The Physics of Networks
D.W. Heermann
Random Graphs
2010
D.W. Heermann (Random Graphs) The Physics of Networks 2010 1 / 9
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
D.W. Heermann (Random Graphs) The Physics of Networks 2010 8 / 9
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
D.W. Heermann (Random Graphs) The Physics of Networks 2010 9 / 9
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