Hiroki Sayama and Yaneer Bar-Yam, Formulating evolutionary dynamics of organism-environment couplings using graph product multilayer networks, an invited talk at PhysPlex II: Second Satellite Symposium on Multilayer and Interconnected Networks: Applications, at Conference on Complex Systems 2017 (CCS 2017), September 21, 2017, Cancun, Mexico.

1. FormulatingEvolutionaryDynamicsofOrganism-
EnvironmentCouplingsUsingGraphProductMultilayer
Networks Hiroki Sayama andYaneer Bar-Yam
sayama@binghamton.edu

2. Evolution
2

3. © 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

8. © nationalgeographic.com
8

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

17. Cartesian
product
17

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

21. Direct
product
21

22. 22
: Kronecker product

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

25. Strong
product
25

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

29. Organism-
Environment
Coupling Space
29

30. env-nodes
30
G

31. org-nodes
31
H

32. δ: spatial diffusion rate
(average link weight in G)
32
µ: type diffusion rate
(average link weight in H)

33. 33
org-env combos G□H

34. 3434
Spatial & type diffusions as one diffusion process
Laplacian of G□H very easy to obtain & analyze

35. Reaction-Diffusion
Evolutionary
Dynamics
35

36. 36
Local population
dynamics
Diffusion on
G□H

37. 37
Local population
dynamics Environment-
independent
fitnesses
Environment-
dependent
fitnesses

38. δ: spatial diffusion rate
38
µ: type diffusion rate
How does δ affect the
fitnesses of organisms?

39. Numerical Study
39

40. 40
G, δ
H, µ, FH
α, Fe
Dominant
eigenvector
of (F – L)

41. 41
Dominant
eigenvector
of (F – L)
α, Fe
H, µ, FH
G, δ
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)
α, Fe
H, µ, FH
G, δ
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
α = 0.0 α = 0.1 α = 0.2
α = 0.4 α = 0.5 α = 0.6
α = 0.8 α = 0.9 α = 1.0

45. 45
µ = 10-4 µ = 10-3 µ = 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 GPMN’s spectral properties
50

51. DegreeSpectra: ExactSolutions
51
ThankYou
@hirokisayama