3. SMALL WORLDS

The Watts-Strogatz model

Watts-Strogatz, Nature 1998

Small world: the average shortest path length in a real network is smallSix degrees of separation (Milgram, 1967)Local neighborhood + long-range friendsA random graph is a small world

Networks in nature (empirical observations)

)ln(network Nl

graph randomnetwork CC

Model proposed

Crossover from regular lattices to random graphsTunableSmall 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 nodesp small: regular latticep large: classical random graph

p=0 Ordered lattice

)1(4

)2(3

12

K

KC

K

Nl

p=1 Random graph

small N

KC

small ln

ln

K

Nl

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 nodesl starts to decrease when p>=2/NK (existence of one short cut)

Npl

Npl

ln)1(

)0(

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

NlNN

NlNN

ln*

*

Scaling

Scaling hypothesis

1 if ln

1 if )( where

)/(),( **

uu

uu uF

NNFNpNl

N*=N*(p)

Crossover length

d

pN

/1

*

d: dimension of the original regular lattice

pN /1* 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

1/)ln(

1)(

function scaling universal )(

)(),(

uuu

uconsuf

uf

pKNfK

NpNl d

Mean-field results

Newman-Moore-Watts

uu

u

uuuf

4tanh

4

4)(

2

1

2

Smallest-world network

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

Distribution of shortest paths

Can be computed exactlyIn the limit L->, p->0, but =pL constant. z=l/L

zezzzpzQplLP 22 )]21(221[2),(),(

different values of pL

Average shortest path length

Clustering coefficient

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

222 48)1(2

)1(3)('

KppKKK

KKpC

Degree distribution

p=0 delta-functionp>0 broadens the distributionEdges left in place with probability (1-p)Edges rewired towards i with probability 1/N

notes

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

Spectrum

() depends on Kp=0 regular lattice () has singularitiesp grows singularities broadenp->1 semicircle law

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