Transcript 3. Small worlds

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)
lnetwork  ln(N )
Cnetwork  Crandom graph
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
N
l
 1
2K
3( K  2)
C
4( K  1)
p=1 Random graph
ln N
l
ln K
K
C
N
small
small
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
l ( p  0)  N
l ( p  1)  ln N
Rapid drop of l, due to the appearance of shortcuts 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:
N  N l  N
*
N  N *  l  ln N
Scaling
Scaling hypothesis
l ( N , p)  N F ( N / N )
*
 u
where F (u )  
ln u
*
if u  1
if
u  1
N*=N*(p)
Crossover length
N p
*

  1/ d
N  1/ p
*
d: dimension of the original regular lattice
for the 1-d ring
Crossover length on p
General scaling form
N
l ( N , p) 
f ( pKN d )
K
f (u ) universalscaling function
 cons u  1
f (u )  
ln(u ) / u u  1
Depends on 3 variables, entirely determined by a single
scalar function.
Not an easy task
Mean-field results
Newman-Moore-Watts
f (u ) 
4
u  4u
2
1
tanh
u
u  4u
2
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
LP(l , p)  Q( z, p)  2[1  2z  2 z(1  2z)]e
2
2 z
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
3K ( K  1)
C ' ( p) 
2
2
2
2 K ( K  1)  8 pK  4 p K
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
3rd moment is high [clustering, large number of triangles]