The Physics of Networks

D.W. Heermann

Random Graphs

2010

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Outline

1 Random GraphsExample: Erdos-Renyi ModelExample: Random Loop PolymerAdjacency Matrix

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Random Graphs Example: Erdos-Renyi Model

Example: Erdos-Renyi Model

p = 0.1 p = 0.3

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Random Graphs Example: Erdos-Renyi Model

p = 1.0

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Random Graphs Example: Erdos-Renyi Model

N = 10

N = 100

Erdös-Renyi ModelFraction of the Largest Cluster

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Random Graphs Example: Random Loop Polymer

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Random Graphs Adjacency Matrix

9 1 0 0 0 1 1 1 1 1 1 1 0 0 01 8 1 0 0 0 1 0 1 1 1 1 1 0 00 1 8 1 0 0 0 1 1 1 1 0 1 1 00 0 1 9 1 0 0 0 1 1 1 1 1 1 10 0 0 1 8 1 0 0 0 1 1 1 1 1 11 0 0 0 1 8 1 0 0 0 1 1 1 1 11 1 0 0 0 1 8 1 0 0 0 1 1 1 11 0 1 0 0 0 1 6 1 0 0 0 1 1 01 1 1 1 0 0 0 1 8 1 0 0 0 1 11 1 1 1 1 0 0 0 1 7 1 0 0 0 01 1 1 1 1 1 0 0 0 1 8 1 0 0 01 1 0 1 1 1 1 0 0 0 1 8 1 0 00 1 1 1 1 1 1 1 0 0 0 1 9 1 00 0 1 1 1 1 1 1 1 0 0 0 1 9 10 0 0 1 1 1 1 0 1 0 0 0 0 1 6

( (1                                                       b-1

d

N

N

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Random Graphs Adjacency Matrix

Eigenvalue distribution of the adjacency matrix

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

ρ(λ)

λ

p=0.1p=0.1p=0.1p=0.1p=0.1p=0.1

N=2025, b=45, d=0, p=0.1N=3025, b=55, d=0, p=0.1N=4225, b=65, d=0, p=0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

ρ(λ)

λ

p=0.9p=0.9

N=2025, b=45, d=0, p=0.9N=3025, b=55, d=0, p=0.9N=4225, b=65, d=0, p=0.9

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Random Graphs Adjacency Matrix

Wigner Semi-Circle Law

0

0.05

0.1

0.15

0.2

0.25

-4 -3 -2 -1 0 1 2 3 4

P(s

)

s

Eigenvalue Probability Distribution Wigner Model with N/b2 = 1

N=100, b=10N=225, b=15N=324, b=18

Wigner semi-circle distribution

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