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

i

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