3. SMALL WORLDS The Watts-Strogatz model

Watts-Strogatz, Nature 1998 Small world: the average shortest path length in a real network is small Six degrees of separation (Milgram, 1967) Local neighborhood + long-range friends A random graph is a small world

Networks in nature (empirical observations)

Model proposed Crossover from regular lattices to random graphs Tunable Small world network with (simultaneously): –Small average shortest path –Large clustering coefficient (not obeyed by RG)

Two ways of constructing

Original model Each node has K>=4 nearest neighbors (local) Probability p of rewiring to randomly chosen nodes p small: regular lattice p large: classical random graph

p=0 Ordered lattice

p=1 Random graph

Small shortest path means small clustering? Large shortest path means large clustering? They discovered: there exists a broad region: –Fast decrease of mean distance –Constant clustering

Average shortest path Rapid drop of l, due to the appearance of short- cuts between nodes l starts to decrease when p>=2/NK (existence of one short cut)

The value of p at which we should expect the transtion depends on N There will exist a crossover value of the system size:

Scaling Scaling hypothesis

N*=N*(p)

Crossover length d: dimension of the original regular lattice for the 1-d ring

Crossover length on p

General scaling form Depends on 3 variables, entirely determined by a single scalar function. Not an easy task

Mean-field results Newman-Moore-Watts

Smallest-world network

L nodes connected by L links of unit length Central node with short-cuts with probability p, of length ½ p=0 l=L/4 p=1 l=1

Distribution of shortest paths Can be computed exactly In the limit L-> , p->0, but  =pL constant. z=l/L

different values of pL

Average shortest path length

Clustering coefficient How C depends on p? New definition C’(p)= 3xnumber of triangles / number of connected triples C’(p) computed analytically for the original model

Degree distribution p=0 delta-function p>0 broadens the distribution Edges left in place with probability (1-p) Edges rewired towards i with probability 1/N notes

only one edge is rewired exponential decay, all nodes have similar number of links

Spectrum  ( ) depends on K p=0 regular lattice   ( ) has singularities p grows  singularities broaden p->1  semicircle law

3 rd moment is high [clustering, large number of triangles]