Formulating Evolutionary Dynamics of Organism-Environment Couplings Using Graph Product Multilayer Networks
- 1. FormulatingEvolutionaryDynamicsofOrganism- EnvironmentCouplingsUsingGraphProductMultilayer Networks Hiroki Sayama andYaneer Bar-Yam sayama@binghamton.edu
- 2. Evolution 2
- 3. 息 McDougal Littell Inc. 3
- 4. Reproduction with variation (crossover, mutation) Selection with competition Parallel search over possibility space by accumulating incremental changes Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Parent Parent Parent Parent Parent 4
- 5. 5 f
- 6. Reproduction with variation (crossover, mutation) Selection with competition Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Offspring Parent Parent Parent Parent Parent 6
- 7. Reproduction with variation (crossover, mutation) Selection with competition Offspring pool Parent pool Mean-field assumption is adopted for both selection and reproduction 7
- 9. 9 PRE 65, 051919 (2002) PRL 88, 228101 (2002) Cons. Biol. 17, 893-900 (2003)
- 10. Here we present a theoretical framework that formulates the evolution of general organism-environment couplings using graph product multilayer networks. Review of graph product multilayer networks Organism-environment coupling space Reaction-diffusion evolutionary dynamics Numerical study 10
- 11. Review of Graph Product Multilayer Networks 11 Sayama (2017) J. Complex Netw., cnx042 https://doi.org/10.1093/comnet/cnx042 https://arxiv.org/abs/1701.01110
- 12. 12 2x2 = 4
- 13. 13 x = ?
- 14. 14 1 2 3 4 a b c {1, 2, 3, 4} {a, b, c}x (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c)Meta nodes
- 15. DegreeSpectra: ExactSolutions 15 (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c) (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c) (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c) Layers Layers Graph Product Multilayer Networks
- 16. Cartesian product Direct product Strong product 16
- 18. 18 1 2 3 4 a b c {1, 2, 3, 4} {a, b, c}x (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c)
- 19. 19 , : Kronecker sum and product
- 20. DegreeSpectra: ExactSolutions 20 lH 1 lH 2 lH n lG 1 lG 2 lG m lH 1 lH 2 lH n lG 1 lG 1+lH 1 lG 1+lH 2 lG 1+lH n lG 2 lG 2+lH 1 lG 2+lH 2 lG 2+lH n lG m lG m+lH 1 lG m+lH 2 lG m+lH n
- 23. 23 1 2 3 4 a b c {1, 2, 3, 4} {a, b, c}x (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c)
- 24. DegreeSpectra: ExactSolutions 24 Sayama, H. (2016) DiscreteApplied Mathematics 205, 160-170. https://doi.org/10.1016/j.dam.2015.12.006 lH 1 lH 2 lH n lG 1 lG 2 lG m lH 1 lH 2 lH n lG 1 lG 1lH 1 lG 1lH 2 lG 1lH n lG 2 lG 2lH 1 lG 2lH 2 lG 2lH n lG m lG mlH 1 lG mlH 2 lG mlH n Approximated Laplacian spectrum
- 26. 26 , : Kronecker sum and product
- 27. 27 1 2 3 4 a b c {1, 2, 3, 4} {a, b, c}x (1, a) (1, b) (1, c)(3, a) (3, b) (3, c) (2, a) (2, b) (2, c) (4, a) (4, b) (4, c)
- 28. DegreeSpectra: ExactSolutions 28 Sayama, H. (2016) DiscreteApplied Mathematics 205, 160-170. https://doi.org/10.1016/j.dam.2015.12.006 Approximated Laplacian spectrum
- 30. env-nodes 30 G
- 31. org-nodes 31 H
- 32. 隆: spatial diffusion rate (average link weight in G) 32 袖: type diffusion rate (average link weight in H)
- 34. 3434 Spatial & type diffusions as one diffusion process Laplacian of GH very easy to obtain & analyze
- 37. 37 Local population dynamics Environment- independent fitnesses Environment- dependent fitnesses
- 38. 隆: spatial diffusion rate 38 袖: type diffusion rate How does 隆 affect the fitnesses of organisms?
- 40. 40 G, 隆 H, 袖, FH 留, Fe Dominant eigenvector of (F L)
- 41. 41 Dominant eigenvector of (F L) 留, Fe H, 袖, FH G, 隆 Fixed parameters Random weighted graph with 20 nodes Random weighted graph with 20 nodes 10-5 Random weighted diagonal matrix (20x20) 0.5 Random weighted diagonal matrix (400x400)
- 42. 42 Dominant eigenvector of (F L) 留, Fe H, 袖, FH G, 隆 Varied parameter 10-5 ~ 102
- 43. 43 n = 20,000 Actual fitness ~ inherent fitness (non-spatial; environment-independent) Actual fitness inherent fitness (spatial; environment- dependent)
- 44. 44 留 = 0.0 留 = 0.1 留 = 0.2 留 = 0.4 留 = 0.5 留 = 0.6 留 = 0.8 留 = 0.9 留 = 1.0
- 45. 45 袖 = 10-4 袖 = 10-3 袖 = 10-2
- 46. Conclusions 46
- 47. 47
- 48. Multilayer networks can be useful to study evolution. 48
- 49. Limitations Considered simple linear dynamics only No inter-species interactions considered Diffusion rates independent of organisms No analytical explanation (yet) 49
- 50. Next Steps Including nonlinear, coupled population dynamics (e.g., replicator equations) Obtaining analytical conditions by exploiting GPMNs spectral properties 50