Anomalous statistics of dynamical systems on networks Stefan Thurner www.complex-systems.meduniwien.ac.at www.santafe.edu trento jul 23 2012
with R. Hanel and M. Gell-Mann PNAS 108 (2011) 6390-6394 Europhys Lett 93 (2011) 20006 Europhys Lett 96 (2011) 50003 trento jul 23 2012 1
Why are networks cool? • Tell you who interacts with whom • Same statistical system on different networks can behave totally different trento jul 23 2012 2
How ? • Simple example: Ising spins on constant-connectency networks • Show: this is not of Boltzmann Gibbs type – give exact statistics trento jul 23 2012 3
Why Statistics ? • Central concept: understanding macroscopic system behavior on the basis of microscopic elements and interactions → entropy • Functional form of entropy: must encode information on interactions too! • Entropy relates number of states to an extensive quantity, plays funda- mental role in the thermodynamical description • Hope: ’thermodynamical’ relations → phase diagrams, etc. trento jul 23 2012 4
3 Ingredients • Entropy has scaling properties → what are entropies for non-ergodic systems? • How does entropy grow with system size? → what n.e. system is realized? • Symmetry in thermodynamic systems → if broken: entropy has no thermodynamic meaning → forget dream about handling system with TD trento jul 23 2012 5
What is the entropy of strongly interacting systems? trento jul 23 2012 6
Appendix 2, Theorem 2 C.E. Shannon, The Bell System Technical Journal 27 , 379-423, 623-656, 1948. trento jul 23 2012 7
Entropy W � S [ p ] = g ( p i ) i =1 p i ... probability for a particular (micro) state of the system, � i p i = 1 W ... number of states g ... some function. What does it look like? trento jul 23 2012 8
The Shannon-Khinchin axioms • SK1: S depends continuously on p → g is continuous • SK2: entropy maximal for equi-distribution p i = 1 /W → g is concave • SK3: S ( p 1 , p 2 , · · · , p W ) = S ( p 1 , p 2 , · · · , p W , 0) → g (0) = 0 • SK4: S ( A + B ) = S ( A ) + S ( B | A ) Theorem: If SK1-SK4 hold, the only possibility is Boltzmann-Gibbs-Shannon entropy W � S [ p ] = g ( p i ) with g ( x ) = − x ln x i =1 trento jul 23 2012 9
Shannon-Khinchin axiom 4 is non-sense for NWs → SK4 violated for strongly interacting systems → nuke SK4 SK4 corresponds to weak interactions or Markovian processes trento jul 23 2012 10
The Complex Systems axioms • SK1 holds • SK2 holds • SK3 holds • S g = � W i g ( p i ) , W ≫ 1 Theorem: All systems for which these axioms hold (1) can be uniquely classified by 2 numbers, c and d (2) have the unique entropy � W � e Γ (1 + d , 1 − c ln p i ) − c � S c,d = e · · · Euler const 1 − c + cd e i =1 trento jul 23 2012 11
The argument: generic mathematical properties of g • Scaling transformation W → λW : how does entropy change ? trento jul 23 2012 12
Mathematical property I: an unexpected scaling law ! S g ( Wλ ) S g ( W ) = ... = λ 1 − c lim W →∞ g ( zx ) Theorem 1: Define f ( z ) ≡ lim x → 0 g ( x ) with (0 < z < 1) . Then for systems satisfying SK1, SK2, SK3: f ( z ) = z c , 0 < c ≤ 1 trento jul 23 2012 13
Theorem 1 Let g be a continuous, concave function on [0 , 1] with g (0) = 0 and let f ( z ) = lim x → 0 + g ( zx ) /g ( x ) be continuous, then f is of the form f ( z ) = z c with c ∈ (0 , 1] . Proof. Note that f ( ab ) = lim x → 0 g ( abx ) /g ( x ) = lim x → 0 ( g ( abx ) /g ( bx ))( g ( bx ) /g ( x )) = f ( a ) f ( b ) . All pathological solutions are excluded by the requirement that f is continuous. So f ( ab ) = f ( a ) f ( b ) implies that f ( z ) = z c is the only possible solution of this equation. Further, since g (0) = 0 , also lim x → 0 g (0 x ) /g ( x ) = 0 , and it follows that f (0) = 0 . This necessarily implies that c > 0 . f ( z ) = z c also has to be concave since g ( zx ) /g ( x ) is concave in z for arbitrarily small, fixed x > 0 . Therefore c ≤ 1 . trento jul 23 2012 14
Mathematical properties II: yet another one !! S ( W 1+ a ) S ( W ) W a (1 − c ) = ... = (1 + a ) d lim W →∞ Theorem 2: Define h c ( a ) ≡ ... trento jul 23 2012 15
Theorem 2 Let g be like in Theorem 1 and let f ( z ) = z c then h c given in Eq. (8) is a constant of the form h c ( a ) = (1 + a ) d for some constant d . Proof. We determine h c ( a ) again by a similar trick as we have used for f . ( x b ) ( a +1 − 1 ) +1 � � b g g ( x a +1 ) g ( x b ) h c ( a ) = lim x → 0 x ac g ( x ) = ( x b ) ( a +1 − 1 ) c g ( x b ) x ( b − 1) c g ( x ) b � a +1 � = h c − 1 h c ( b − 1) , b for some constant b . By a simple transformation of variables, a = bb ′ − 1 , one gets h c ( bb ′ − 1) = h c ( b − 1) h c ( b ′ − 1) . Setting H ( x ) = h c ( x − 1) one again gets H ( bb ′ ) = H ( b ) H ( b ′ ) . So H ( x ) = x d for some constant d and consequently h c ( a ) is of the form (1 + a ) d . trento jul 23 2012 16
Summary Strongly interacting systems → SK1-SK3 hold S g ( W λ ) S g ( W ) = λ 1 − c → lim W →∞ 0 ≤ c < 1 S ( W 1+ a ) S ( W ) W a (1 − c ) = (1 + a ) d → lim W →∞ d real Remarkable: • all systems are characterized by 2 exponents: ( c, d ) – universality class • Which S fulfills above? → S c,d = � W i =1 re Γ (1 + d , 1 − c ln p i ) − rc � � 1 � � − d B (1+ x W k r ) d − W k ( B ) 1 − c • Which distribution maximizes S c,d → p c,d ( x ) = e � ∞ 1 − c + cd , B = 1 − c 1 � 1 − c � dt ta − 1 exp( − t ) ; Lambert- W : solution to x = W ( x ) eW ( x ) r = cd exp , Γ( a, b ) = b cd trento jul 23 2012 17
Holds very generically • for all non-ergodic systems • for all non-Markovian systems (complex systems) trento jul 23 2012 18
Examples • S 1 , 1 = � i g 1 , 1 ( p i ) = − � i p i ln p i + 1 (BG entropy) i p q 1 − � • S q, 0 = � i g q, 0 ( p i ) = i + 1 (Tsallis entropy) q − 1 i g 1 ,d ( p i ) = e i Γ (1 + d , 1 − ln p i ) − 1 • S 1 ,d> 0 = � � d (AP entropy) d • ... trento jul 23 2012 19
Classification of entropies: order in the zoo entropy c d S BG = � i p i ln(1 /p i ) 1 1 1 − � pq i • S q< 1 = ( q < 1) c = q < 1 0 q − 1 i p i ( p κ i − p − κ • S κ = � ) / ( − 2 κ ) ( 0 < κ ≤ 1 ) c = 1 − κ 0 i 1 − � pq i • S q> 1 = ( q > 1) 1 0 q − 1 i (1 − e − bpi ) + e − b − 1 • S b = � ( b > 0) 1 0 pi − 1 pi ) • S E = � i p i (1 − e 1 0 i Γ( η +1 η , − ln p i ) − p i Γ( η +1 • S η = � η ) ( η > 0) 1 d = 1 /η i p i ln 1 /γ (1 /p i ) • S γ = � 1 d = 1 /γ i p β • S β = � i ln(1 /p i ) c = β 1 S c,d = � i er Γ( d + 1 , 1 − c ln p i ) − cr c d trento jul 23 2012 20
Distribution functions of CS • p (1 , 1) → exponentials (Boltzmann distribution) • p ( q, 0) → power-laws ( q -exponentials) • p (1 ,d> 0) → stretched exponentials • p ( c,d ) all others → Lambert- W exponentials NO OTHER POSSIBILITIES trento jul 23 2012 21
q -exponentials Lambert-exponentials (c) r=exp( − d/2)/(1 − c) (b) d=0.025, r=0.9/(1 − c) 0 0 10 10 c=0.2 − 10 10 (0.3, − 4) c=0.4 (0.3, − 2) p(x) p(x) (0.3, 2) c=0.6 − 20 − 20 (0.3, 4) 10 10 (0.7, − 4) (0.7, − 2) (0.7, 2) − 30 10 (0.7, 4) c=0.8 0 5 0 5 10 10 10 10 x x trento jul 23 2012 22
The world beyond Shannon violates K2 compact support BG − entropy (1,0) of distr. function 1 Stretched exponentials − asymptotically stable violates K2 (c,d) − entropy, d<0 (c,d) − entropy, d>0 c Lambert W − 1 exponentials Lambert W 0 exponentials q − entropy, 0<q<1 (c,0) 0 (0,0) violates K3 − 1 0 1 2 d trento jul 23 2012 23
Scaling property opens door to ... • ...bring order in the zoo of entropies through universality classes • ...understand ubiquity of power laws (and extremely similar functions) • ...understand where Tsallis entropy comes from • ...understand statistical systems on networks trento jul 23 2012 24
The requirement of extensivity trento jul 23 2012 25
Needed for TD program to work: extensive entropies System has N elements → W ( N ) ... phasespace volume (system property) Extensive: S ( W A + B ) = S ( W A ) + S ( W B ) = · · · [use scaling property I] → � � �� 1 d Can proof: extensive is equivalent to W ( N ) = exp 1 − c W k µ (1 − c ) N d W ′ ( N ) N →∞ 1 − 1 c = lim N W ( N ) � 1 � W d = N →∞ log W lim W ′ + c − 1 N Message: Growth of phasespace volume determines entropy and vice versa trento jul 23 2012 26
Examples • W ( N ) = 2 N → ( c, d ) = (1 , 1) and system is BG • W ( N ) = N b → ( c, d ) = (1 − 1 b , 0) and system is Tsallis • W ( N ) = exp( λN γ ) → ( c, d ) = (1 , 1 γ ) • ... Can explicitly verify statements in theory of binary processes and spin- systems on networks trento jul 23 2012 27
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