Introduction to complex systems. Training course. Lecture 3.

1. Complex Systems
Session 3.
Self-organization and criticality

2. The Matthew Effect
Robert K. Merton (1968)
Herbert A. Simon (1955) Nobel M. Prize (1978)
Udny Yule (1925)
Yule-Simon process and distribution
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3. Yule-Simon
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4. Preferential attachment
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5. Scaling in networks
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6. Complex Networks
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7. Degree distribution of Internet
routers
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α= 1.1

8. Degree distribution of WWW pages
α= 1.1

9. Degree distribution in Social
Networks

10. Gibrat’s law
Robert Gibrat (1931)
Growth rate (x=company- asset- or city size)

11. Growth is independent of size
Under this assumption, size distribution is power law again
with α approximately 1.
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12. Firm size distribution
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13. Sales &
Profit
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14. The Black Swan
SzEEDSM Complex Systems 2017 Nassim Taleb

15. Fat Tail
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16. Earthquake size distribution
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17. Phase transitions

18. Ferromagnetism and the Ising
model

19. Cluster size distribution

20. Mean field approximation

21. Mean field diagram

22. Bistability and bifurcation

23. Percolation

25. SzEEDSM Complex Systems 2017

26. Epidemic processes

27. SIS model

28. Traffic congestion

30. Inverse U curve

32. Social collapse

33. Collective intelligence

35. Power outages

36. 10 4 10 5 10 6 10 7
10
-2
10
-1
10
0
10
1
N= # of customers affected by outage
Frequency
(per year) of
outages > N
1984-1997
August 10, 1996

37. Square
site
percolation
or
simplified
“forest
fire”
model.
The simplest possible toy model of cascading
failure.

38. connected
not
connected
Connected clusters

39. A “spark” that hits
a cluster causes
loss of that
cluster.

40. yield
=
density
- loss
Assume: one randomly
located spark
(average)

41. yield
=
density
- loss
Think of (toy) forest fires.
(average)

42. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(avg.)
yield
density
“critical point”
N=100

45. Critical point

46. criticality
This picture is very generic.

47. 10
0
10
1
10
2
10
3
10
4
10
-1
10
0
10
1
10
2
Power laws Criticality

48. 10
0
10
1
10
2
10
3
10
4
10
-1
10
0
10
1
10
2
Power
laws: only
at the
critical point
low density
high density

49. Life, networks, the brain, the universe and
everything are at “criticality” or the “edge of
chaos.”
Does anyone really believe
this?

50. Self-organized criticality:
dynamics have critical point as global attractor
Simpler explanation: systems that
reward yield will naturally evolve to
critical point.

51. Would you
design a
system this
way?

52. Maybe random
networks
aren’t so great

53. High yields

54. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

55. isolated
critical
tolerant

56. Why power laws?
Almost any
distribution
of sparks
Optimize
Yield
Power law
distribution
of events

57. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
“optimized”
density
yield

58. 5 10 15 20 25 30
5
10
15
20
25
30
Probability distribution (tail of normal)
High probability region

59. Optimal “evolved”
“Evolved” = add one site at
a time to maximize
incremental (local) yield
Very local and limited optimization, yet still
gives very high yields.
Small events likely
large events
are unlikely

60. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
“optimized”
density
High yields.

61. Optimized grid
Small events likely
large events
are unlikely

62. Optimized grid
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
random
grid
High yields.

63. This source of power
laws is quite universal.
Almost any
distribution
of sparks
Optimize
Yield
Power law
distribution
of events

64. Tolerance is very different
from criticality.
• Mechanism generating power laws.
• Higher densities.
• Higher yields, more robust to sparks.
• Nongeneric, won’t arise due to random
fluctuations.
• Not fractal, not self-similar.
• Extremely sensitive to small perturbations that were
not designed for, “changes in the rules.”

65. Extreme robustness and extreme hypersensiti
Small
flaws

66. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

67. 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1