1. 1 Satellite Meeting of STATPHYS 22(KIAS) 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. 2 Satellite Meeting of STATPHYS 22(KIAS) I. Introduction Self-organized critical steady state S.J.Gould (1972) 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 The "punctuated equilibrium" theory

3. 3 Satellite Meeting of STATPHYS 22(KIAS) The Bak-Sneppen evolution model 0.20.30.150.40.450.70.90.350.10.550.750.50.80.650.60.25 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. 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 PBC P.Bak and K.sneppen PRL 71,4083 (1993) 0.20.30.150.40.450.70.90.950.470.220.750.50.80.650.60.25 Lowest fitness (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. 4 Satellite Meeting of STATPHYS 22(KIAS) Gap and Critical fitness : The lowest fitness at step s

5. 5 Satellite Meeting of STATPHYS 22(KIAS) Avalanche - subsequent sequences of mutations through fitness below a certain threshold Distribution of avalanche sizes in the critical state Punctuated equilibria - long periods of passivity interrupted by sudden bursts of activity The activity versus time in a local segment of ten consecutive sites. 1d2d 1.07(1)1.245(10)

6. 6 Satellite Meeting of STATPHYS 22(KIAS) (1) Study for a characteristic of evolution when interacting structures of biospecies are Scale-free Networks or Random Networks Motivation of this study (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. 7 Satellite Meeting of STATPHYS 22(KIAS) Exactly solvable model 0.20.30.150.40.450.70.90.350.10.550.750.50.80.650.60.25 - 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. II. Random Neighbor Model 0.20.30.150.770.450.340.90.350.520.550.750.50.220.650.60.89 Lowest fitness New lowest fitness : the i-th smallest fitness value : the distribution for : the distribution of all fitness values in the ecology where

8. 8 Satellite Meeting of STATPHYS 22(KIAS) The evolution equation for for (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) Avalanches 0 1 The limit is necessary to obtain the tree structure.

9. 9 Satellite Meeting of STATPHYS 22(KIAS) - 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. III. BS Evolution Model on Network Structures 0.2 0.75 0.6 0.7 0.8 0.9 0.4 0.3 0.5 0.1 0.25 0.45 0.2 0.21 0.6 0.7 0.62 0.9 0.4 0.3 0.5 0.31 0.25 0.98

10. 10 Satellite Meeting of STATPHYS 22(KIAS) Scale-free network : degree distribution - We predict the critical behavior of random network is similar to random neighbor model. Random network Scale-free network Mean degree : Num. Nodes : - Condition

11. 11 Satellite Meeting of STATPHYS 22(KIAS) IV. Results Random Network Random Neighbor model Consistent with PRL 81,2380 (1998)

12. 12 Satellite Meeting of STATPHYS 22(KIAS) Scale-free Network Is consistent with Europhys. Lett., 57, 765 (2002)

13. 13 Satellite Meeting of STATPHYS 22(KIAS)

14. 14 Satellite Meeting of STATPHYS 22(KIAS)

15. 15 Satellite Meeting of STATPHYS 22(KIAS) Europhys. Lett., 57, 765 (2002)

16. 16 Satellite Meeting of STATPHYS 22(KIAS)

17. 17 Satellite Meeting of STATPHYS 22(KIAS) ** Star networks

18. 18 Satellite Meeting of STATPHYS 22(KIAS) IV. Summary 2.15 (two power-law regimes) (star network) 2.3+1.2 2.40 2.27+1.32 2.75 2.22+1.47 3.00logarithmic 2.06+1.59 3.50 1.65 4.30 1.65 5.70 1.5 ◆ 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)