Fluctuation scaling in complex systems Zoltn Eisler 1,2 , Jnos - - PowerPoint PPT Presentation

Fluctuation scaling in complex systems

Zoltán Eisler1,2, János Kertész2

1Capital Fund Management, Paris, France 2Department of Theoretical Physics, Budapest

University of Technology and Economics

arXiv:0708.2053, accepted to Advances in Physics

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Outline

Taylor’s law a.k.a. Fluctuation scaling Empirical data and “theory” in parallel Random walks Forests Coins Humans

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Taylor’s law or fluctuation scaling

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(Temporal) Fluctuation scaling

Take (stable) populations i of some species, and

• bserve them in time

Calculate the mean and the variation of the specimen count Plot the two σi fi

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Fluctuation scaling

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σi ∝ fiα

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Fluctuation scaling

Take similar systems i, and observe them in time Calculate the mean and the variation of a positive additive signal Then vary i σi fi σi ∝ fiα

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Fluctuation scaling

Take similar systems, and observe them in time Calculate the mean and the variation of some positive signal Then vary i σi fi σi ∝ fiα

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Why do we care?

the value of α varies mostly in [1/2, 1]

σi ∝ fiα

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Why do we care?

the value of α varies mostly in [1/2, 1] simple dynamical rules?

σi ∝ fiα

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Why do we care?

the value of α varies mostly in [1/2, 1] simple dynamical rules? it is NOT a universal exponent

σi ∝ fiα

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The possible values of α

Random walks Forests Coins (?) Humans

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Random walks

walkers (many) N

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f1(t) f2(t)

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Random walks

walkers (many)

Vn,i(t) =    1 if walker n is on node i at time t, if not.

N

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f1(t) f2(t)

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Random walks

walkers (many)

Vn,i(t) =    1 if walker n is on node i at time t, if not.

N

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fi(t) =

N

• n=1

Vi,n(t)

f1(t) f2(t)

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Random walks

walkers (many)

Vn,i(t) =    1 if walker n is on node i at time t, if not.

N fi = N Vn,i = Npi ∝ ki

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fi(t) =

N

• n=1

Vi,n(t)

f1(t) f2(t)

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Random walks

walkers (many)

Vn,i(t) =    1 if walker n is on node i at time t, if not.

N fi = N Vn,i = Npi ∝ ki σ2

i = Npi

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fi(t) =

N

• n=1

Vi,n(t)

f1(t) f2(t)

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Random walks

walkers (many)

Vn,i(t) =    1 if walker n is on node i at time t, if not.

N fi = N Vn,i = Npi ∝ ki σ2

i = Npi

α = 1/2

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fi(t) =

N

• n=1

Vi,n(t)

f1(t) f2(t)

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Random walks

walkers N(t) α → 1

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fi(t) =

N(t)

• n=1

Vi,n(t)

f1(t) f2(t)

σ2

i = Npi +

ΣN N 2 (Npi)2

fi = N Vn,i = N pi ∝ ki

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Random walks

walkers N(t) α → 1

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fi(t) =

N(t)

• n=1

Vi,n(t)

f1(t) f2(t)

σ2

i = Npi +

ΣN N 2 (Npi)2

fi = N Vn,i = N pi ∝ ki

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Random walks

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Random walks

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Classification by α

α = 1/2: central limit theorem α = 1: strongly driven system Universality classes? Any value between the two is a crossover?

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σi ∝ fiα

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Forests

Consider a forest i of trees Tree n produces seeds in year t The total seed production of year t Vn,i(t) Ni fi(t) =

Ni

• n=1

Vn,i(t)

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Forests

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Masting

fi(t) =

Ni

• n=1

Vn,i(t)

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Masting

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Forests

HV = 1 − 0.4 2 = 0.8

σ2

i = Σ2 V i

• N 2HV i

i

• +

α = HV

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Forests

HV = 1 − 0.4 2 = 0.8

σ2

i = Σ2 V i

• N 2HV i

i

• +

α = HV

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Forests

α = HV Synchronization phase transition Satake-Iwasa model

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Animals?

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Animals?

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Animals?

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Classification by α

α = 1/2: central limit theorem α = 1: strongly driven system 1/2 < α < 1: sums of correlated random variables

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σi ∝ fiα

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Coin flipping

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Coin flipping

mean: 1/2, 1, 3/2, 2 variance: 1/4, 1/2, 3/4, 1 α = 1/2

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Coin flipping

α = 1

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Coin flipping

α = 3/4

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Classification by α

α = 1/2: central limit theorem α = 1: strongly driven system 1/2 < α < 1: sums of correlated random variables 1/2 < α < 1: “coin flipping”

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σi ∝ fiα

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Time window dependence

σi ∝ fiα(∆t)

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Time window dependence

σi(∆t) =

• f ∆t

i

(t) −

• f ∆t

i

(t) 21/2 ∝ ∆tHi σi ∝ fiα(∆t)

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Time window dependence

σi(∆t) =

• f ∆t

i

(t) −

• f ∆t

i

(t) 21/2 ∝ ∆tHi ∆tHi ∝ fiα(∆t) σi ∝ fiα(∆t)

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Time window dependence

σi(∆t) =

• f ∆t

i

(t) −

• f ∆t

i

(t) 21/2 ∝ ∆tHi ∆tHi ∝ fiα(∆t) dHi d(log fi) ∼ dα(∆t) d(log ∆t) ∼ γ σi ∝ fiα(∆t)

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Time window dependence

σi(∆t) =

• f ∆t

i

(t) −

• f ∆t

i

(t) 21/2 ∝ ∆tHi α(∆t) = α∗ + γ log ∆t Hi = H∗ + γ log fi σi ∝ fiα(∆t)

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Human dynamics

σi ∝ fiα(∆t)

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Human dynamics

α(∆t) = α∗ + γ log ∆t Hi = H∗ + γ log fi

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Human dynamics

α(∆t) = α∗ + γ log ∆t Hi = H∗ + γ log fi

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Time window dependence

FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters!

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Time window dependence

FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters!

• rder of magnitude matters → non-universality

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Time window dependence

FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters!

• rder of magnitude matters → non-universality

α can take any value depending on the time resolution

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Time window dependence

FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters!

• rder of magnitude matters → non-universality

α can take any value depending on the time resolution

• ne must map a range in Δt

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Conclusions

Fluctuation scaling: in any field with positive, additive quantities The exponent α can be used to gain hints about dynamics Empirical observation of limit theorems? Hurst exponents change logarithmically with size?

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References

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References

L.R. Taylor, Nature 189, 732 (1961)

• M. de Menezes and A.-L. Barabási, PRL 92, 29701 (2004)
• W. Koenig and J. Knops, The American Naturalist 155, 59 (2000)
• Z. Eisler and J. Kertész, PRE 71, 057104 (2005)
• Z. Eisler et al., arXiv:0708.2053, to appear in Advances in Physics

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