Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes

MACSPro'2019 - Modeling and Analysis of Complex Systems and Processes, Vienna
21 - 23 March 2019

Dayana Mukhametshina, Alexander Shapoval, Mikhail Shnirman

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1. Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes D. Mukhametshina, S. Shapoval, M. Shnirman National Research University Higher School of Economics, Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS Marth 23, 2019 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 1 / 20
2. Self-Organized Criticality Figure: Earthquackes Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 2 / 20
3. Self-Organized Criticality Figure: Distribution of daily changes in financial indeсes Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 3 / 20
4. Self-Organized Criticality Figure: Slow loading and quick stress release Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 4 / 20
5. Self-Organized Criticality Key questions: Scarcity of universality classes Prediction of big events Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 5 / 20
6. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
7. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
8. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
9. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
10. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
11. Plan of the talk Introduction Self-similar lattice Model Dynamics Power-Law Size-Frequency Relationship Waiting Time Distribution Further developments Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 6 / 20
12. Self-similar lattice n – depth of self-similarity Figure: Self similar lattice with (a) symmetrical and (b) knight patterns; n = 3, d = 3. The number of neighbors is written inside each cell. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 7 / 20
13. Self-similar lattice Notations: Pattern – an arbitrary partition of the lattice (d × d) cells into two (marked and unmarked) sets. The case d = 3 is considered. Figure: Symmetrical (a) and chess knight (b) patterns Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 8 / 20
14. Dynamics Notations: hi – number of "grains" in the cell ci N(i) – the set of the cells that are adjacent to the cell ci by a common edge; these cells are called neighbours. ci – number of edges located on the lattice boundary. Hi = |N(i)| + ci Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 9 / 20
15. Dynamics Notations: hi – number of "grains" in the cell ci N(i) – the set of the cells that are adjacent to the cell ci by a common edge; these cells are called neighbours. ci – number of edges located on the lattice boundary. Hi = |N(i)| + ci Accumulation: 1 A cell c of the self-similar lattice is chosen at random with the probability being proportional to the cell’s area. 2 hi → hi + 1. 3 If hi < Hi, we repeat accumulation process. Otherwise, the avalanche starts. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 9 / 20
16. Self-similar lattice n – depth of self-similarity Figure: Self similar lattice with (a) symmetrical and (b) knight patterns; n = 3, d = 3. The number of neighbors is written inside each cell. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 10 / 20
17. Dynamics Avalanche: 1 hi → hi − Hi 2 hj → hj + 1, ∀j ∈ N(i) 3 If at least one neighbor becomes unstable, the transfers continue. The size of the avalanche – the number of sand grains that are displaced during the avalanche. 1 3 0 3 3 3 1 3 1 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 11 / 20
18. Dynamics Avalanche: 1 hi → hi − Hi 2 hj → hj + 1, ∀j ∈ N(i) 3 If at least one neighbor becomes unstable, the transfers continue. The size of the avalanche – the number of sand grains that are displaced during the avalanche. 1 3 0 3 4 3 1 3 1 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 11 / 20
19. Dynamics Avalanche: 1 hi → hi − Hi 2 hj → hj + 1, ∀j ∈ N(i) 3 If at least one neighbor becomes unstable, the transfers continue. The size of the avalanche – the number of sand grains that are displaced during the avalanche. 1 3 0 3 4 3 1 3 1 1 4 0 4 0 4 1 4 1 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 11 / 20
20. Dynamics Avalanche: 1 hi → hi − Hi 2 hj → hj + 1, ∀j ∈ N(i) 3 If at least one neighbor becomes unstable, the transfers continue. The size of the avalanche – the number of sand grains that are displaced during the avalanche. 1 3 0 3 4 3 1 3 1 1 4 0 4 0 4 1 4 1 3 0 2 0 4 0 3 0 3 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 11 / 20
21. Dynamics Avalanche: 1 hi → hi − Hi 2 hj → hj + 1, ∀j ∈ N(i) 3 If at least one neighbor becomes unstable, the transfers continue. The size of the avalanche – the number of sand grains that are displaced during the avalanche. 1 3 0 3 4 3 1 3 1 1 4 0 4 0 4 1 4 1 3 0 2 0 4 0 3 0 3 3 1 2 1 0 1 3 1 3 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 11 / 20
22. Power-Law Size-Frequency Relationship Figure: Symmetrical (a) and knight (b) patterns, n = 5, 6, 7, d = 3 The power-law part of the graphs turns to a downward bend at the right. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 12 / 20
23. Power-Law Size-Frequency Relationship F(s) = σ∈[s/∆s,s∆s) f(σ), s∆s s/∆s s−1 = 2s0 ln ∆s. (1) 1 f(s) – the fraction of the avalanches in the catalogue with size s 2 ∆s is a parameter Figure: Symmetrical (a) and knight (b) patterns, n = 5, 6, 7, d = 3, ∆s = 1.6 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 13 / 20
24. Waiting Time Distribution We will call an avalanche the rare event if its size is bigger than some S∗. Figure: Symmetrical pattern, S∗ = 32000 (a) and knight pattern, S∗ = 20000 (b) , n = 5 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 14 / 20
25. Waiting Time Distribution Simple algorithm t0 – time moment, then rare event occurs T – alarm duration Alarm lasts till t1, then the new event occurs or for (t0, t0 + T] Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 15 / 20
26. Waiting Time Distribution Simple algorithm t0 – time moment, then rare event occurs T – alarm duration Alarm lasts till t1, then the new event occurs or for (t0, t0 + T] Type I and type II errors η – the fraction of the unpredicted rare events τ – the sum of the intervals with the raised alarm divided by length of the whole time interval ε = η + τ Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 15 / 20
27. Waiting Time Distribution Figure: Symmetrical pattern, S∗ = 32000 (a) and knight pattern, S∗ = 20000 (b) , n = 5 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 16 / 20
28. Futher Developments Figure: Symmetrical (a) and knight (b) patterns, n = 5, d = 3 Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 17 / 20
29. THANK YOU Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 18 / 20
30. Self-similar lattice Prerequisites: n – depth of self-similarity C0,L – lattice with L = 3n cells on the side r = 0, 1, . . . , n − 2 1 Split each cell c ∈ Cr,L into 9 equal squares with 3n−r−1 cells on the side. Four out of nine cells form the pattern Pr+1(c). For r = n − 1: Split Cn−1,L into 9 equal squares with 1 cell on the side. 2 ˆ Cr+1,L denotes the set of all cells appeared at sub-step 1 that form the patterns Pr+1(c), c ∈ Cr,L. For r = n − 1: Cells corresponding to the pattern will be denoted as ˆ Cn,L. Algorithm terminates. 3 Cr+1,L denotes the set of all cells appeared at sub-step 1 that does not belong to the patterns Pr+1(c), c ∈ Cr,L. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 19 / 20
31. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 20 / 20