VIEWS: 18 PAGES: 164 CATEGORY: Personal Finance POSTED ON: 10/22/2009
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Why Impossible Things Happen so Often? The emergence of (Social) Complex Adaptive Dynamics from Microscopic Noise Physicists view: complex from simple rather then more complex (herds, nests) from complex (humans, ants)) The Importance of Being Discrete; PNAS 97 (2000) 10322 Nadav Shnerb, Yoram Louzoun, Eldad Bettelheim, Sorin Solomon Henri Atlan (biology), J Goldenberg (marketing), Irun Cohen (immunology), Hanoch Lavee (desert/ecology) H and M Levy (finance econo), D Mazursky (Soc. Sci., Bus. Adm.) Contents: - define the problem (emergence of complexity) and our proposed solution (take discreteness seriously) - define the simplest model that makes the point - continuous approximation erroneous, trivial, death - discrete representation emergence, adaptation, survival - role of discreteness -emergence of collective adaptive features - applications to economic systems -trade globalization=> wealth localization ; instability - noise+logistic => scaling (Pareto, Fat Tails) spatial patterns biology COMPLEXITY life functions Cells EMERGENCE ? reactions Molecules chemistry 4100 4050 4000 3950 3900 3850 3800 Pareto -Zipf economics 3750 9 9.5 The Guardian COMPLEXITY Collective Trends 10 10.5 11 11.5 EMERGENCE ? sell-buy Individuals individual behavior Reality Discrete Individuals SAME SYSTEM Models Continuum Functions Complex ----------------------------------Trivial Localized features ----------------------Uniformity Adaptive ----------------------------------Fixed laws Development -----------------------------Decay Resilience ---------------------------------Death Misfit was always assigned to the neglect of specific details. We show it was rather due to the neglect of the discreteness. Once taken in account=>complex adaptive collective dynamics emerges even in the worse conditions Discrete Individuals microscopic noise Autocatalytic proliferation amplification Collective Macroscopic Objects -Emergent Properties : Adaptability -Life always wins in 2 D eternal agents A, - initial uniform density - diffusion coefficient a(x,t=0)=a0 Da b(x,t=0)= b0 Db mortals B - initial uniform density - diffusion rate - death rate µ: B - birth rate l: B+AB+B+A [when at the same location with a "catalyst," A], Diffusion of A at rate Da A Diffusion of A at rate Da A Diffusion of A at rate Da A Diffusion of B at rate Db B Diffusion of B at rate Db B Diffusion of B at rate Db B A+B A+B+B; Birth of new B at rate l B A A+B A+B+B; Birth of new B at rate l A B A+B A+B+B; Birth of new B at rate l AB A+B A+B+B; Birth of new B at rate l AB A+B A+B+B; Birth of new B at rate l ABB A+B A+B+B; Birth of new B at rate l ABB A+B A+B+B; Birth of new B at rate l ABB BB A+B A+B+B; Birth of new B at rate l ABB BB B Death of B at rate m B B Death of B at rate m B B Death of B at rate m Diffusion of A at rate Da A Diffusion of A at rate Da A Diffusion of A at rate Da A Diffusion of A at rate Da A Diffusion of B at rate Db B Diffusion of B at rate Db B Diffusion of B at rate Db B Diffusion of B at rate Db B A+B A+B+B; Birth of new B at rate l B A A+B A+B+B; Birth of new B at rate l A B A+B A+B+B; Birth of new B at rate l AB A+B A+B+B; Birth of new B at rate l AB A+B A+B+B; Birth of new B at rate l ABB A+B A+B+B; Birth of new B at rate l ABB A+B A+B+B; Birth of new B at rate l ABB BB A+B A+B+B; Birth of new B at rate l ABB BB A+B A+B+B; Birth of new B at rate l Another Example B AA A+B A+B+B; Birth of new B at rate l Another Example B AA A+B A+B+B; Birth of new B at rate l Another Example B AA A+B A+B+B; Birth of new B at rate l Another Example B AA A+B A+B+B; Birth of new B at rate l Another Example BBB AA A+B A+B+B; Birth of new B at rate l Another Example BBB AA B Death of B at rate m B B Death of B at rate m B B Death of B at rate m B B Death of B at rate m Interpretations in Various Fields Finance: sites= individuals, companies, B = capital A= wealth generating conditions jobs, good location, good managers, customers What will happen? The naive lore: e.g. reaction-diffusion (Partial) differential equations=> - A density a(x,t) ao - B reproduction rate l a0 - B death rate m +l a0) b(x,t) < 0 +l a0) t no wealth no life.... .(x,t) = (-m - B change:b b (x,t) decays as b (x,t) ~e (-m Continuum, Partial Differential Equations Approximation a ( x,t ) = the number of A’s ; The number of B‟s born b ( x,t )= the number of B „s The number of B‟s dying ~ ~ a ( x,t ) b ( x,t ) l d t In total: the number of A-B pairs: ~ proportional to their number ~ - m b ( x,t ) d t d b ( x,t ) ~ ( a ( x,t ) l - m ) b ( x,t ) d t+diffusion Continuum d t 0: . =(al-m)b+D b . =D a a b Db Da . =D diffusion a a D a => a ( x,t ) a 0 Substituting (and starting with constant b) => linear ordinary differential equation in b: . =(a l-m)b+D Db b b b = (a0 l - m ) b . [Malthus 1798 ] the asymptotic solution: b (x,t) ~ e (l a0 – m ) t =>for l a 0– m < 0 the B population vanishes (the death rate m overwhelms the average birth rate l a 0 ) surprize snapshots wealth, life decay Angels and Mortals by my students Eldad Bettelheim and Benny Lehmann The continuum view: a(x,t) = 2. Wherever the B‟s are, they - generate a new B at rate 2 l and - disappear at rate m. AA AA AA AAB AA AA If 2 l -m < 0, b(x,t) decreases as e (2l -m ) t E.g m = 5/2 l e(2l - 5/2 l) t = e-1/2lt AA AA AA AA AA AA AA AA AA AA B AA AA AA e-1/2lt +e-1/2lt 0 AAB AA AA AA AA AA AA AA B AA e(2l - 5/2 l) t = eAA AA 1/2lt The discrete view: a(x,t) 2. But < a(x,t) > = 2. Now B‟s - generate new B at rate a(x,t) l and - disappear at rate m. AA AA A AA AAB AA Even if 2 l -m < 0, b(x,t) may still increase as e (a(x,t)l -m ) t E.g m = 5/2 l e(3l - 5/2 l) t = e+1/2lt AA AA AA AA AA AA AA AA AA AA AA AA e+1/2lt +e-3/2lt AAA AAB AA AA AA AA AA AA AA BA B AA AA e( l - 5/2 l) t = e-3/2lt AA AA A one –dimensional simple example continuum prediction a = 2/14 l= 1 m= 1/2 A B level A level 11 12 13 14 1 2 3 4 5 6 7 8 9 10 A one –dimensional < b > simple example continuum prediction continuum prediction (1-5/14) t t a × l1- m 1/2 = -5/14 2/14 × A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 B level A level A one –dimensional < b > simple example discrete prediction continuum prediction (1-5/14) t t b4(t+1) = (1 + 1 × l – m) b4(t) b13(t+1) = (1 + 0 × l – m) b13(t) A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 B level A one –dimensional < b > Initial exponential decay simple example discrete prediction continuum prediction (9/14) t t (3/2) (3/2) (½) (½) (½) A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 B level A one –dimensional < b > simple example discrete prediction continuum prediction (3/2) t (9/14) t t (3/2)2 (3/2)2 (½)2 (½)2 (½)2 A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 B level A one –dimensional < b > simple example discrete prediction continuum prediction (3/2) t growth (9/14) t t (3/2)3 (3/2)3 (½)3 (½)3 (½)3 A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 B level A one –dimensional < b > simple example discrete prediction continuum prediction (3/2) t growth (9/14) t t (3/2)4 (3/2)4 (½)4 (½)4 (½)4 A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 A one –dimensional < b > simple example discrete prediction (3/2) t growth continuum (9/14) t prediction t (3/2)5 (3/2)5 (½)5 (½)5 (½)5 A 1 2 3 4 5 6 7 8 9 A 10 11 12 13 14 In principle A –diffusion could smear the effect away. BUT IT DOESN’T: one can prove rigorously (Renormalization Group (2000), Branching Random Walks Theorems (2002)) that:No matter how fast the death rate m, how low the A density, how small the proliferation rate l -On a large enough 2 dimensional surface, the B population always grows! - In all dimensions d: l/Da > 1-Pd always suffices Pd = Polya‟s constant ; P2 = 1 New theorem: for A death rate ma : l > Da + ma suffices ! Emergence of Adaptive B islands Example: single A a0=1/V ; l=2m ; - m +l a 0 ~ - m < 0 e ( l - m ) t =e m A 1 2 3 4 5 6 7 8 t e-mt 9 10 11 12 13 14 A B diffusion A A AA A A A A B diffusion A A A A Growth stops again when A jumps again A A A AA A A A A A A A Factor lost at each A jump (DB /l ) DA t ( l – m -DB) t e Growth in between jumps Number of A jumps during the time t A A A A A ln b at current A location [ln (DB /l ) DA + ( l – m -DB)] t e TIME ln b (A location (unseen) E and C b distribution) A TIME SP Emergent Collective Dynamics: B-herds /capital search, follow, exploit and grow on fortuitous fluctuations in A density. A A A A A Emergent Collective Dynamics: B-herds /capital search, follow, exploit and grow on fortuitous fluctuations in A density. This collective knowledge and adaptive behavior at the market level is in apparent contradiction to the individual level where B’s don’t follow anybody A A A A A Emergent Collective Dynamics: B-herds /capital search, follow, exploit and grow on fortuitous fluctuations in A density. This collective knowledge and adaptive behavior at the market level is in apparent contradiction to the individual level where B’s don’t follow anybody Emergent Collective Dynamics: B-herds /capital search, follow, exploit and grow on fortuitous fluctuations in A density. This collective knowledge and adaptive behavior at the market level is in apparent contradiction to the individual level where B’s don’t follow anybody the market has more knowledge and success than its trading agents Emergent Collective Dynamics: B-islands search, follow, adapt to, and exploit fortuitous fluctuations in A density. This collective knowledge and adaptive behavior at the market level is in apparent contradiction to the individual level where B’s don’t follow anybody the market has more knowledge and success than its trading agents The strict innocence/ stupidity of the individual traders and the emergence of complex adaptive dynamics with self-serving behavior do not interfere one with another. Yet they determine one another. Emergent Collective Dynamics: B-islands search, follow, adapt to, and exploit fortuitous fluctuations in A density. This collective knowledge and adaptive behavior at the market level is in apparent contradiction to the individual level where B’s don’t follow anybody the market has more knowledge and success than its trading agents The strict innocence/ stupidity of the individual traders and the emergence of complex adaptive dynamics with self-serving behavior do not interfere one with another. Yet they determine one another. Is this a mystery? Not in the AB model where all is on the table ! patches= momentarity advantaged professions, industries, regions, “herds”, strategies CONCLUSION Discretization=> micro-inhomogeneity Auto-catalysis (b ~ b)=> amplification Complex Collective objects with emergent properties . Identity Spatio-temporal localization Adaptation Increase chances of survival Sustainability Cognition Searching, Finding and Exploiting A fluctuations Applies to many real life systems New Scientist Collaborations that identified and studied systems in biology, finance and social science that are naively non-viable (decay to extinction) when viewed macroscopically but perfectly viable in reality (and when simulated / analyzed correctly at the microscopic individual level). In particular, most of the species in nature could be in this regime: --- - negative naive average growth rate but - actual survival and proliferation. Ordinary miracles Michael Brooks New Scientist magazine, May 2000: << Jeff Kirkwood, a population dynamics researcher at Imperial College, London, says this close look is particularly valuable when predicting population growth in a diverse environment. "If you looked 'on average', the conditions are just hopeless and no one has any right to survive," he says. But if there are patches where it is possible to survive,>> “Chris Melhuish … unconscious adaptation occurs in very simple robot systems …create swarms of cheap, small "dumb" robots that move and …perform their small tasks without being encumbered with senses, computing power or communication devices. These little robots might herd around more complex "angelic" control units with more senses and intelligence, …power. [AB] simulation shows that these higher beings could be few and far between, and the dumb mortals could be very dumb indeed.” Lipid raft (influences diffusion) Information carriers with (autocatalytic adsorbtion) cell membrane Biological specificity and ”cognitive immunology” Interpretations in Various Fields: Origins of Life: - individuals =chemical molecules, - spatial patches = first self-sustaining proto-cells. Speciation: - Sites: various genomic configurations. - B= individuals; Jumps of B= mutations. - A= advantaged niches (evolving fitness landscape). - emergent adaptive patches= species Immune system: - B cells; A antigen B cells that meet antigen with complementary shape multiply. (later in detail the AIDS analysis) Similar auto-catalytic mechanisms for Spread of: INTERNET LINKS, INTERNET SUBJECTS, viruses communities, herds products, technologies, languages, ideas Finance:- sites: investment instruments - B = capital units, A= profit opportunities. Newton (after loosing 20 K Pounds in stock market) “I can calculate the motions of heavenly bodies, but not the madness of people." Financial markets don't need wise/ intelligent investors to work: Capital can survive and even proliferate simply by being autocatalytic Adam Smith’s invisible hand… doesn’t even need investor’s self-interest What if l, m, etc. are arbitrary ? Expressing the AB system formally as a Statistical Field Theory model and applying Renormalization Group Analysis to obtain its phases. For more details see: Reaction-Diffusion Systems with Discrete Reactants , Eldad Bettelheim , MSc Thesis, Hebrew University of Jerusalem 2001 http://racah.fiz.huji.ac.il/~eldadb/masters/masters2.html One describes the dynamics of the system in terms of the probabilities of the various configurations: Pnm (x) = the probability that there are m B’s and n A’s at the site x . The death of B’s and the birth of B’s in the presence of A’s are represented by the first and respectively second term in the Master Equation: d Pnm / dt = - m [ m Pnm – (m+1) P n,m+1 ] - l [ mn P nm – n (m-1) P n,m-1] l2s B AB l2s B AB l2s B AB l2s AB B l2s AB B l2s AB B l2s AB B l2s AB B l2s AB B l2s A B B l2s B A B l2s AB l2s AB l2s AB B l2s AB B l2s AB B l2s AB B ls AB B ls/2 AB B l2s AB B m = m – na l l Positive Naïve Effective B decay rate = (m-l a0) Each point represents another AB system: the coordinates represent its parameters: naive effective B decay rate (m-la0) and B division rate l Death B DivisionRate Life; Negative Decay Rate = Growth =l Positive Naïve Effective B decay rate = (m-l a0) Each point represents another AB system: the coordinates represent its parameters: naive effective B decay rate (m-la0) and B division rate l Initially, at small scales, B effective decay rate increases At larger scales B effective decay rate decreases B DivisionRate Life Wins! Death 2 p (d-2) D Life; Negative Decay Rate = Growth =l (m-l a0) l (m-l a0) l (m-l a0) l Ordinary miracles Michael Brooks, New Scientist magazine, May 2000 << According to John Beringer, an expert on microbial biology at the University of Bristol: "Microbes that need oxygen will be found close to the surface of soil, and microbes that are very fastidious about oxygen concentration will be found in bands at the appropriate oxygen concentration." Microbes concentrating on a two-dimensional resource may have been more successful than their cousins who tried exploiting a three-dimensional feast.>> Is our passing to multi-floor 3D habitat destroying social coherence? Finite cut-off What if A and B have finite size? (I.e. finite momentum cut-off ) In the finite cut-off extreme region: Life always wins - for any A configuration but never at your location - at all locations in statistical average (over all A configurations) but never in your particular stochastic realization. Rigorous proof using Polya Theorem Pólya 's Random Walk Constant What is the probability Pd ( ) that eventually an A returns to its site of origin? Pólya : but for d>2 P1 ( ) = P2 ( ) =1 Pd () < 1; P3 ( ) = 0.3405373 using it Kesten and Sidoravicius studied the AB model (preprint 75 p): On large enough 2 dimensional surfacesl, m, Db,Da,a0 Total/average B population always grows. Study the effect of one A on b(x,t) on its site of origin x Ignore for the moment the death and emigration and other A‟s Probability of one A return by time t (d-dimesional grid):Pd(t) Typical duration of an A visit: 1/Da e l/ Da Expected increase in b(x,t) due to 1 return events: e l/ Da Pd (t) Average increase of b(x,t) per A visit: Study the effect of one A on b(x,t) on its site of origin x Ignore for the moment the death and emigration and other A‟s Probability of one A return by time t (d-dimesional grid):Pd(t) Typical duration of an A visit: 1/Da e l/ Da Expected increase in b(x,t) due to 1 return events: e l/ Da Pd (t) Average increase of b(x,t) per A visit: But for d= 2 Pd ( ) =1 so el/Da Pd (t ) > eh el/Da Pd() eh 1 e l/ Da Pd (t) finite positive expected growth in finite time ! >0 t ~ eDa / 2l < h~ l/2Da t t {Probability of n returns before time t = n t } > P nd(t ) Growth induced by such an event: e nl/ Da Expected factor to b(x,t) due to n return events: > [el/ Da Pd(t )]n = e nh = eh t/t exponential time growth! Taking in account death rate m, emigration rate Db and that there are a(x,0) such A‟s: < b(x,t) > -increase is expected at all x’s where: a(x,0) > (m+Db)t/h There is a finite density of such a(x,0)‟s => <b(x,t)> > b(x,0) e - (m +Db) t e a(x,0)h t/t AB is extremely simple: quality - isolate minimal conditions for emergence, complex dynamics and adaptation. - allows rigorous proofs so necessary for such unbelievable properties. But to apply to real systems one has to take into account additional facts. E.g. saturation / competition. Does the AB effect survive? B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g BB B+B B; Competition of B’s at rate g BB B+B B; Competition of B’s at rate g B Finite Competition Range R y X B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g B B B+B B; Competition of B’s at rate g B Finite competition radius In each region of radius R there is only one active center. For equal g ‟s finite R gives greatly more B „s than R=0. Competition increases efficiency ! Polish spatial economic map since 89 (Andrzej Nowak ). Polish spatial economic map since 89 (Andrzej Nowak ). Polish spatial economic map since 89 (Andrzej Nowak ). Phys. Rev. Lett. 90, 38101 (2003) Measurements of organic matter distribution (every season) Mediterranean; Semi-arid; uniform 500mm patchy Desert; uniform 200mm s B+B B; Competition of B’s at rate g B So for R-> ∞ (globalization of trade / competition) => one island (wealth localization) => and large fluctuations Instability and Colapse of over-globalized civilizations Infinite Range (efficient but unstable) Intermediate Range Zero Range (unefficent but very stable) Conclusion? Regionalize but do not Globalize More reality tests: Pareto? No one however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern at least approximately. Davis; Cowles Commission for Research in Economics Pareto’s curve is destined to take its place as one of the great generalizations of human knowledge. Snyder 1939 Does the autocatalytic discrete AB system exhibit Pareto Law? (People with that wealth) Data Dragulescu 2003 (wealth) Number of sites with that B occupation number Data Model YES Dragulescu 2003 Number of B’s on the site Go To The Generalized Lotka-Volterra • Presentation for a Detailed Explanation ; computer room; 3513 etc •