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

6. The XY model H(G, {θi}) = –J∑edges (i,j) cos(θi–θj), θi are angles

7. https://www.complexity-explorables.org/explorables/if-you-ask-your-xy/

8. The XY model

9. XY model on WS networks Kim & al., PRE 64:056135 2001.

12. Dynamic XY model on WS networks Kim & al., PRE 64:056135 2001.

13. 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?

14. 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

15. 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:

16. 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

17. 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

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 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. Thank you http://petterhol.me