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

Fluctuation scaling in complex systems Zoltán Eisler 1,2 , János Kertész 2 1 Capital Fund Management, Paris, France 2 Department of Theoretical Physics, Budapest University of Technology and Economics

arXiv:0708.2053, accepted to Advances in Physics

Outline Taylor’s law a.k.a. Fluctuation scaling Empirical data and “theory” in parallel Random walks Forests Coins Humans 3 /40

Taylor’s law or fluctuation scaling 4 /40

(Temporal) Fluctuation scaling Take (stable) populations i of some species, and observe them in time Calculate the mean and the variation of the specimen count � f i � Plot the two σ i 5 /40

Fluctuation scaling σ i ∝ � f i � α 6 /40

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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 � f i � Then vary i σ i ∝ � f i � α σ i 8 /40

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Fluctuation scaling Take similar systems, and observe them in time Calculate the mean and the variation of some positive signal � f i � Then vary i σ i ∝ � f i � α σ i 10 /40

Why do we care? the value of α varies mostly in [1/2, 1] σ i ∝ � f i � α 11 /40

Why do we care? the value of α varies mostly in [1/2, 1] simple dynamical rules? σ i ∝ � f i � α 11 /40

Why do we care? the value of α varies mostly in [1/2, 1] simple dynamical rules? it is NOT a universal exponent σ i ∝ � f i � α 11 /40

The possible values of α Random walks Forests Coins (?) Humans 12 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t ) 13 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t )  1 if walker n is on  node i at time t , V n,i ( t ) = 0 if not.  13 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t )  1 if walker n is on  node i at time t , V n,i ( t ) = 0 if not.  N � f i ( t ) = V i,n ( t ) n =1 13 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t )  1 if walker n is on  node i at time t , V n,i ( t ) = 0 if not.  N � f i ( t ) = V i,n ( t ) n =1 � f i � = N � V n,i � = Np i ∝ k i 13 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t )  1 if walker n is on  node i at time t , V n,i ( t ) = 0 if not.  N � f i ( t ) = V i,n ( t ) n =1 � f i � = N � V n,i � = Np i ∝ k i σ 2 i = Np i 13 /40

Random walks f 1 ( t ) walkers (many) N f 2 ( t )  1 if walker n is on  node i at time t , V n,i ( t ) = 0 if not.  N � f i ( t ) = V i,n ( t ) n =1 � f i � = N � V n,i � = Np i ∝ k i σ 2 α = 1 / 2 i = Np i 13 /40

Random walks f 1 ( t ) walkers N ( t ) f 2 ( t ) N ( t ) � f i ( t ) = V i,n ( t ) n =1 � f i � = � N � � V n,i � = � N � p i ∝ k i � Σ N � 2 σ 2 ( Np i ) 2 i = Np i + α → 1 � N � 14 /40

Random walks f 1 ( t ) walkers N ( t ) f 2 ( t ) N ( t ) � f i ( t ) = V i,n ( t ) n =1 � f i � = � N � � V n,i � = � N � p i ∝ k i � Σ N � 2 σ 2 ( Np i ) 2 i = Np i + α → 1 � N � 14 /40

Random walks 15 /40

Random walks 15 /40

Classification by α α = 1/2: central limit theorem α = 1: strongly driven system Universality classes? Any value between the two is a crossover? σ i ∝ � f i � α 16 /40

Forests Consider a forest i of trees N i Tree n produces seeds in year t V n,i ( t ) The total seed production of year t N i � f i ( t ) = V n,i ( t ) n =1 17 /40

Forests 18 /40

Masting N i � f i ( t ) = V n,i ( t ) n =1 19 /40

Masting 20 /40

Forests H V = 1 − 0 . 4 2 = 0 . 8 � � N 2 H V i σ 2 i = Σ 2 + V i i α = H V 21 /40

Forests H V = 1 − 0 . 4 2 = 0 . 8 � � N 2 H V i σ 2 i = Σ 2 + V i i α = H V 22 /40

Forests Synchronization phase transition Satake-Iwasa model α = H V 23 /40

Animals? 24 /40

Animals? 24 /40

Animals? 25 /40

Classification by α α = 1/2: central limit theorem α = 1: strongly driven system 1/2 < α < 1: sums of correlated random variables σ i ∝ � f i � α 26 /40

Coin flipping 27 /40

Coin flipping mean: 1/2, 1, 3/2, 2 α = 1 / 2 variance: 1/4, 1/2, 3/4, 1 28 /40

Coin flipping α = 1 29 /40

Coin flipping α = 3 / 4 30 /40

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” σ i ∝ � f i � α 31 /40

Time window dependence σ i ∝ � f i � α ( ∆ t ) 32 /40

Time window dependence σ i ∝ � f i � α ( ∆ t ) �� 2 � 1 / 2 �� f ∆ t f ∆ t ∝ ∆ t H i � σ i ( ∆ t ) = ( t ) − ( t ) i i 32 /40

Time window dependence σ i ∝ � f i � α ( ∆ t ) �� 2 � 1 / 2 �� f ∆ t f ∆ t ∝ ∆ t H i � σ i ( ∆ t ) = ( t ) − ( t ) i i ∆ t H i ∝ � f i � α ( ∆ t ) 32 /40

Time window dependence σ i ∝ � f i � α ( ∆ t ) �� 2 � 1 / 2 �� f ∆ t f ∆ t ∝ ∆ t H i � σ i ( ∆ t ) = ( t ) − ( t ) i i ∆ t H i ∝ � f i � α ( ∆ t ) d α ( ∆ t ) dH i d (log � f i � ) ∼ d (log ∆ t ) ∼ γ 32 /40

Time window dependence σ i ∝ � f i � α ( ∆ t ) �� 2 � 1 / 2 �� f ∆ t f ∆ t ∝ ∆ t H i � σ i ( ∆ t ) = ( t ) − ( t ) i i α ( ∆ t ) = α ∗ + γ log ∆ t H i = H ∗ + γ log � f i � 33 /40

Human dynamics σ i ∝ � f i � α ( ∆ t ) 34 /40

Human dynamics α ( ∆ t ) = α ∗ + γ log ∆ t H i = H ∗ + γ log � f i � 35 /40

Human dynamics α ( ∆ t ) = α ∗ + γ log ∆ t H i = H ∗ + γ log � f i � 36 /40

Time window dependence FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters! 37 /40

Time window dependence FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters! order of magnitude matters → non-universality 37 /40

Time window dependence FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters! order of magnitude matters → non-universality α can take any value depending on the time resolution 37 /40

Time window dependence FS enforces a logarithmic relationship on correlation strength → only the order of magnitude matters! order of magnitude matters → non-universality α can take any value depending on the time resolution one must map a range in Δ t 37 /40

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? 38 /40

References 39 /40

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 40 /40