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3. SMALL WORLDS The Watts-Strogatz model
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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
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Networks in nature (empirical observations)
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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)
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Two ways of constructing
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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
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p=0 Ordered lattice
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p=1 Random graph
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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
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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)
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The value of p at which we should expect the transtion depends on N There will exist a crossover value of the system size:
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Scaling Scaling hypothesis
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N*=N*(p)
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Crossover length d: dimension of the original regular lattice for the 1-d ring
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Crossover length on p
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General scaling form Depends on 3 variables, entirely determined by a single scalar function. Not an easy task
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Mean-field results Newman-Moore-Watts
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Smallest-world network
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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
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Distribution of shortest paths Can be computed exactly In the limit L-> , p->0, but =pL constant. z=l/L
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different values of pL
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Average shortest path length
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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
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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
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only one edge is rewired exponential decay, all nodes have similar number of links
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Spectrum ( ) depends on K p=0 regular lattice ( ) has singularities p grows singularities broaden p->1 semicircle law
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3 rd moment is high [clustering, large number of triangles]
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