Universalities in Complex Systems:
Models of Nature
Konrad Hoppe
Department of Mathematics
Universalities
- Albeit the complexity of the systems under
investigation, one of the main goals is to find
Universalities
- Curiously, one can find many of these...
- In this talk, I will present
- Examples of universal patterns
- Reasons for their existence
- Various modelling attempts
Fibonacci Sequence
- Motivated in 1202 by Leonardo of Pisa (aka
Fibonacci) for rabbit populations
- Assume there exists one pair of rabbits (one
male, one female)
- After one month each pair mates and each
female produces one pair (f & m)
- How many rabbit pairs exists after one year?
- 1, 1, 2, 3, 5, 8, 13, 21,...
- The sequence follows a simple rule:
Fibonacci Sequence
- For Rabbit populations unrealistic
- No death
- Always pairs of female and male
- However, frequently found in nature
(www.goldenratiomyth.weebly.com)
(www.fractalfoundation.org)
Fibonacci Sequence
- The sunflower spirals are created by starting in
the middle and placing seeds by rotating
about a constant angle of 360/1.618 degrees
before placing the next seed
- 1.618 is a special number, it is the Golden
Ratio
- Directly related to the Fibonacci numbers
- This placing is optimal in terms of space
usage!
Self Similar Romanesco Broccoli
- Spirals
- Self Similarity
- Zooming in shows similar structures on each level
- Again, very rich macroscopic structure
- Also self similarity can be explained with simple tools
Self Similarities in Mathematics
- The Mandelbrot Set
- Described by all c, for
which the sequence
is bounded
- The Mandelbrot set is
self-similar around
certain points, for
example around
(-1.40115…,0)
Social Segregation
-
Our cities are segregated (www.blogs.lse.ac.uk)
Example of London
Can we understand this?
Yes! Cellular Automata are a powerful tool, to
model 2D evolution
Cellular Automata
- Rules are defined
Moore neighbourhood
depending on the
neighbourhood of a cell
- Cells are usually binary:
on/off
- E.g. If every cell around
a given cell is switched Von Neumann neighbourhood
on, then switch this cell
off
Schelling’s Model of Segration
- Two groups, ethnicities, classes,…
- Define a critical threshold F*, such that if the
fraction of neighbours from the owngroup is <F*,
then move to a free space
- Schelling did the experiment with coins on a grid
(1974):
Schelling’s Model of Segregation
- Separation of two phases:
- F<1/3: Random initial configuration remains
random
- F>1/3: Random configuration converges to a
segregated pattern
Cellular Automata in Nature
- A simple Automaton is defined in 1D
- State of a cell depends on its previous value and the
cells with offset -1,+1
- The rules are of the form (1,0,1) -> 1, (1,1,1) -> 0
- Since there are 23 =8 possible pattern in a
neighborhood
- Since each rule leads either to 1 or 0, there are 28 =
256 possible rules
Cellular Automata in Nature
- Rule 30 is considered to be interesting
111
110
101
100
011
010
001
000
0
0
0
1
1
1
1
0
Complex Networks
- Networks are complex structures that are
mostly self organized
- Albeit their very heterogeneous structures,
also here universalities can be found
Snapshot of the internet
(www.newmedialab.cuny.edu)
Complex Networks
- The degree distribution P(k) states the probability
that a randomly chosen node has k neighbours
- Many real-world networks have power law
degree distributions: 𝑃 𝑘 = 𝐴𝑘 −𝑎
The structure and function of complex networks, Newman (2003)
Complex networks
- If we understand the underlying principles that give rise to these
universal patterns
- Then we can maybe alter network structures
- My interest is in fitness dependent networks
- Each node is endowed with a static fitness value x
- Links in the networks are formed according to some function f(x,y)
- 𝑓 𝑥, 𝑦 = 1
- 𝑓 𝑥, 𝑦 = 𝑥𝑦
- 𝑓 𝑥, 𝑦 = 𝑥 − 𝑦
- If we can understand network connectivity in terms of f(x,y) then
we can understand also more complicated aspects
- Because just one function gives rise to all aspects of the network
- First thing: Check whether we can describe topology P(k) in terms
of f(x,y)
World trade network internals
Fitness density
Degree density
Fitness conditional degrees.
Model vs. reality
Degree density
from fitness model
The degree distribution
• Central quantity is the fitness and entry time conditional
degree distribution
• Evolves as
• Where lambda is an attractor in terms of f(u,w)
• Solution can be found using generating functions
Cascading failures
• Exposure to failures in the
neighbourhood lead to own
failure
• Fitness describes…
– attractiveness
– resilience
• This coupling makes sense!
– Interbank lending markets
• The attractiveness of a
partnership is coupled to its
possible endurance
• Exposure runs against edge
direction
Cascading failures cont’d.
•
•
•
•
A node fails if a critical fraction of its neighbours has failed
This critical fraction is the node’s fitness
A node is vulnerable if x < 1/k
Using previous technology, we know the fitness conditional
degree distribution p(k|x)
• This leads to the probability that a randomly chosen node
has degree k and is vulnerable
• And finally the percolation threshold
Size of the vulnerable
component
•
•
What is the effect of…
•
different fitness
distributions?
•
different attachment kernels?
•
different network densities?
This knowledge is valuable to
policy makers and regulators