Statistics of non-equilibrium systems from the theory of - PowerPoint PPT Presentation

SSR is a random walk on directed ordered NW a) p exit =1 ⇒ φ Start p exit =0 ⇒ φ ∞ 1/5 5 1/4 4 1/3 3 1/2 2 1 (1 - p exit )/5 1 p exit Stop b) 1/2 0 10 Node occupation probability Path probability -1 − 2 10 -0.65 − 4 10 1/4 0 2 10 10 Path rank 1/8 acyclic p exit =0.3 p exit =0.3 1 2 3 4 5 Node rank note fully connected torino mar 14 2019 61

SSR = targeted random walk on networks simple example Directed Acyclic Graph (no cycles) torino mar 14 2019 62

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Simple routing algorithm • take directed acyclic network fix it • pick start-node • perform a random walk from start-node to end-node (1) • repeat many times from other start-nodes • prediction visiting frequency of nodes follows Zipf law torino mar 14 2019 66

All diffusion processes on DAG are SSR sample ER graph → direct it → pick start and end → diffuse torino mar 14 2019 67

Exponential NW HEP Co-authors torino mar 14 2019 68

prior probabilities are practically irrelevant! torino mar 14 2019 69

What happens if introduce weights on links? ER Graph poisson weights power weights torino mar 14 2019 70

prior probabilities are practically irrelevant! torino mar 14 2019 71

What happens if we introduce cycles? ER → direct it → change link to random direction with 1 − λ “driving” λ = 0 . 8 λ = 0 . 5 torino mar 14 2019 72

Zipf’s law is an immense attractor! torino mar 14 2019 73

Zipf’s law is an attractor • no matter what the network topology is → Zipf • no matter what the link weights are → Zipf • if have cycles → exponent is less than one torino mar 14 2019 74

all good search is SSR torino mar 14 2019 75

What is good search? a search process is good if ... • ... at every step you eliminate more possibilities than you actually sample • ... every step you take eliminates branches of possibilities if eliminate fast enough → power law in visiting times if eliminate too little → sample entire space (exhaustive search) if no cycles: expect Zipf’s law torino mar 14 2019 76

clicking on web page is often result of search process torino mar 14 2019 77

adamic & hubermann 2002 torino mar 14 2019 78

breslau et al 99 torino mar 14 2019 79

what about exponents > 1 ? torino mar 14 2019 80

1 2 3 4 5 6 7 8 9 1011 12 1314151617181920 Multiplication factor µ → p ( i ) = i − µ torino mar 14 2019 81

0 10 µ =1 µ =1.5 µ =2 −2 10 µ =2.5 Relative Frequency −4 10 0 10 µ =2.5 Relative Frequency µ =3.5 −2 µ =4.5 10 −6 10 −4 10 −8 10 −6 10 0 2 4 10 10 10 Energy −10 10 0 1 2 3 4 10 10 10 10 10 State torino mar 14 2019 82

what if we introduce conservation laws? torino mar 14 2019 83

Conservation laws in SSR processes assume you have duplication at every jump, µ = 2 if you are at i → duplicate → one jumps to j , the other to k conservation means: i = j + k . conservation means: f ( i ) = f (state 1 ) + f (state 2 ) + · · · + f (state µ ) → p ( i ) = i − 2 for all µ same result was found by E. Fermi for particle cascades torino mar 14 2019 84

−1 10 µ =2.5 µ =3.5 −2 µ =4.5 10 Relative frequency −3 −2 10 −4 10 −5 10 −6 10 −7 10 0 1 2 3 4 10 10 10 10 10 Energy (arbitrary units) torino mar 14 2019 85

complex systems are driven – always! torino mar 14 2019 86

Complex systems are driven non-equilibrium systems • only driven systems produce non-trivial structures • without driving: just ground state or equilibrium • every driven system: relaxing part + driving part • every relaxing part is a SSR torino mar 14 2019 87

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where do all the distributions come from? torino mar 14 2019 89

Assume that driving rate depends on state λ ( i ) torino mar 14 2019 90

→ λ ( x ) = − x d dx log p ( x ) that can be proved torino mar 14 2019 91

Proof transition probabilities from state k to i are � λ ( k ) q i g ( k − 1) + (1 − λ ( k )) q i if i < k p SSR ( i | k ) = (1 − λ ( k )) q i if i > k g ( k ) is the cdf of q i , g ( k ) = � i ≤ k q i . Observing that � � p λ,q ( i + 1) 1 + λ ( i + 1) q i +1 = p λ,q ( i ) q i +1 g ( i ) q i we get � − 1 � q i q j ∼ q ( i ) q ( j ) − � j ≤ i λ ( j ) � p λ,q ( i ) = 1 + λ ( j ) e g ( j − 1) Z λ,q g ( j − 1) Z λ,q 1 <j ≤ i Z λ,q is the normalisation constant. For uniform priors, taking logs and going to continuous variables gives the result, λ ( x ) = − x d dx log p λ ( x ) . torino mar 14 2019 92

the driving process determines distribution torino mar 14 2019 93

Special cases λ ( x ) = − x d dx log p ( x ) p ( x ) = x − 1 • Zipf: slow driving ( λ = 1 ) → p ( x ) = x − α • Power-law: constant driving λ ( x ) = α → p ( x ) = e − β ( x − 1) • Exponential: λ ( x ) = βx → p ( x ) = x − α e − βx • Power-law + cut-off: λ ( x ) = α + βx → p ( x ) = x α − 1 e − βx • Gamma: λ ( x ) = 1 − α + βx → torino mar 14 2019 94

Special cases λ ( x ) = − x d dx log p ( x ) p ( x ) = e − β 2 ( x − 1) 2 • Normal: λ ( x ) = 2 βx 2 → p ( x ) = e − β α ( x − 1) α • Stretched exp: λ ( x ) = αβ | x | α → x e − (log x − β )2 σ 2 + log x • Log-normal: λ ( x ) = 1 − β 1 → 2 σ 2 σ 2 • Gompertz: λ ( x ) = ( βe αx − 1) βx p ( x ) = e βx − αe βx → � α � α − 1 � x � • Weibull: λ ( x ) = β − α αx α + α − 1 → p ( x ) = − x e β β 1 βx • Tsallis: λ ( x ) = → p ( x ) = (1 − (1 − Q ) βx ) 1 − Q 1 − βx (1 − Q ) torino mar 14 2019 95

Driving determines statistics of driven systems slow → Zipf’s law constant → power law extreme driving → prior distribution (uniform) driving state dependent → any distribution torino mar 14 2019 96

Examples that are of SSR–nature • self-organized critical systems • driven systems (with stationary distributions) • search • fragmentation • propagation of information in laguage: sentence formation • sequences of human behavior • games: go, chess, life ... • record statistics torino mar 14 2019 97

• do we now have a statistical theory for statistics of driven, dissipative, (stationary) non-equilibrium systems? • are SOC processes a sub-set of SSR processes? • is there a maximum configuration principle for arbitrarily driven systems? • can we do thermodynamics with these systems? torino mar 14 2019 98

Conclusions • most complex systems are driven and show power laws • path-dependent processes of SSR-type abound in nature • SSR has extremely robust attractors – priors don’t matter • relaxation is usually SSR • details of driving + SSR → explain statistics torino mar 14 2019 99