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Stochastic Adjacency● p of going from
v to u
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Stochastic Adjacency● p of going from
v to u
● A / A.sum(axis = 1)
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Stochastic Adjacency● p of going from
v to u
● A / A.sum(axis = 1)
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Stochastic Adjacency● p of going from v
to u
● A / A.sum(axis = 1)
● Useful eigen-properties
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Walk Probabilities
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Walk Probabilities
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Walk Probabilities
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Walk Probabilities
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Walk Probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
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Walk probabilities
![Page 61: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/61.jpg)
Walk probabilities
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Walk probabilities
p(1→2) in two steps?
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Walk probabilities
p(1→2) in two steps?
(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))
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Walk probabilities
p(1→2) in two steps?
(p(1→5) * p(5→2)) + (p(1→9) * p(9→2))
p(u→v)?
![Page 65: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/65.jpg)
Powers of Stochastic
A
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Powers of Stochastic
A
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Powers of Stochastic
A
![Page 68: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/68.jpg)
Powers of Stochastic
A A2
![Page 69: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/69.jpg)
Powers of Stochastic
A A2
![Page 70: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/70.jpg)
Powers of Stochastic
A A2
![Page 71: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/71.jpg)
Powers of Stochastic
A A2
![Page 72: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/72.jpg)
Powers of Stochastic
A A2
![Page 73: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/73.jpg)
Powers of Stochastic
![Page 74: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/74.jpg)
Powers of Stochastic
![Page 75: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/75.jpg)
Powers of Stochastic
![Page 76: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/76.jpg)
Powers of Stochastic
![Page 77: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/77.jpg)
Powers of Stochastic
1
![Page 78: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/78.jpg)
Powers of Stochastic
![Page 79: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/79.jpg)
Powers of Stochastic
![Page 80: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/80.jpg)
Powers of Stochastic
![Page 81: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/81.jpg)
A2
![Page 82: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/82.jpg)
A2
![Page 83: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/83.jpg)
A2 A3
![Page 84: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/84.jpg)
A2 A30
![Page 85: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/85.jpg)
Stationary Distribution
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Stationary Distribution● A∞ = π
![Page 87: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/87.jpg)
Stationary Distribution● A∞ = π
![Page 88: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/88.jpg)
Stationary Distribution● A∞ = π
● πA = π
![Page 89: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/89.jpg)
Stationary Distribution● A∞ = π
● πA = π
● ~ PageRank
![Page 90: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/90.jpg)
Disconnected Mazes
![Page 91: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/91.jpg)
Disconnected Mazes
![Page 92: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/92.jpg)
Disconnected Mazes
![Page 93: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/93.jpg)
Disconnected Mazes
?
![Page 94: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/94.jpg)
Disconnected Mazes
?
![Page 95: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/95.jpg)
Multiple Components
A
![Page 96: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/96.jpg)
Multiple Components
A30
![Page 97: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/97.jpg)
Multiple Components
A30
![Page 98: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/98.jpg)
Multiple Components
A30
![Page 99: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/99.jpg)
Multiple Components
A30
![Page 100: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/100.jpg)
Multiple Components
A30
![Page 101: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/101.jpg)
Google’s Solution
![Page 102: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/102.jpg)
Google’s Solution
![Page 103: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/103.jpg)
Google’s Solution
Beamme up,Scotty!
![Page 104: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/104.jpg)
Google’s Solution
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Google’s Solution
A
![Page 106: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/106.jpg)
Google’s Solution
A
![Page 107: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/107.jpg)
Google’s Solution
A+ε
![Page 108: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/108.jpg)
Stationary Distribution vs Degree
![Page 109: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/109.jpg)
Stationary Distribution vs Degree
![Page 110: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/110.jpg)
Stationary Distribution vs Degree
Corr = 1
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Stationary Distribution vs Degree
Corr = 1π = k / k.sum()
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Stationary Distribution vs PageRank
![Page 113: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/113.jpg)
Stationary Distribution vs PageRank
![Page 114: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/114.jpg)
Stationary Distribution vs PageRank
![Page 115: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/115.jpg)
Stationary Distribution vs PageRank
![Page 116: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/116.jpg)
Stationary Distribution vs PageRank
![Page 117: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/117.jpg)
Graph Spectrum
![Page 118: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/118.jpg)
A Recap on Eigenvectors
v
(x, y)
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A Recap on Eigenvectors
v
w = Av
(x’, y’)
(x, y)
![Page 120: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/120.jpg)
A Recap on Eigenvectors
v
w = Av
(x’, y’)(λx, λy)
(x, y)
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A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
(x, y)
![Page 122: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/122.jpg)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
Eigenvector
(x, y)
![Page 123: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/123.jpg)
A Recap on Eigenvectors
v
w = Av
(x’, y’)
w = Av = λv
(λx, λy)
Eigenvector
Eigenvalue
(x, y)
![Page 124: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/124.jpg)
Right & Left
Av = λv
![Page 125: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/125.jpg)
Right & Left
Av = λv Right Eigenvector
![Page 126: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/126.jpg)
Right & Left
Av = λv Right Eigenvector
![Page 127: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/127.jpg)
Right & Left
Av = λv
vA = vλ
Right Eigenvector
![Page 128: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/128.jpg)
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
![Page 129: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/129.jpg)
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
![Page 130: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/130.jpg)
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
![Page 131: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/131.jpg)
Right & Left
Av = λv
vA = vλ
Right Eigenvector
Left Eigenvector
![Page 132: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/132.jpg)
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 133: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/133.jpg)
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
![Page 134: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/134.jpg)
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
![Page 135: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/135.jpg)
Eigenvalues & Components
![Page 136: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/136.jpg)
Eigenvalues & Components
![Page 137: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/137.jpg)
Eigenvalues & Components
![Page 138: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/138.jpg)
Eigenvalues & Components
![Page 139: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/139.jpg)
Eigenvalues & Components
![Page 140: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/140.jpg)
Eigenvalues & Components
![Page 141: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/141.jpg)
Eigenvalues & Components
![Page 142: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/142.jpg)
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 143: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/143.jpg)
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1
![Page 144: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/144.jpg)
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
![Page 145: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/145.jpg)
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
![Page 146: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/146.jpg)
Eigenvalues & Components
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
If both equal to 1Graph has two components
λ1
![Page 147: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/147.jpg)
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
![Page 148: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/148.jpg)
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 149: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/149.jpg)
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 150: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/150.jpg)
Eigenvalues & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 151: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/151.jpg)
Eigenvectors
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 152: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/152.jpg)
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
![Page 153: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/153.jpg)
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
![Page 154: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/154.jpg)
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
![Page 155: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/155.jpg)
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
![Page 156: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/156.jpg)
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
![Page 157: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/157.jpg)
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
![Page 158: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/158.jpg)
Eigenvectors
vi
a
b
c
d
e
f
g
h
i
l
m
n
o
![Page 159: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/159.jpg)
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
![Page 160: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/160.jpg)
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
![Page 161: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/161.jpg)
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
![Page 162: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/162.jpg)
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
![Page 163: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/163.jpg)
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1
![Page 164: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/164.jpg)
Eigenvectors & Components
λ1 > λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
v1
λ1
![Page 165: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/165.jpg)
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
![Page 166: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/166.jpg)
Left Largest Eigenvector
![Page 167: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/167.jpg)
Left Largest Eigenvector
![Page 168: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/168.jpg)
Left Largest Eigenvector
![Page 169: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/169.jpg)
Left Largest Eigenvector
vL1 = π
![Page 170: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/170.jpg)
Left Largest Eigenvector
vL1 = π
πA = π
![Page 171: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/171.jpg)
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
![Page 172: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/172.jpg)
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
λ1
![Page 173: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/173.jpg)
Left Largest Eigenvector
vL1 = π
πA = π
πA = π1
λ1
![Page 174: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/174.jpg)
Left & Right
![Page 175: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/175.jpg)
Left & RightvL
1
![Page 176: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/176.jpg)
Left & RightvL
1
![Page 177: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/177.jpg)
Left & RightvL
1
v1
a
a
a
a
a
a
a
a
a
![Page 178: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/178.jpg)
Left & RightvL
1
vR1
v1
a
a
a
a
a
a
a
a
a
![Page 179: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/179.jpg)
Left & RightvL
1
vR1 A
![Page 180: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/180.jpg)
Left & RightvL
1
vR1 AT
![Page 181: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/181.jpg)
Left & RightvL
1
vR1 AT
A / A.sum(axis = 0)
![Page 182: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/182.jpg)
Left & Right
AT
A / A.sum(axis = 0)
![Page 183: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/183.jpg)
Left & Right
AT
A / A.sum(axis = 0)
vL1
vR1
![Page 184: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/184.jpg)
Graph Laplacian
![Page 185: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/185.jpg)
Degree Matrix
![Page 186: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/186.jpg)
Degree MatrixZeros everywhere
![Page 187: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/187.jpg)
Degree MatrixZeros everywhere
i-th diagonal element...
![Page 188: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/188.jpg)
Degree MatrixZeros everywhere
i-th diagonal element... … is the degree of node i
![Page 189: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/189.jpg)
Degree MatrixZeros everywhere
i-th diagonal element... … is the degree of node i
D
![Page 190: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/190.jpg)
Laplacian Matrix
D A
![Page 191: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/191.jpg)
Laplacian Matrix
D AL = -
![Page 192: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/192.jpg)
Laplacian Matrix
L = D - A
![Page 193: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/193.jpg)
Laplacian Matrix
L = D - A
![Page 194: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/194.jpg)
Laplacian Matrix
L = D - A
![Page 195: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/195.jpg)
Laplacian Matrix
L = D - A
0
![Page 196: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/196.jpg)
Eigenvalues
λ1 ≥ λ
2 ≥ λ
3 ≥ … ≥ λ
n-2 ≥ λ
n-1 ≥ λ
n
1
A
L
![Page 197: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/197.jpg)
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
![Page 198: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/198.jpg)
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
![Page 199: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/199.jpg)
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
![Page 200: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/200.jpg)
Eigenvectors
![Page 201: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/201.jpg)
Eigenvectors
v1
![Page 202: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/202.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
a
![Page 203: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/203.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
![Page 204: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/204.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
![Page 205: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/205.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
![Page 206: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/206.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
![Page 207: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/207.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
![Page 208: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/208.jpg)
Eigenvectors
v1
a
a
a
a
a
a
a
a
av
2
.37
.37
.28
.28
.12
-.23
-.38
-.38
-.43
![Page 209: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/209.jpg)
Eigenvectors
![Page 210: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/210.jpg)
Eigenvectors
+
![Page 211: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/211.jpg)
Eigenvectors
+-
![Page 212: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/212.jpg)
Eigenvectors
![Page 213: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/213.jpg)
Eigenvectors
2nd Eigenvector
![Page 214: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/214.jpg)
Eigenvectors
2nd Eigenvector
3rd Eigenvector
![Page 215: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/215.jpg)
Eigenvectors
![Page 216: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/216.jpg)
Eigenvectors
![Page 217: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/217.jpg)
Eigenvectors
![Page 218: Network Analysis Graphs as Matrices - Michele Coscia · Graphs as Network Analysis Matrices Michele Coscia & Luca Rossi ITU København. Background: Background: Looking at a problem](https://reader036.vdocuments.mx/reader036/viewer/2022062307/5fc5c36bd951d42aad3d1c34/html5/thumbnails/218.jpg)
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