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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Transcript

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
   

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2. Self-Organized Criticality
   Figure: Earthquackes
   Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 2 / 20
   

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3. Self-Organized Criticality
   Figure: Distribution of daily changes in financial indeсes
   Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 3 / 20
   

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4. Self-Organized Criticality
   Figure: Slow loading and quick stress release
   Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 4 / 20
   

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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
   

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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
   

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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
   

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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
   

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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
   

   View Slide

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
    

    View Slide

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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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29. THANK YOU
    Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 18 / 20
    

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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
    

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31. Mukhametshina et al (HSE) Self-Organized Criticality Marth 23, 2019 20 / 20
    

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