Observing Self-Organized Criticality - an approach to evolution dynamics

1. Observing Self-OrganizedCriticality- an approach to evolution dynamics By Won-Min Song

2. Inspiration & Background -The ‘Jenga’ Experiment : Though it showed some emergent phenomena as a complex system, still fails to capture definite SOC features. -’Real’ biological approach : “Biological networks often are scale-free networks.” (H. Jeong et al(2000):The large-scale organization of metabolic networks)

3. So, Ask, • “What can SOC tell about scale-free organization”? To answer the question, • Bak-Sneppen model : Adopts landscape function in SOC model. Eliminate the least fit species and modify fitness of co-evolunary artners species. Replace their fitness values by giving births to new species with random fitness • Cellular automaton : For a regular network, the following algorithm has been used. (x,y)t=coordinate of least fit species in the cellular automaton F(x,y)t, F(x±1,y)t, F(x,y±1)t-> random number between 0 and 1.

4. Generation of a random network with desired degree distribution : By assigning even number of ‘spokes’ for each node according to desired probability density function, one can create a desired random network. Caution! • When connecting, avoid loops by joining spokes from another node if loops are not sought after.

5. Confirmation of SOC phenomenain regular lattice • Maximum critical value : sets ultimate boundary between death and survival(Wills et al 2004). improves as the system accumulates ‘memory’ of balance between births and deaths. Represented by maximum of minimum fitness values till t iterations. -100000 iterations 40X40 regular lattice. x-axis = log(t) y-axis = maximum struck until t iterations.

6. Observing power-law behavior : probability distribution of eliminating a species of age t has been plotted for the same simulation. => Power-law behavior observed with p(t)~t-1.3. => Displays SOC characteristics in the dynamics.

7. BS model on different random networks(Poisson and Power-law degree distributions) -By Renyi and Erdos’s study on random network, Poisson degree distribution effectively generates a random network. -λ~5.3 used to generate the network for the random network. The parameter is chosen to match mean connectivity of the scale-free network.

8. Power-law degree distribution (scale-network) -Power-law degree distribution (scale-free network) : p(k)~k-2.2 used to generate scale-free network.

9. Random network outcomes -Probability distribution of striking a cell of age t -Maximum of minimum fitness values upto time t. -noise distribution in minimum fitness fluctuation -Gaussian fit to the fluctuation

10. Scale-free network Outcomes -Probability distribution of striking a cell of age t -Maximum of minimum fitness values upto time t. -noise distribution in minimum fitness fluctuation -Gaussian fit to the fluctuation

11. General features • Skewed noise fluctuation : Because the boundary for minimum value develops as the critical value develops, it is ‘biased’ as the system ‘evolves’. Gaussian fit is thus not valid. • Fail to see power-law behavior in the age probability distribution : A possible explanation for the change is change in the critical value behavior. I.e. the system has not settled into critical states.

12. -Moreover, two networks settles to the same critical value that draws a line between survival and death => Given the same size of system, mean connectivity decides the ultimate fate of survival or death. Q)Then what can be told about the two different networks? => Efficiency of network. Scale-free network needs only a few number of highly connected nodes to achieve the same level of stability that a random network does by distributing the ‘weight’ over the entire system.

13. Scale-free network with reasons - It has been shown the maximum critical value tends to zero in a scale free network as N->inf with modification to adapt the real situations. (Moreno et al(2002), Wills et al(2004)) I.e. gets more stable with increasing system size. - Normally biological networks are huge, ~millions. They may have evolved by ‘finding’ power-law efficient during evolution period.

14. Tolerance to external attack is achieved by heterogeneity of the system. • Cost for tolerance: If the highly connected nodes are targeted, the result would be ‘devastating’.

15. Bibliography • [1] H. Jeong, B. T., R. Albert, Z. N. Oltvai & A. L. Barabasi (2000). "The large-scal organization of metabolic networks." Nature407. • [2] Per Bak, C. T., Kurt Wiesenfeld (1987). "Self-Organized Criticality: An Explanation of 1/f Noise." Physical Review Letters59(4). • [3] P. R. Wills, J. M. M., P. J. Smith (2004). "Genetic information and self-organized criticality." Europhys. Lett.68(6): 901-907. • [4] Moreno Y., V. A. (2002). Europhys. Lett.57. • [5] Matt Hall, K. C., Simone A. di Collobiano, and Henrik Jeldtoft Jensen (2002). "Time-dependent extinction rate and species abundance in a tnagled-nature model of biological evolution." Physical Rewiew E66. • [6] H. Jeong, S. P. M., A-L Barabasi, Z.N. Oltvai (2001). "Lethality and Cetrality in protein networks." nature411. • [7] Wikipedia(en.wikipedia.org) • [8] Per Bak, K. S. (1993). "Puntuated Equilibrium and Criticality in a Simple Model of Evolution." Physical Review Letters71(24): 4083-4086.