evolvable by design panos oikonomou james franck institute institute of biophysical dynamics the...

Evolvable by Design

Panos Oikonomou

James Franck InstituteInstitute of Biophysical Dynamics

The University of Chicago

Philippe Cluzel

How topology affects network evolution

NetSci07

Introduction

• Some features are ubiquitous in nature and artificial systems • Which are the consequences/advantages of such organizations?

• How do such systems evolve?

Is there an evolutionary advantage in topological features?

US Human Genome Project Yeast protein net, Jeong et al (2001)Internet map

Random Topology Scale-free Topology

!/)( kKekP kK CkkP )(

• Dynamical rules for each node

• Dynamics of network

• Evolutionary Game: Genotype, Phenotype, Fitness,

Mutations & Selection • Results:

Random vs. scale-free

• Interpretation and heuristic explanation

Outline

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0)( if 0

0)( if 1

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0

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0

i

i

i

K

jijiji

K

jijij

K

jijij

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htwt

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Boolean Threshold Dynamics

Network Dynamics

• N nodes in two states: ON/OFF

• Updated according to boolean rules

• Starting from random initial conditions

• Performs Cycle of length L

Target “Phenotype”

Output Signal:Boolean time series

The target Perform robustly a cycle behavior of length Lc

The fitness average hamming distance over time

ParametersNet. Size~500 nodesLc= 1-50μ= 0.001-0.1

Evolutionary Algorithm

Parameters

Pop. Size ~50 netsNet. Size~500 nodesLc= 1-50μ= 0.001-0.1

Evolutionary Path

Random networks

Discontinuous evolution:

• Long fitness plateaus & sudden advantageous jumps

• Networks change by neutral mutations

• Convergence depends on rare advantageous mutative events. Each independent population converges differently from the average.

Scale-free networks

Continuous evolution:

• Diversity: the population consists of many different phenotypes

• Evolvability capacity to produce many different heritable phenotypes.

• All populations follow the same trend and are able to converge

Continuous vs. Discontinuous Evolutionary paths

Probability that a mutation affects an output node:

xdynPP

2.5 5 7.5 10 12.5 15 17.5

0.2

0.4

0.6

0.8

1Random

‹K›

‹P›

‹P›

‹K›

Scale-free

2.5 5 7.5 10 12.5 15 17.5

0.0001

0.001

0.01

0.1

1

)()( iisdyn kPkpP

Different topologies give different evolutionary behaviors!

Topology pre-determines the evolutionary paths of networks

evolution "at the edge of chaos“?

random networks exhibit chaotic behavior for K > Kc= 3.83 and scale-free networks exhibit chaotic behavior for exponents γ < γc= 2.42.

Conclusions

Homogeneous random networks and scale-free networks exhibit drastically different evolutionary paths.

Topology pre-determines the evolutionary paths of networks.

Possible implications in design and evolutionary strategies…

Oikonomou et al, Nature Physics, 2 (8), 2006.

Philippe Cluzel (Univ. of Chicago)

Leo Kadanoff (Univ. of Chicago)

Max Aldana (UNAM)

Acknowledgements