Bak-Sneppen Evolution models on Random and Scale-free Networks

1. Bak-Sneppen Evolution models on Random and Scale-free Networks I. Introduction II. Random Neighbor Model III. BS Evolution Model on Network Structures IV. Results V. Summary Sungmin Lee, Yup Kim Kyung Hee Univ.

2. I. Introduction The "punctuated equilibrium" theory Instead of a slow, continuous movement, evolution tends to be characterized by long periods of virtual standstill ("equilibrium"), "punctuated" by episodes of very fast development of new forms S.J.Gould (1972) Self-organized critical steady state

3. Fitness - An entire species is represented by a single fitness - The ability of species to survive - The fitness of each species is affected by other species to which it is coupled in the ecosystem. The Bak-Sneppen evolution model P.Bak and K.sneppen PRL 71,4083 (1993) Lowest fitness PBC At each time step, the ecology is updated by (i) locating the site with the lowest fitness and mutating it by assigning a new random number to that site, and (ii) changing the landscapes of the two neighbors by assigning new random numbers to those sites New lowest fitness Snapshot of the stationary state M.Paczuski, S.Maslov, P.Bak PRE 53,414 (1996)

4. Gap and Critical fitness : The lowest fitness at step s

5. Punctuated equilibria - long periods of passivity interrupted by sudden bursts of activity The activity versus time in a local segment of ten consecutive sites. Avalanche - subsequent sequences of mutations through fitness below a certain threshold Distribution of avalanche sizes in the critical state

6. Motivation of this study (1) Study for a characteristic of evolution when interacting structures of biospecies are Scale-free Networks or Random Networks (2) Self-Organized Criticality of Evolution and Punctuated Equilibrium on Network Structures (3) What is the best structure for the adaptation of species-correlation? (Is there evolution-free network?)

7. : the i-th smallest fitness value : the distribution of all fitness values in the ecology II. Random Neighbor Model Exactly solvable model Lowest fitness New lowest fitness - At each time step, the ecology is updated by (i) locating the node with the lowest fitness and mutating it by assigning a new random number to that site (ii) changing the landscapes of randomly selected K-1 sites by assigning new random numbers to those sites. : the distribution for where

8. for The limit is necessary to obtain the tree structure. for The evolution equation for Avalanches (1) identifying each burst with a node (2) and each of K new fitness values resulting from a burst - with either a branch rooted in that node (if the fitness value is less than the threshold value) - with a leaf rooted in the same node (if the fitness value is above threshold) 1 0

9. 0.2 0.2 0.9 0.3 0.9 0.25 0.3 0.25 0.8 0.1 0.62 0.31 0.7 0.7 0.45 0.75 0.98 0.21 0.4 0.4 0.6 0.5 0.6 0.5 III. BS Evolution Model on Network Structures - generate network structures with N nodes - A random fitness equally distributed between 0 and 1, is assigned to each node. - At each time step, the ecology is updated by (i) locating the node with the lowest fitness and mutating it by assigning a new random number to that site (ii) changing the landscapes of the linked neighbors by assigning new random numbers to those nodes.

10. Random network - We predict the critical behavior of random network is similar to random neighbor model. Scale-free network Scale-free network : degree distribution - Condition Mean degree : Num. Nodes :

11. IV. Results Random Network Random Neighbor model Consistent with PRL 81,2380 (1998)

12. Scale-free Network Is consistent with Europhys. Lett., 57, 765 (2002)

15. Europhys. Lett., 57, 765 (2002)

17. ** Star networks

18. IV. Summary ◆ Random Network ◆ Scale-free Network ◆ We would like to remark that two power-law regimes are shown in BS model on small world (cond-mat/9905066)