1. The Watts-Strogatz model
3. SMALL WORLDS
The Watts-Strogatz model

2. 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

3. Networks in nature (empirical observations)

4. 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)

5. Two ways of constructing

6. 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

7. p=0 Ordered lattice

8. p=1 Random graph

9. 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

11. 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)

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

13. Scaling
Scaling hypothesis

14. N*=N*(p)

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

17. Crossover length on p

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

19. Mean-field results
Newman-Moore-Watts

20. Smallest-world network

22. 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

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

24. different values of pL

25. Average shortest path length

26. 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

27. 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

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

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

30. 3rd moment is high [clustering, large number of triangles]