Spin models on networks revisited

Petter Holme
Petter HolmeSungkyunkwan University
Spin models on networks
revisited
Petter Holme
Tokyo Institute of Technology
Spin models of statistical physics
1. An underlying graph G.

Traditionally a d-dimensional

lattice.
2. A spin variable θi associated

with every node in the graph.
3. A function H (the “Hamiltonian”)

that maps G and {θi} to a number.

Typically (always?) ∑f(θi–θj) where the

sum is over edges (i,j).
4. The probability of {θi} is exp(–H/kBT).
↑ ↓ ↓ ↓ ↑ ↓ ↓ ↑

↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑

↓ ↓ ↓ ↑ ↑ ↓ ↓ ↑

↓ ↑ ↑ ↓ ↓ ↓ ↑ ↓
Why put spin models on networks
Why put spin models on networks
Why put spin models on networks
Spin models on networks revisited
The XY model
H(G, {θi}) = –J∑edges (i,j) cos(θi–θj), θi are angles
https://www.complexity-explorables.org/explorables/if-you-ask-your-xy/
The XY model
XY model on WS networks
Kim & al., PRE 64:056135 2001.
XY model on WS networks
Kim & al., PRE 64:056135 2001.
XY model on WS networks
Kim & al., PRE 64:056135 2001.
Dynamic XY model on WS networks
Kim & al., PRE 64:056135 2001.
The YX model
Just like XY, but keep (randomly sampled) spins fixed
and vary the links of the graph.
Holme, Wu, Minnhagen, Multiscaling in an YX model
of networks, Phys. Rev. E. 80, 036120 (2009).
Magnetic transitions no longer possible, but maybe
some transition in network structure?
The YX model
Just like XY, but keep (randomly sampled) spins fixed
and vary the links of the graph.
(a) H = −199.52
π/2
0π
−π/2
(b) H = −195.86 (c) H = −192.05
The YX model
0.4
0.6
0.8
1
1.2
1.4
10 100 103
104
TN
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 100 103
104
TN
DNε
δδ
10.110.1
DNε
(b)(a)
400
800
N = 1600
200
δ =1.52, ε = –0.74
The diameter of the largest connected component:
The YX model
The largest connected component:
0
0.2
0.4
0.6
0.8
1
10−5
10−4
10−3
0.01
T
1−s1
(a)
400
800
1600
N = 3200
(b)
0.05
0.1
0.15
0.2
3
10100
(1−s1)Nβ
TNα
α =1.6, β = 0.22
The YX model
The 2nd largest connected component:
10 100
TN
γ10−3
10−5
10−4
0.01 0.1
0
T
s2
0.1
0.2
(b)(a)
s2
0.1
0.2
N = 3200
1600
800
400
0
γ =1.44
The YX model
The 2nd largest connected component:
10 100
TN
γ10−3
10−5
10−4
0.01 0.1
0
T
s2
0.1
0.2
(b)(a)
s2
0.1
0.2
N = 3200
1600
800
400
0
γ =1.44
The free XY model
Magnetization (avg. degree 8).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
magnetization
temperature
16
32
64
128
256
The free XY model
Magnetization (avg. degree 8).
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0.01 0.1 1
magnetization*Nnu
temperature
16
32
64
128
256
ν = 0.30
The free XY model
Magnetization (avg. degree 8).
0.0000001
0.0000010
0.0000100
0.0001000
0.0010000
0.0100000
0.1000000
1.0000000
10.0000000
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
The free XY model
Size of the largest component (avg. degree 8).
0.97
0.975
0.98
0.985
0.99
0.995
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
The free XY model
Number of components (avg. degree 8).
1
10
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
The free XY model
Magnetization (avg. degree 4).
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.01 0.1 1
magnetization*Nnu
temperature
8
16
32
64
128
256
ν = 0.30
The free XY model
Size of the largest component (avg. degree 4).
0.7
0.75
0.8
0.85
0.9
0.95
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
The free XY model
Number of components (avg. degree 4).
1
10
100
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0.01 0.1 1
magnetization*Nnu
temperature
16
32
64
128
256
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.01 0.1 1
magnetization*Nnu
temperature
8
16
32
64
128
256
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.01 0.1
magnetization*Nnu
temperature
8
16
32
64
128
256
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0.001 0.01 0.1
magnetization*Nnu
temperature
8
16
32
64
128
256
512
k = 8, ν = 0.30 k = 4, ν = 0.30
k = 2, ν = 0.18 k = 1, ν = 0.02
Magnetization crossing plots
0.97
0.975
0.98
0.985
0.99
0.995
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
0.7
0.75
0.8
0.85
0.9
0.95
1
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
512
Size of LCC
k = 8 k = 4
k = 2 k = 1
2
2.5
3
3.5
4
4.5
5
5.5
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
16
32
64
128
256
2
3
4
5
6
7
8
9
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
2
4
6
8
10
12
14
16
18
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
0
2
4
6
8
10
12
14
16
18
20
22
0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000
8
16
32
64
128
256
512
Diameter
k = 8 k = 4
k = 2 k = 1
The freer XY model
Let both the links and spins be free to update; don’t
conserve the number of links.
π/23π/2
π
0
θ
T = 10–3
T = 1 T = 103
Thank you
http://petterhol.me
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Spin models on networks revisited

  • 1. Spin models on networks revisited Petter Holme Tokyo Institute of Technology
  • 2. Spin models of statistical physics 1. An underlying graph G.
 Traditionally a d-dimensional
 lattice. 2. A spin variable θi associated
 with every node in the graph. 3. A function H (the “Hamiltonian”)
 that maps G and {θi} to a number.
 Typically (always?) ∑f(θi–θj) where the
 sum is over edges (i,j). 4. The probability of {θi} is exp(–H/kBT). ↑ ↓ ↓ ↓ ↑ ↓ ↓ ↑
 ↑ ↑ ↓ ↑ ↑ ↓ ↑ ↑
 ↓ ↓ ↓ ↑ ↑ ↓ ↓ ↑
 ↓ ↑ ↑ ↓ ↓ ↓ ↑ ↓
  • 3. Why put spin models on networks
  • 4. Why put spin models on networks
  • 5. Why put spin models on networks
  • 7. The XY model H(G, {θi}) = –J∑edges (i,j) cos(θi–θj), θi are angles
  • 10. XY model on WS networks Kim & al., PRE 64:056135 2001.
  • 11. XY model on WS networks Kim & al., PRE 64:056135 2001.
  • 12. XY model on WS networks Kim & al., PRE 64:056135 2001.
  • 13. Dynamic XY model on WS networks Kim & al., PRE 64:056135 2001.
  • 14. The YX model Just like XY, but keep (randomly sampled) spins fixed and vary the links of the graph. Holme, Wu, Minnhagen, Multiscaling in an YX model of networks, Phys. Rev. E. 80, 036120 (2009). Magnetic transitions no longer possible, but maybe some transition in network structure?
  • 15. The YX model Just like XY, but keep (randomly sampled) spins fixed and vary the links of the graph. (a) H = −199.52 π/2 0π −π/2 (b) H = −195.86 (c) H = −192.05
  • 16. The YX model 0.4 0.6 0.8 1 1.2 1.4 10 100 103 104 TN 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 100 103 104 TN DNε δδ 10.110.1 DNε (b)(a) 400 800 N = 1600 200 δ =1.52, ε = –0.74 The diameter of the largest connected component:
  • 17. The YX model The largest connected component: 0 0.2 0.4 0.6 0.8 1 10−5 10−4 10−3 0.01 T 1−s1 (a) 400 800 1600 N = 3200 (b) 0.05 0.1 0.15 0.2 3 10100 (1−s1)Nβ TNα α =1.6, β = 0.22
  • 18. The YX model The 2nd largest connected component: 10 100 TN γ10−3 10−5 10−4 0.01 0.1 0 T s2 0.1 0.2 (b)(a) s2 0.1 0.2 N = 3200 1600 800 400 0 γ =1.44
  • 19. The YX model The 2nd largest connected component: 10 100 TN γ10−3 10−5 10−4 0.01 0.1 0 T s2 0.1 0.2 (b)(a) s2 0.1 0.2 N = 3200 1600 800 400 0 γ =1.44
  • 20. The free XY model Magnetization (avg. degree 8). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 magnetization temperature 16 32 64 128 256
  • 21. The free XY model Magnetization (avg. degree 8). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.01 0.1 1 magnetization*Nnu temperature 16 32 64 128 256 ν = 0.30
  • 22. The free XY model Magnetization (avg. degree 8). 0.0000001 0.0000010 0.0000100 0.0001000 0.0010000 0.0100000 0.1000000 1.0000000 10.0000000 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256
  • 23. The free XY model Size of the largest component (avg. degree 8). 0.97 0.975 0.98 0.985 0.99 0.995 1 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 16 32 64 128 256
  • 24. The free XY model Number of components (avg. degree 8). 1 10 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 16 32 64 128 256
  • 25. The free XY model Magnetization (avg. degree 4). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.01 0.1 1 magnetization*Nnu temperature 8 16 32 64 128 256 ν = 0.30
  • 26. The free XY model Size of the largest component (avg. degree 4). 0.7 0.75 0.8 0.85 0.9 0.95 1 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256
  • 27. The free XY model Number of components (avg. degree 4). 1 10 100 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256
  • 28. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.01 0.1 1 magnetization*Nnu temperature 16 32 64 128 256 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.01 0.1 1 magnetization*Nnu temperature 8 16 32 64 128 256 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.01 0.1 magnetization*Nnu temperature 8 16 32 64 128 256 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 0.001 0.01 0.1 magnetization*Nnu temperature 8 16 32 64 128 256 512 k = 8, ν = 0.30 k = 4, ν = 0.30 k = 2, ν = 0.18 k = 1, ν = 0.02 Magnetization crossing plots
  • 29. 0.97 0.975 0.98 0.985 0.99 0.995 1 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 16 32 64 128 256 0.7 0.75 0.8 0.85 0.9 0.95 1 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 512 Size of LCC k = 8 k = 4 k = 2 k = 1
  • 30. 2 2.5 3 3.5 4 4.5 5 5.5 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 16 32 64 128 256 2 3 4 5 6 7 8 9 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 2 4 6 8 10 12 14 16 18 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 0 2 4 6 8 10 12 14 16 18 20 22 0.00001 0.00010 0.00100 0.01000 0.10000 1.00000 10.00000 100.00000 8 16 32 64 128 256 512 Diameter k = 8 k = 4 k = 2 k = 1
  • 31. The freer XY model Let both the links and spins be free to update; don’t conserve the number of links. π/23π/2 π 0 θ T = 10–3 T = 1 T = 103