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Stochastic models of innovation processes
Werner Ebeling *Werner Ebeling *
with R.Feistel*, I. Hartmann-Sonntag*, A.Scharnhorst**
* Humboldt-Universität, Berlin**NIWI, KNAW, Amsterdam
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1. Introduction
• Stochast. effects play important role in biological and socioeconomical processes,
• examples: innovations and technology transfer,transfer,
• the simple picture: the new is the better and replaces the bad old is not always true !!!
• Role of chance, of stochastic effects! • We consider two simple math models:
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1) Discrete Urn-model:what happens if new technologies appear
on the market, result of competition
• stochastic effects are important if the advantage of the NEW is small
• selection is vague with a broad region of • selection is vague with a broad region of neutrality; in order to win the competition the NEW needs big advantage.
• technologies with nonlinear growth rates have only a chance to win in niches or with external support.
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2) Models based on continuous Brownian dynamics: Transitions to other technologies
• Technologies are modelled as active Brownian particles with velocity-dependent friction, collective interactions and external confinement. interactions and external confinement.
• We simulate the dynamics of such transitions by Langevin equations and estimate the transition rates.
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2. Stochastic Urn Model
• Evolution as dynamics in a network
• A special role play transitions to new technologies (node 10).
NEW
technologies (node 10). • By changing the old, by
new ideas, inventions =formally a transition to a new node
• Fate of the NEW = stochastics on nodes
• Urn models !!!
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On history of stochastic urn models
• Paul & Tatjana Ehrenfest 1907: Urn models (flees jump from dog to dog) .
• Bartholomay 1958/59, Bartlett 1960: Birth and
First biophys. Appl.!!!
death processes, survival probabilities• Kimura/Eigen: Applications to problems of
evolution Applications to genetics population dynamics, etc.
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Stochastic change in occupation of nodes
Error reproduction of other nodes j (species)
Transitions to other nodes MUTATIONS
N_i occupation of node i is changing in
time 0, 1, 2, ...
stochastic death + other effects
Reproduction of the node i (species)
MUTATIONS
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Transformation of given d.e. of Volterra type to stochastic models. Recipe is clear only
for polynoms (transition probs ~ coeff.)
Special cases: Lotka-Volterra, Eigen-Schuster,..
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Network: Use edges between the nodes fo characterizing processes like self-reproduction, mutations, catalytic reproductions, decay etc.
Loop = selfreproduction
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When we need stochastic analysis ?
• As a rule stochast effects are small since (N >> 1). However there are other cases (N=0,1): Innovations!
• Of special interest innov with hypercycle • Of special interest innov with hypercycle charactkter (see theory of HC by Eigen/Schuster)
• HC are ring nets of species/ technologies with hyperbolic growth (WINDOWS, GOOGLE, all or nothing )
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Hypercycles of technology nets
Node (species) 1
Node (species) 2
Node (species) 3
Node (species) 3
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Stochastic models (birth& death): define nodes for species and
occupation numbers
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Occupation number space
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Def transition probs dep on coefficients
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Formulate a master eq as balance of elementary processes, simplex cond N = const
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How sharp is stochastic selection? What is stochastic neutrality ?
Node (species) 1
Node (species) 2
The message is:
Stochastic selection is very weak, nearly always neutral
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Study binary competition : 1=OLD, 2=NEW
• Consider a two-component system:• The MASTER with dominant
occupation:• The NEW species with one, or a few,
representatives which try to survive and (if possible) to win the competition.
• In general we will assume that e NEW is better with respect to reproductive rates
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Binary competition N_1 + N_2 = N = const
Only 1 independent variable N_2 (represent of the NEW)
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Linear rates, prob of survival (Bartholomay, Bartlett)
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Prob. of survival infinite generations in dep. on pop. size N=3,5,100 + determ. result
N = 35
100100
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Prob of survival n=10,3,1 generations (from below) and determin. result
as function of relative advantage (t-large)
n=13
10 generations
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Traditional conclusions get vague: Bad/Neutral/Better
• Deleterious?• neutral ??????• neutral ??????• Advantageous?
• Neutrality gets a new dynamic meaning (depending on N and n) !!!
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Nonlinear rates = hyperzyclic techn nets (selfacceleration)
• DETERMINISTIC picture:• growth is hyperbolic ! (singular at a finite
time)• Result depends not only on advantage but • Result depends not only on advantage but
also on initial conditions !• The (untercritical) NEW has no Chance !
(once-forever selection)• Ex: modern Infotec (Windows,Google,..)
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Simplest model: lin+quad rate terms
Separatrix
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Stochastic problems with nonlinear rates: New results !
Simple for N_2 (0) = 1 1 in numerator remains
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Special case of quadr growthb_i x_i^2
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N= 10, 4070
100
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Das Neue (auch HC) hat eine Chance (survival prob > 0)
Summary of stochastic effects:
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Hypercyclic nets of technol are qualitat different from linear nets!
• Deterministic picture: If a separatrix exists, the NEW has no chance
• Stochastic picture: New hypertechns with better rates have a good
at all.• Exception: the NEW
gets support, to cross the separatrix
chance.• However this is true
only for small niches
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A few references: discrete m.• Feistel/Ebeling: Evolution of Complex
Systems. Kluwer Dordrecht 1989• Ebeling/Engel/Feistel: Physik der
Evolutionsprozesse. Berlin 1990• J.Theor.Biol. 39, 325 (1981)• Phys. Rev. Lett. 39, 1979 (1987)• BioSystems 19,91(1986), in press(2006)• Physica A 287, 599 (2000)• arXiv:cond-mat/0406425 18 Jun 2004
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3. Brownian agents modelling transitions to new technologies
• Idea: Describe Techn by a set of cont Parameters: Heigth, weigth, size, power, techn data , ...,
• LANDSCAPE• LANDSCAPE• Space of cont. Charakteristika
(Metcalfe, Saviotti seit 1984)• Scharnhorst: G_O_E_THE (geometrical
oriented Evolution theory)
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Idee aus der Biologie: Wright Fitness landscapeEvolution as Optimization Process
Wright 1932: Population in an Adaptive Landscape/Fitness Landscape
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Size
size Basics
Speesd
Technological Evolution: Characteristics Space of Output Indicators
Metcalfe, Saviotti 1984
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The Occupation Landscape Changes According to the Fitness Landscape
Valuation
Occupation
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Evolutionary theory (Eigen/Schuster): d.e. corr to overdamped Langevin-eq. or
diffusion eq for conc.
x );(2D1
xd
vtdUd =⋅+=+ ξγ
!!! driving and
inertiaconsider s,coordinatet independen as
s velocitieheconsider t wemodel-ABM In the
x );(2Dm
x0 dtvt
drdt=⋅+=+ ξγ
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)(2Dv)(m
1v 2 tv
dr
dU
dt
d ξγ ⋅+−=+
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Dynamik von Techn, die linear zum Zentrum getrieben werden
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Rotationen (links/rechts): (limit cycles)
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10000 aktive Teilchen um linear anziehendes Zentrum: Einschwingprozess !
Swaest1.gif
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Rotations around a center
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2000 Agenten mit paarweise linearer Anziehung
Swharm.gif
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Transitions between two attractors (from good to better)
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Landscape with 2 hills
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Simulation of the transition of agents
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Statistical data, transition time etc dep on parameters
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time of transition (well to well) for increasing strength of drive
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the effect of collective relative attraction
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Wertelandschaft mit 3 Maxima
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Übergänge zwischen 3 Werte Maxima
3max46.exe
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Landscapes with many extrema
• 1. Ratchets (Saw tooth -Potentials)
• 2. Landscapes with randomly distr extrema
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Evolution of networks of agents (illustration by Erdmann)
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Referenzen zu Brown-Agenten
• Phys. Rev. Lett. 80, 5044-5047 (1998)• BioSystems 49, 5044-5047 (1999)• Eur. Phys. Journal B 15,105-113 (2000)• Phys. Rev. E 64, 021110 (2001)• Phys. Rev. E 64, 021110 (2001)• Schweitzer: Brownian agents..Berlin 2002• Phys. Rev E 65, 061106 (2002)• Phys. Rev E 67, 046403 (2003)• Complexity 8, No. 4 (2003) • Fluctuation & Noise Lett., (2004)
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Conclusions
• Stochastic effects may be important for socio-economic processes:
• In the framework of linear rate theory, stochastic selection is quite neutral, to win the competition, the NEW needs big advantage !!!competition, the NEW needs big advantage !!!
• Hypercyclic systems can win in small niches,• Complex transition/evolution processes may be
described by dynamics on landscapes.• Mathematical difficulties are relatively high !!!
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Solve if possible
• Analytical solutions. This is possible only for a few examples as:
• Simulations by means of a fast computer with sufficient memory
• The Fisher-Eigen-Schuster problem
• Survival probabilities
• Formulate efficient algorithms
• Extrapolate and compare with analytical results
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Size
Engine size
Basics 1
Speed
Technological Evolution: Characteristics Space of Output Indicators
Metcalfe, Saviotti 1984