This document presents a theoretical framework for modeling the evolution of organism-environment couplings using graph product multilayer networks. It describes how spatial and type diffusions between organism and environment nodes can be modeled as a single diffusion process on the graph product. The framework then models evolutionary dynamics as a reaction-diffusion process on this network, where local population dynamics are influenced by both inherent and environment-dependent fitnesses, and diffusion allows spatial and type mixing. Results demonstrate how the actual fitness of organisms depends on both their inherent fitness and the strength of spatial diffusion between environments.

1. Graph Product
Representationof
Organism-Environment
Couplings in Evolution
Hiroki Sayama
sayama@binghamton.edu

2. 2
MATH EVOLUTION

3. Math
3

4. What Are Graph
Product Multilayer
Networks?
4
Sayama (2018) J. Complex Netw., cnx042
https://doi.org/10.1093/comnet/cnx042
https://arxiv.org/abs/1701.01110

5. 5
2x2 = 4

6. 6
x = ?

7. 7
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

8. DegreeSpectra: ExactSolutions
8
(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

9. Cartesian
product
9

10. 10
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)

11. 11
, : Kronecker sum and product

12. DegreeSpectra: ExactSolutions
12
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

13. Direct
product
13

14. 14
: Kronecker product

15. 15
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)

16. DegreeSpectra: ExactSolutions
16
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

17. Strong
product
17

18. 18
, : Kronecker sum and product

19. 19
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)

20. DegreeSpectra: ExactSolutions
20
Sayama, H. (2016) DiscreteApplied Mathematics 205, 160-170.
https://doi.org/10.1016/j.dam.2015.12.006
Approximated Laplacian spectrum

21. DegreeSpectra: ExactSolutions
21
Generalization of
strong products
Sayama (2018) J. Complex Netw., cnx042 https://doi.org/10.1093/comnet/cnx042
https://arxiv.org/abs/1701.01110

22. DegreeSpectra: ExactSolutions
22

23. 23

24. Evolution
24

25. 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
25

26. Reproduction
with variation
(crossover, mutation)
Selection
with competition
Offspring
pool
Parent
pool
Mean-field
assumption is adopted
for both selection and
reproduction
26

27. © nationalgeographic.com
27

28. 28
PRE 65, 051919 (2002)
PRL 88, 228101 (2002)
Cons. Biol. 17, 893-900 (2003)

29. Finally, in this talk:
We present a theoretical framework
that formulates the evolution of
organism-environment couplings
using graph product multilayer
networks
29

30. Organism-
Environment
Coupling Space
30

31. env-nodes
31
G

32. org-nodes
32
H

33. δ: spatial diffusion rate
(average link weight in G)
33
µ: type diffusion rate
(average link weight in H)

34. 34
org-env combos G□H

35. 3535
Spatial & type diffusions as one diffusion process
Laplacian of G□H very easy to obtain & analyze

36. Reaction-Diffusion
Evolutionary
Dynamics
36

37. 37
Local population
dynamics
Diffusion on
G□H

38. 38
Local population
dynamics Environment-
independent
fitnesses
Environment-
dependent
fitnesses

39. δ: spatial diffusion rate
39
µ: type diffusion rate
How does δ affect the
fitnesses of organisms?

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
Results

44. 44
n = 20,000
Actual fitness ~ inherent fitness
(non-spatial;
environment-independent)
Actual fitness ≠
inherent fitness
(spatial;
environment-
dependent)

45. 45
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

46. 46
µ = 10-4 µ = 10-3 µ = 10-2

47. Conclusions
47

48. 48

49. When spatial mixing is weak, it is
not adequate to attribute fitness
to individual organisms alone.
What is actually evolving is the
org-env coupling.
49

50. DegreeSpectra: ExactSolutions
50
ThankYou
@hirokisayama