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 (

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 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 ( …,0)

Social Segregation -Our cities are segregated ( -Example of London -Can we understand this? -Yes! Cellular Automata are a powerful tool, to model 2D evolution

Cellular Automata -Rules are defined depending on the neighbourhood of a cell -Cells are usually binary: on/off -E.g. If every cell around a given cell is switched on, then switch this cell off Moore neighbourhood Von Neumann neighbourhood

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

-Rule 30 is considered to be interesting

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 (

Complex Networks The structure and function of complex networks, Newman (2003)

Complex networks

World trade network internals Fitness densityDegree 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