=Paper= {{Paper |id=Vol-2478/paper11 |storemode=property |title=Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes |pdfUrl=https://ceur-ws.org/Vol-2478/paper11.pdf |volume=Vol-2478 |authors=Dayana Mukhametshina,Alexsander Shapoval,Mikhail Shnirman }} ==Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes== https://ceur-ws.org/Vol-2478/paper11.pdf

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

Dayana Mukhametshina1 , Alexsander Shapoval1,2[0000−0001−5340−1930] , and
Mikhail Shnirman2
1
National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa,
101000 Moscow, Russia
hse@hse.ru
https://www.hse.ru/en/
2
Institute of Earthquake Prediction Theory and Mathematical Geophysics,
Profsoyuznaya 84/32, 117997 Moscow, Russia

Abstract. In 1987, Bak, Tang, and Wiesenfeld introduced a mechanism
(hereafter, the BTW mechanism) that underlies self-organized critical
systems. Extreme events generated by the BTW mechanism are be-
lieved to exhibit an unpredictable occurrence. In spite of this general
opinion, the largest events in the original BTW model are efficiently
predictable by algorithms that exploit information that is hidden in ap-
plications. Intending to relate the predictability of self-organized critical
systems with the level of its asymmetry, we examine the inter-event dis-
tribution of extreme avalanches generated by the BTW mechanism on
symmetrical and asymmetrical self-similar lattices. Initially, we claim
that the main part of the size-frequency relationship is power-law in-
dependent of the asymmetry, but the asymmetry reduces the range of
scale-free avalanches in the domain of small avalanches. Further, we turn
to extremes and claim that they are located on the downward bend of the
distribution of the avalanches over their sizes. Finally, we compare the
probability distribution of waiting time between two successive extremes
with the exponential distribution. The latter gives the reference point of
the complete unpredictability naturally measured in terms of the sum
of two rates related to type I and II statistical errors: the rate of the
unpredicted avalanches and the alarm time rate. We posit that the devi-
ations of the observed probability distribution from the exponential one
do not affect the unpredictability of extremes drawn from the waiting
time between them.

Keywords: sandpile · power-law · waiting time distribution.

1 National Research University Higher School of Economics.
* Copyright c 2019 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0)
2 D. Mukhametshina , A. Shapoval, M. Shnirman

1 Introduction
Extreme events constitute an essential phenomenon of complex systems. Their
economic and social consequences are difficult to overestimate. Nevertheless, yet
several decades ago, scholars primary focused on regular behavior of complex
systems. Only recently the scenarios of extremes become better understood,
but the prediction of extremes still remains a challenge for researchers, espe-
cially if the system exhibits so called self-organized criticality associated with
a power-law size-frequency distribution of “normal” events [12]. Bak et al. [2]
introduced a simple mechanism (hereafter, the BTW mechanism) that gener-
ates self-organized criticality. Two multi-scale processes: slow loading and quick
stress release, which balance the stress on average, characterize the BTW mecha-
nism. The stress release is modeled as an avalanche that redistributes the stress
over the underlying system and carries the energy out at the boundary. The
system attains a critical state observed through the power-law distribution of
the avalanches over their size [9]. Typically, the existence of the power-law size-
frequency distributions signals that the prediction of extremes are hardly pos-
sible [1]. Under the BTW mechanism, a slight increase of loading affects the
system alike its local redistribution. However, the BTW mechanism generates
an out-of-equilibrium system whose observed variables oscillate around average
values. Such oscillations, in general, could open a door for an effective prediction
of extremes that topple the system from the super- to sub-critical state, if the
oscillation exhibits a certain quasi-periodicity.
Bak et al. [2] gave the first simple explanation of numerous critical phe-
nomena reported by that time and hugely extended by nowadays [36]. The
(truncated) power-laws describe the probability distributions of earthquakes [13],
landslides [34], solar flares [7], sizes of large cities [10,15], city fires [33], and fi-
nancial crashes [17]. Surprisingly, the power-law exponent observed in the BTW
model is hardly tuned with the transformation of the BTW mechanism [3]. The
random version of the model proposed by [16] results in a similar but distinct
exponent [6,4]. However other numerous modifications leave the models inside
the two classes of universality determined by these two exponents [29].
The BTW mechanism itself primary exhibits a concept of multiscale processes
that exhibits self-organized criticality, but does not describe details of observed
systems. The impossibility to adjust the exponent of the power-law and leave
the (very) limited set of the universality classes reduces the range of the direct
applicability of the BTW model and its adjacent generalizations. The exponent
is tuned, if the BTW mechanism is realized on fractals [8,5], site-percolation
lattice [20], and self-similar lattice [31].
Typically, a downward bend follows the scale-free range of the size-frequency
relationship at the right part of the graph. In the BTW model, the transition is
explained by the finite-size effect. The occurrence of the large events can exhibit
time-clustering [28]. In this case, the largest events follow a foreshock activity
or trigger aftershocks, or demonstrate the both phenomema [32]. Such time-
clustering underlies the algorithms that predict the largest events. These events
are characterized by specific waiting time probability distribution [22]. The BTW
Self-Organized Criticality on Self-Similar Lattice 3

model is dissipative on the boundary, and large avalanches drive energy out of the
system. Therefore, a large loading seems to be required for a consecutive extreme
avalanche to occur. The existence of the quiescent episodes in the dynamics
generates a certain quasi-periodicity of the largest avalanches. These arguments
are confirmed for the BTW model [30] and the laboratory experiment imitating
the model [25]. As far as the waiting time probability distribution deviates from
the exponential one, the waiting time itself serves as a precursor of the main
event in the models and their applications [26,11,27,35]. However, the extension
of the set of the largest events to smaller ones destroys the efficiency of this
precursor, tending the waiting time distribution to be exponential [23,11]. The
avalanches that are located on the power-law part of the size-frequency relation
graph are considered as unpredictable [24,1]. [19] constructed the framework
based on the fraction of the unpredicted events and the alarm time rate in order
to quantify the prediction algorithms. The fall of the prediction efficiency with
the size of the forecasted events remains a general feature of the system [14]
The goal of this paper is to extend the applicability of the BTW-like models
to real-life processes by testing the existence of new universality classes gener-
ated by asymmetry of the underlying system and inspecting the predictability
of extremes within the universality classes. Even if the universality classes are
insensitive to the asymmetry of the system, the asymmetry itself may affect
the scenario of extremes. We examine the inter-event distribution of extreme
avalanches generated by the BTW mechanism on the self-similar lattice. Ini-
tially, we address the question of how the asymmetry of the lattice affects the
size-frequency relationship of the avalanches, finding the scale-free part of the
relationship and establishing that its exponent is insensitive to the asymmetry.
Further, we turn to extremes and claim that they are located on the downward
bend of the distribution of the avalanches over their sizes. Finally, we compare
the probability distribution of waiting time between two successive extremes
with the exponential distribution. The latter gives the reference point of the
complete unpredictability naturally measured in terms of type I and II errors.
We posit that the deviations of the observed probability distribution from the
exponential one do not affect the unpredictability of extremes drawn from the
waiting time between them.
The rest of the paper is organized in the following way. We define the model
in Section 2, discussing the size-frequency relationship in details. The waiting
time probability distribution is studied in Section 3. The last section concludes.

2 Model
2.1 Self-Similar Lattice:
In order to define a model we introduce the following notation. Consider d × d
lattice. Let a pattern be an arbitrary partition of the lattice cells into two (marked
and unmarked) sets. We note that it is enough to specify the marked set to
determine the partition. The case d = 3 is considered. We deal with the two
patterns that consist of four cells illustrated by Figure 1. The first pattern is
4 D. Mukhametshina , A. Shapoval, M. Shnirman

symmetrical. It consists of four corner cells, colored in grey in Figure 1a. The
second pattern consists of three cells located along a side, whereas the forth cell
adjoins the corner one, Figure 1b. This pattern can be treated as an example of
the largest asymmetry. As the pattern repeats the move of the chess knight, we
further refer to it as a knight

Fig. 1. Symmetrical (a) and chess knight (b) patterns

We define a self-similar lattice that consists of cells of different sizes. The smallest
cells determine the unit of measurements. The (linear) length of the lattice is
L = 3n , where n is the depth of self-similarity realized in computations. In
theoretical constructions, n tends to +∞. The recursive algorithm, constructing
the self-similar lattice, at each step takes all cells that does not belong to the
pattern, divide them into 9 equal squares, and specify the pattern among these
new, smaller squares. The patterns are pre-defined; they are identical at each
step.
More precisely, let a pattern be fixed. We denote C0,L a lattice with L cells
on the side. At every step r = 0, 1, . . . , n − 2 of the recursion, the algorithm
repeats the following sub-steps:
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).
2. Ĉr+1,L denotes the set of all cells appeared at sub-step 1 that form the
patterns Pr+1 (c), c ∈ Cr,L .
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 .

Note that step 1 is not applied to the cells from Ĉr+1,L . The final step r = n − 1
differs from the previous ones:
1. Split Cn−1,L into 9 equal squares with 1 cell on the side.
2. Cells corresponding to the pattern will be denoted as Ĉn,L .
Self-Organized Criticality on Self-Similar Lattice 5

The set ∪nr=1 Ĉr,L is further referred to as self-similar lattice. Figure 2 illustrates
the self-similar lattices obtained with n = 3 and d = 3

Fig. 2. Self similar lattice with (a) symmetrical and (b) knight patterns; n = 3, d = 3.
The number of neighbors is written inside each cell.

2.2 Dynamics
The cells, i. e., c ∈ Ĉr,L , where r = 1, . . . , n, are assumed to be enumerated in
some way c1 , c2 , . . .. We use the following notation.
– hi is a number associated with the cell ci . Following tradition [2], the quantity
hi is called a grain of sand, whereas the model is referred to as sandpile.
– N (i) is the set of the cells that are adjacent to the cell ci by a common edge;
these cells are called neighbors. |N (i)| denotes the number of the neighbors
illustrated in Figure 2.
– If the cell ci is located inside the lattice C0,L (i. e., neither edge belong to
the lattice boundary) then we put Hi = |N (i)|. Otherwise, Hi is the sum of
|N (i)| and the number of the ci ’s edges located on the lattice boundary.

The cell ci is called stable if hi < Hi . At the beginning all cells are stable. For
the sake of simplicity we put hi = 0 for all i. Our computer simulations give
evidence that what follows is insensitive to the initial conditions.
The model dynamics consists of two stages: accumulation and avalanche.
Accumulation. A grain falls on a randomly chosen cell:
1. A cell c of the self-similar lattice is chosen at random with the probability
being proportional to the cell’s area. If c ∈ Ĉr,L , r = 1, . . . , n, then the
probability is p = 32(n−r) /32n = 3−2r .
2. Let i be the number of the just chosen cell c, i. e., c = ci . Then a grain of
sand “falls” on ci : hi → hi + 1.
6 D. Mukhametshina , A. Shapoval, M. Shnirman

3. We perform a stability check. If hi < Hi , we repeat accumulation process.
Otherwise, the avalanche starts.
Avalanche. The unstable cell ci topples transferring grains to the neighbors
equally:

hi → hi − Hi (1)
hj → hj + 1, ∀j ∈ N (i) (2)

The transfer (1) and (2) conserves the total amount of grains in the lattice when
an inner cell topples. If at least one neighbor becomes unstable, the transfers
continue. In other words, while there are unstable cells ci , the rules (1) and (2)
are applied. Each avalanche is finite because the transfers at the boundary are
dissipative. We call the number of sand grains that are displaced during the
avalanche the size of the avalanche.

2.3 Power-Law Size-Frequency Relationship
First, we investigate the probability distribution of the avalanches over their
sizes. Let N (s) be the number of avalanches that have the size s. Then we define
1
Q(s) = (N (s) + N (s − 1) + N (s − 2) + N (s − 3))
4
We substitute Q(s) for N (s) to stabilize the graph in the domain of small sizes.
Figure 3 displays the graphs of Q(s) for the symmetrical (a) and knight (b)
patterns. The power-law part of the graphs turns to a downward bend at the
right.

Fig. 3. Symmetrical (a) and knight (b) patterns, n = 5, 6, 7, d = 3
Self-Organized Criticality on Self-Similar Lattice 7

The power-law part and the structure of the downward bend can be studied in
more details when the exponential binning is applied. Let f (s) be the fraction
of the avalanches in the catalogue that have the size s;
X
F (s) = f (σ), (3)
σ∈[s/∆s,s∆s)

where ∆s is a parameter. We stress that if f (s) follows a power-law function,
then F (s) also does, but the exponent of the power-law is increased by 1. Indeed,
the summation in equation (3) serves as the integral that transforms 1/sβ into
1/sβ−1 . For instance, if β = 1, the integral over the exponential bin is
Z s∆s
s∆s
s−1 = ln s s/∆s = 2 ln ∆s = 2s0 ln ∆s.
s/∆s

Fig. 4. Symmetrical (a) and knight (b) patterns, n = 5, 6, 7, d = 3, ∆s = 1.6

Figures 4 a and b exhibit the size-frequency relation for the both patterns. The
both graphs have a power-law part followed by a bend down that might be
characterized by a steeper line in the log-log scale. The asymmetry affects the
left part of the graph that corresponds to small avalanches. On the asymmetrical
lattice, a significant share of the avalanches reaches the boundary and stops
almost immediately.

3 Waiting Time Distribution
In this section, we investigate the waiting time probability distribution and relate
it to the prediction of rare events. Rare events are defined parametrically. We will
8 D. Mukhametshina , A. Shapoval, M. Shnirman

call an avalanche the rare event if its size is bigger than some S ∗ . The inter-event
probability distribution rather accurately follows the exponential function while
S ∗ belongs to the power-law range of sizes (we do not support this statement
by a graph). If S ∗ is located on the right part of the downward bend, this
probability distribution deviates from the exponential function; see Figure 5,
where the complement cumulative distribution function, 1 − cdf is displayed.
Under symmetrical pattern, the deviation is in the domain of a large waiting
time: a large time gap between successive rare events occurs more frequency
then the exponential random probability distribution predicts. In the case of the
knight pattern, the observed waiting time distribution is convex in the linear-log
scale, but the convexity is small.

Fig. 5. Symmetrical pattern, S ∗ = 32000 (a) and knight pattern, S ∗ = 20000 (b) ,
n=5

We introduce a simple algorithm predicting the next rare event based on the
record of the previous one. If a rare event occurs at the time moment t0 , then
an alarm is raised for the interval (t0 , t0 + T ], where the parameter T > 0 will
be adjusted later. The alarm means that the algorithm predicts the occurrence
of the consecutive rare event during time window (t0 , t0 + T ]. The unit of time is
associated with the grain falling. It falls one grain per the time unit. Each alarm
continues either up to the occurrence of a new rare event (and then a new alarm
is raised) or T units of time.
The rare events that occurs when the alarm is raised are predicted. The
other rare events are unpredicted. Let η be the fraction of the unpredicted rare
events. The first rare event is unpredicted by the construction of the algorithm.
Therefore, it is ignored when η is computed. Further, let τ be the alarm rate
(the sum of the intervals with the raised alarm divided by length of the whole
Self-Organized Criticality on Self-Similar Lattice 9

time interval) and ε = η + τ . The quantities η and τ are related to type I and
II statistical errors [18]. Their sum ε describes the efficiency of the prediction
algorithm. The equation ε ≈ 1 characterizes unpredictability. If the underlying
probability distribution of the waiting time is exponential then our algorithm
results in ε = 1 (this is the result of an elementary mathematics skipped here).
The both errors as functions of the parameter T are displayed in Figure 6 whereas
their sum is shown in 7. In our computer simulations, S ∗ is chosen as 32000
(symmetrical pattern) and 20000 (knight pattern), that leads to the amount of
rare events equalled to 193 and 139 respectively.

Fig. 6. Symmetrical pattern, S ∗ = 32000 (a) and knight pattern, S ∗ = 20000 (b) ,
n=5

Fig. 7. Symmetrical pattern, S ∗ = 32000 (a) and knight pattern, S ∗ = 20000 (b) ,
n=5
10 D. Mukhametshina , A. Shapoval, M. Shnirman

The quantity ε oscillates around 1. This suggests that the waiting time proba-
bility distribution deviates from the power one insignificantly.

4 Conclusion

We constructed the BTW mechanism on the self-similar lattice. The existence of
the asymmetry in the geometry of the self-similar lattice does not affect the self-
organized criticality of the underlying system. The size-frequency relationship
follows the power-law and the power-law exponents are identical. Their value
was earlier predicted by [31]. The power-law part of the size-frequency relation
graphs is transformed to a downward bend located at the right. There are traces
of another power-law in this bend whose exponent is (very) roughly estimated
as 3. The best approximation to the downward bend deserves a separate study.
We found that the waiting time distribution between rare events deviates
from the exponential one, if the low border of these events is sufficiently large.
However we did not detect a power-law inter-event distribution as [22] identified
for the classical BTW sandpile. In the sandpile on the self-similar lattice studied
here, the prediction of the rare avalanches drawn from their waiting time distri-
bution fails completely. We construct the algorithm that expects the occurrence
of a new extreme within T time units after the previous extreme. This algorithm
is ineffective for any value of the parameter T .
This result does not give evidence against the prediction of extremes in the
sandpile model on the self-similar lattice in general. We believe that the pre-
dictability of large avalanches found by [28] is translated onto the case of the
BTW-mechanism on the self-similar lattice. At least, it is worth trying to con-
struct a prediction algorithm that switches on the alarm when the surplus of
sand grains on the lattice is observed. This algorithm could be effective since
large avalanches are expected upon the system becomes supercritical.
The BTW-like models considered in the paper can be used when study-
ing earthquake formation processes, as the exponent in the power law is tuned
through an appropriate choice of the self-similarity determinant. Narkunskaya &
Shnirman [21] scrutinized long-term dynamics of the distribution of the earth-
quakes over their magnitude and constructed a precursor of strong earthquakes
drawn from an upward bend of this distribution. The existence of a similar
precursor in the model would further justify its applicability to the earthquake
formation process. Then the prediction algorithms adjusted on our artificial sys-
tem can be further tested when predicting strong earthquakes.

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