Observing Self-Organized
Criticality
- an approach to evolution dynamics
By Won-Min Song
Inspiration & Background
-The ‘Jenga’ Experiment
: Though it showed some emergent
phenomena as a complex system, still fails
to capture definite SOC features.
-’Real’ biological approach
: “Biological networks often are scale-free
networks.”
(H. Jeong et al(2000):The large-scale organization of metabolic networks)
So, Ask,
- “What can SOC tell about scale-free
organization”?
To answer the question,
- Bak-Sneppen model
: Adopts landscape function in SOC model. Eliminate the least fit
species and modify fitness of co-evolunary artners species. Replace
their fitness values by giving births to new species with random
fitness
- Cellular automaton
: For a regular network, the following algorithm has been used.
(x,y)t=coordinate of least fit species in the cellular automaton
F(x,y)t, F(x±1,y)t, F(x,y±1)t -> random number between 0 and 1.
-Generation of a random network with desired
degree distribution
: By assigning even number of
‘spokes’ for each node according
to desired probability density
function, one can create a desired
random network.
Caution!
- When connecting, avoid loops by
joining spokes from another node
if loops are not sought after.
Confirmation of SOC phenomena
in regular lattice
- Maximum critical value
: sets ultimate boundary between death and survival(Wills et al 2004).
improves as the system accumulates ‘memory’ of balance between births and
deaths.
Represented by maximum of minimum fitness values till t iterations.
log(t) -vs- maximum value reached for minimum fitness upto t iterations
-100000 iterations
0.35
40X40 regular lattice.
0.3
0.25
0.2
x-axis = log(t)
0.15
y-axis = maximum struck
until t iterations.
0.1
0.05
0
0
2
4
6
8
10
12
- Observing power-law behavior
: probability distribution of eliminating a species of age t has been plotted for
the same simulation.
probability of a global minimum occuring at a lattice point of age t with alpha~-1.3657
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
=> Power-law behavior observed with p(t)~t-1.3.
=> Displays SOC characteristics in the dynamics.
10
BS model on different random networks
(Poisson and Power-law degree distributions)
-By Renyi and Erdos’s study on random
network, Poisson degree distribution
effectively generates a random network.
-λ~5.3 used to generate the network for the
random network. The parameter is chosen
to match mean connectivity of the scalefree network.
Power-law degree distribution
(scale-network)
-Power-law degree distribution (scale-free network)
: p(k)~k-2.2 used to generate scale-free network.
degree distribution with alpha~-2.1747
9
8
7
6
5
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Random network outcomes
probability of a global minimum occuring at a lattice point of age t with alpha~-2.2348
noise in minimum fitness
20
0.12
0.1
15
0.08
10
0.06
0.04
5
0.02
0
0
1
2
3
4
5
6
7
evolution of maximum minimum fitness upto t iterations
0
-0.2
-0.1
-0.05
0
0.05
0.1
0.15
0.2
comparison to a gaussian fit in loglog space
0
0.25
-0.15
10
0.2
-2
10
Fc
0.15
0.1
-4
10
0.05
-6
0
10
0
2
4
6
8
10
12
log(t)
-Probability distribution of striking a
cell of age t
-Maximum of minimum fitness values
upto time t.
-3
10
-2
-1
10
10
x=data,- = gaussian fit
-noise distribution in minimum
fitness fluctuation
-Gaussian fit to the fluctuation
0
10
Scale-free network Outcomes
probability of a global minimum occuring at a lattice point of age t with alpha~-2.2017
noise in minimum fitness
20
0.2
15
0.15
10
0.1
5
0.05
0
0
1
2
3
4
5
6
0
-0.25
7
evolution of maximum minimum fitness upto t iterations
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
comparison to a gaussian fit in loglog space
0
0.25
-0.2
10
0.2
-5
10
0.15
Fc
-10
10
0.1
-15
10
0.05
-20
0
10
0
2
4
6
8
10
12
log(t)
-3
10
-2
-1
10
10
x=data,- = gaussian fit
-Probability distribution of striking a
cell of age t
-noise distribution in minimum
fitness fluctuation
-Maximum of minimum fitness values
upto time t.
-Gaussian fit to the fluctuation
0
10
General features
- Skewed noise fluctuation
: Because the boundary for minimum value
develops as the critical value develops, it is
‘biased’ as the system ‘evolves’. Gaussian fit is
thus not valid.
- Fail to see power-law behavior in the
age probability distribution
: A possible explanation for the change is change
in the critical value behavior. I.e. the system has
not settled into critical states.
-Moreover, two networks settles to the same
critical value that draws a line between
survival and death
=> Given the same size of system, mean connectivity decides the ultimate
fate of survival or death.
Q)Then what can be told about the two different
networks?
=> Efficiency of network. Scale-free network needs only
a few number of highly connected nodes to achieve the
same level of stability that a random network does by
distributing the ‘weight’ over the entire system.
Scale-free network with reasons
- It has been shown the maximum critical value
tends to zero in a scale free network as N->inf
with modification to adapt the real situations.
(Moreno et al(2002), Wills et al(2004))
I.e. gets more stable with increasing
system size.
- Normally biological networks are huge,
~millions. They may have evolved by
‘finding’ power-law efficient during
evolution period.
- Tolerance to external attack is achieved by
heterogeneity of the system.
- Cost for tolerance: If the highly connected
nodes are targeted, the result would be
‘devastating’.
Bibliography
[1] H. Jeong, B. T., R. Albert, Z. N. Oltvai & A. L. Barabasi (2000). "The large-scal
organization of metabolic networks." Nature 407.
[2] Per Bak, C. T., Kurt Wiesenfeld (1987). "Self-Organized Criticality: An Explanation
of 1/f Noise." Physical Review Letters 59(4).
[3] P. R. Wills, J. M. M., P. J. Smith (2004). "Genetic information and self-organized
criticality." Europhys. Lett. 68(6): 901-907.
[4] Moreno Y., V. A. (2002). Europhys. Lett. 57.
[5] Matt Hall, K. C., Simone A. di Collobiano, and Henrik Jeldtoft Jensen (2002).
"Time-dependent extinction rate and species abundance in a tnagled-nature model of
biological evolution." Physical Rewiew E 66.
[6] H. Jeong, S. P. M., A-L Barabasi, Z.N. Oltvai (2001). "Lethality and Cetrality in
protein networks." nature 411.
[7] Wikipedia(en.wikipedia.org)
[8] Per Bak, K. S. (1993). "Puntuated Equilibrium and Criticality in a Simple Model of
Evolution." Physical Review Letters 71(24): 4083-4086.