1. Evolvable by Design Panos Oikonomou James Franck Institute Institute of Biophysical Dynamics The University of Chicago Philippe Cluzel How topology affects network evolution NetSci07

2. 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 ProjectYeast protein net, Jeong et al (2001)Internet map

3. Random Topology Scale-free Topology

4. 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

5. Boolean Threshold Dynamics

6. Network Dynamics N nodes in two states: ON/OFF Updated according to boolean rules Starting from random initial conditions Performs Cycle of length L

7. Target “Phenotype” Output Signal: Boolean time series The target Perform robustly a cycle behavior of length L c The fitness average hamming distance over time Parameters Net. Size~500 nodes L c = 1-50 μ= 0.001-0.1

8. Evolutionary Algorithm Parameters Pop. Size ~50 nets Net. Size~500 nodes L c = 1-50 μ= 0.001-0.1

9. Evolutionary Path

10. 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.

11. 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

12. Probability that a mutation affects an output node: Random ‹K› ‹P› ‹K› Scale-free

13. 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 > K c = 3.83 and scale- free networks exhibit chaotic behavior for exponents γ < γ c = 2.42.

14. 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.

15. Philippe Cluzel (Univ. of Chicago) Leo Kadanoff (Univ. of Chicago) Max Aldana (UNAM) Acknowledgements