Observing Self-Organized Criticality - an approach to evolution dynamics By Won-Min Song

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

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 (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. F(x,y) t, F(x±1,y) t, F(x,y±1) t -> random number between 0 and 1.

: 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. - Generation of a random network with desired degree distribution

Confirmation of SOC phenomena in 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. improves as the system accumulates ‘memory’ of balance between births and deaths. Represented by maximum of minimum fitness values till t iterations. Represented by maximum of minimum fitness values till t iterations iterations 40X40 regular lattice. x-axis = log(t) y-axis = maximum struck until t iterations.

- 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 => Displays SOC characteristics in the dynamics.

-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. BS model on different random networks (Poisson and Power-law degree distributions)

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

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

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

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.

-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.

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.

- 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’.

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