self-organized criticality in models of complex system...

Self-Organized Criticality in Modelsof Complex System Dynamics

Yury M.Pis’mak

Department of Theoretical Physics,State University Saint-Petersburg

Dual Year Spain-RussiaParticle Physics, Nuclear Physics

and Astroparticle Physics,Barcelona, Spain, 5-12 November 2011

Yury M.Pis’mak SOC in Models of Complex System Dynamics

Summary

The Bak-Sneppen model of self-organized biological evolution [P.Bak and K. Sneppen, Phys. Rev. Lett. 71 (1993) 4083] in thecase of randomly interacting species can be investigatedanalytically. The master equation obtained in [J. de Boer,B.Derrida, H.Flyvbjerg, A.D.Jackson, and T.Wettig, Phys.Rev.Lett. 73 (1994) 906] is solved exactly for finite system[Yu.M.Pis’mak, Phys. Rev. E 56 (1997) R1325], and for theinfinite system it is represented in terms of an infinite set ofvariables which can be considered as an analog to the set ofintegrals of motion of completely integrable system. Each of thisvariables remains to be constant but its influence on the evolutionprocess is restricted in time and after definite point in time itsvalue is excluded from description of the system dynamics[Yu.M.Pis’mak, Phys. Rev. Lett. 83 (1999) 4892].


Summary

The possibility of application of the Bak-Sneppen approach formodeling of evolution of real complex systems is demonstrated inthe case of investigation of evolution of large open codesprogramm products. [A.A.Gorshenev, Yu.M.Pis’mak Phys. Rev. E,V.70, P. 067103-067103-4, 2004]


Formulation of model

In the framework of the BSM the evolution of biological ecosystemis described as follows. The state of the ecosystem of N species ischaracterized by a set {x1, ..., xN} of N number, 0 ≤ xi ≤ 1. Thenumber xi represents the barrier of the i-th species toward furtherevolution. Initially, each xi is set to a randomly chosen value. Ateach time step the barrier xi with minimal value and the barriers ofits ”nears neighbors” are replaced by new random numbers. In therandom neighbor model (RNM) [5] the K − 1 of the specie withminimal barriers are chosen at random. The interaction structurein the local model nears neighbors is given a near neighbor graph(for the lattice in D-dimensional space). Thus, for each species inthe LM the nearest neighbors are assumed to be fixed.


Summary

For the LM t he most of results are obtained by numericalexperiments or in the framework of mean field approximation. TheRNM is more convenient for analytical studies. The masterequations obtained in [6] for RNM are very useful for this aim.These equations appeared to be exact solvable. The stationarysolution was found in [6]. The time dependent solutions wereobtained in [7] ( infinite system), [8] (finite syst


State of system after large time of evolution

00.10.20.30.40.50.60.70.80.9

1

0 1000 2000 3000 4000 5000

b(i)

i


Master equations for RNMThe master equations for the RNM are obtained in [3]. They are ofthe form:

Pn(t + 1) = AnPn(t) + Bn+1Pn+1(t) + Cn−1Pn−1(t)+

+Dn+2Pn+2(t) + (B1δn,0 + A1δn,1 + C1δn,2)P0(t).

Here, Pn(t) is the probability that n is the number of barriers havingvalues less than a fixed parameter λ at the time t; 0 ≤ n ≤ N,0 ≤ λ ≤ 1, t ≥ 0. The initial probability distribution Pn(0) is supposed tobe given. For 0 < n ≤ N

An = 2λ(1− λ) +n − 1N − 1λ(3λ− 2),

Bn = (1− λ)2 +n − 1N − 1(1− λ)(3λ− 1),

Cn = λ2 − n − 1N − 1λ

2, Dn = (1− λ)2 n − 1N − 1 . (1)

An = Bn = Cn = Dn = 0 for n = 0 and n > N .


Master equations for RNMThe master equations for the RNM are obtained in [3]. They are ofthe form:

Pn(t + 1) = AnPn(t) + Bn+1Pn+1(t) + Cn−1Pn−1(t)+

+Dn+2Pn+2(t) + (B1δn,0 + A1δn,1 + C1δn,2)P0(t).

Here, Pn(t) is the probability that n is the number of barriers havingvalues less than a fixed parameter λ at the time t; 0 ≤ n ≤ N,0 ≤ λ ≤ 1, t ≥ 0. The initial probability distribution Pn(0) is supposed tobe given. For 0 < n ≤ N

An = 2λ(1− λ) +n − 1N − 1λ(3λ− 2),

Bn = (1− λ)2 +n − 1N − 1(1− λ)(3λ− 1),

Cn = λ2 − n − 1N − 1λ

2, Dn = (1− λ)2 n − 1N − 1 . (1)

An = Bn = Cn = Dn = 0 for n = 0 and n > N .Yury M.Pis’mak SOC in Models of Complex System Dynamics

Dynamical variables and parameters

It holds

Pn(t) ≥ 0,N∑

n=0Pn(t) = 1.

For analysis of master equations it is convenient to introduce thegenerating function q(z , u):

q(z , u) ≡∞∑

t=0

N∑n=0

Pn(t)znut .

In virtue of definition, q(z , t) is polynomial in z , analytical in u for|u| < 1 and

q(1, u) =1

1− u .


Master equation for generating function q(z , u)

The master equations can be rewritten for the generating functionq(z , u) as follows:

1u (q(z , u)−q(z , 0)) = (1−λ+λz)2

(1z

(1− 1− z

N − 1

(1z −

∂

∂z

))×

×(q(z , u)− q(0, u)) + q(0, u)

).

The function q(z , 0) =∑N

n=0 Pn(0)zn in (8) is assumed to begiven.


Exact solution of the master equations

The generating function q(z , u) can be presented in the form

q(z , u) = z N − 1u eR(z,u)

λ−1λ∫

z

e−R(x ,u) q(x , 0)dx(1− λ+ λx)2(1− x)

+

+q(0, u)(1 + z N − 1u eR(z,u)

λ−1λ∫

z

e−R(x ,u) (1− u(1− λ+ λx)2)dx(1− λ+ λx)2(1− x)

),

where

q(0, u) =

1∫λ−1λ

e−R(x ,u) q(x ,0)dx(1−λ+λx)2(1−x)

1∫λ−1λ

e−R(x ,u) (1−u(1−λ+λx)2)dx(1−λ+λx)2(1−x)

.


Infinite system with K random neighbors

The master equation in the case N =∞ and K interacting randomneighbors is of the form

Pn(t + 1) =K∑

l=0C l

nθ(K − l)λl(1− λ)K−lPn−l+1(t) +

+θ(K − n)CnKλ

n(1− λ)K−nP0(t).

Here, Pn(t) is the probability that n is the number of barriershaving values less than a fixed parameter λ at the time t;0 ≤ λ ≤ 1, t ≥ 0 n ≥ 0. For compact writing we used thethreshold function θ(n) defined as followsθ(n) = 1 for n ≥ 0 and θ(n) = 0 for n < 0. The initialprobability distribution Pn(0) is supposed to be given.


Master equation for q(z , u)

The generating function q(z , u) fulfil the equation(z−u(1−λ+λz)K )q(z , u) = (z−1)u(1−λ+λz)K q(0, u)+zq(z , 0).

If α(u) is the analytical in u solution of equationα(u)− u(1 + λ(α(u)− 1))K = 0.

andβ(z) =

z(1− λ+ λz)K ,

thenα(β(z)) = z , β(α(u)) = u,

andd(y , u) =

q(α(y), u)

1− α(y)

is analytical in y and u in the neighborhood of y = 0 and u = 0:

d(y , u) =∞∑

n,t=0Cn(t)ynut .


Equations for Cn(t)

The coefficients Cn(t) fulfill the equations

Cn(t + 1) = Cn+1(t) for n ≥ 0, t ≥ 0.

The initial conditions c0 = Cn(0) are defined by the functionq(z , 0).

cn =1

2πi

∮|z|=ε

q(z , 0)β(z)−n

(1− z)

∂

∂z lnβ(z)dz

(2)

The equation for Cn(t) has the simple solution:

Cn(t) = cn+t .


Asymptotic of cn for large n

cn =

√K − 12πKn

{1− K

2n(K − 1)

[M2 +

K 2 − K + 16K 2

]+O

( 1n2

)}for λ = 1/K ≡ λcr ,

cn = θ(λcr −λ)(1−Kλ)+z0√

Ke−nγ(z0)√2π(K − 1)n3

{φ0 −

z20 K

2n(K − 1)[φ2+

+2(K + 1)

z0K φ1 +K 2 + 11K + 1

6z20 K 2 φ0

]+O

( 1n2

)}if λ 6= λcr . Here, we have used the following notations:

M2 ≡∞∑

n=0Pn(0)n2, φn =

∂n+1

∂zn+1

(q(z , 0)

1− z

)∣∣∣∣∣z=z0

.

We have denoted γ(z) ≡ ln(β(z)). In the saddle point z0 we haveγ′(z0) = 0.


Self-Organized Criticality in Software EvolutionThe better sources of information about changes in computerprograms are version control systems. One of them is ConcurrentVersions System (CVS). It keeps information about changeshappened in short time intervals.Using the CVS in our work, we studied the histories of threesoftware projects: Mozilla web browser, Free-BSD OperatingSystem and Gnu Emacs text editor. For each of these projects weanalyzed only files written in the basic for the project language.They are: C++ for Mozilla , C for Free-BSD and Lisp for Emacs.Header files for C/C++ were not studied. Total amounts of theprocessed files are approximately in order 9000, 11000, 900 forMozilla, Free-BSD, Emacs . Total lengths of RCS-files are1 · 107,1 · 107 and 2 · 106 lines. Total amount of the processed dataexceeds 2 Gigabytes. Due to some resource limitations only part ofthe Free-BSD CVS storage ware processed. Histories of all threeprojects are stored under control of the CVS and were publiclyavailable during our research period from the correspondingInternet servers. Yury M.Pis’mak SOC in Models of Complex System Dynamics

Self-Organized Criticality in Software Evolution.Distribution P(A) for Emacs.

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Self-Organized Criticality in SoftwareEvolution.Distribution P(D) for Emacs.

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Self-Organized Criticality in Software Evolution.Distribution P(A) for Free-BSD.

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Self-Organized Criticality in Software Evolution.Distribution P(D) for Free-BSD.

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Distribution P(S) of temporal durations of avalanches for aone dimensional lattice model

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Distribution P(S) of temporal durations of avalanches for arandom neighbor model

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Self-Organized Criticality in Software Evolution

In our work we studied two modifications of the BS-model: onone-dimensional lattice with periodic boundary conditions and withone random neighbor. The α coefficient always taken as 1

2 .The initial size of the system was 8000 elements. The experimentwent on until one million of avalanches registered.For the one-dimensional lattice with periodic boundary conditionsversion of model we got the following results:

1 P(S) distribution of temporal durations of avalanches. P(S)in log-log scale. is a power law with exponentτ = −1.358± 0.005

2 Distribution P(A) of number of inserted nodes is a power lawwith exponent µa − 1.45± 0.01

3 Distribution P(D) of number of deleted nodes is a power lawwith exponent µd = −1.47± 0.02


For the one random neighbor version of model we got the followingresults:

1 P(S) distribution of temporal durations of avalanches. At fig.?? P(S) is shown in log-log scale. P(S) is a power law withexponent τ = −1.901± 0.008

2 Distribution P(A) of number of inserted nodes is a power lawwith exponent µa − 1.98± 0.01

3 Distribution P(D) of number of deleted nodes is a power lawwith exponent µd = −2.10± 0.02


Conclusions

Self-organized criticality seems to be fruitful approach forinvestigations of evolution of natural complex systems.


Thank you for your attention!


self-organized criticality in models of complex system...

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