3. SMALL WORLDS The Watts-Strogatz model. Watts-Strogatz, Nature 1998 Small world: the average...
Transcript of 3. SMALL WORLDS The Watts-Strogatz model. Watts-Strogatz, Nature 1998 Small world: the average...
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
Model proposed
Crossover from regular lattices to random graphsTunableSmall world network with (simultaneously):– Small average shortest path– Large clustering coefficient (not obeyed by RG)
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
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*
*
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
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),(),(
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
Spectrum
() depends on Kp=0 regular lattice () has singularitiesp grows singularities broadenp->1 semicircle law