c09_netw_mod_handout - Network models Properties common to...

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1 Properties common to many large-scale networks, independently of their origin and function: 1. The degree and betweenness distribution are decreasing functions, usually power-laws. 2. The distances scale logarithmically with the network size Network models N log scale - free ll ld 3. The clustering coefficient does not seem to depend on the network size, and is larger than the clustering coefficient of comparable random graphs There are two model families proposed to explain these properties: Small world network models and scale-free network models. k log l small world Benchmark 1: regular lattices One-dimensional lattice: 2 / 1 N L l const C const k N l = = , , Two-dimensional lattice: nodes inside for const. = = 15 6 C nodes inside for const. = = 6 k The average path-length varies as Constant degree (coordination number), constant clustering coefficient. D N l / 1 D-dimensional lattice: Benchmark 2: random graph theory Erdös-Rényi algorithm - Publ. Math. Debrecen 6, 290 (1959) • fixed node number N • connecting pairs of nodes with probability p Expected clustering coefficient: Expected path length: k log N log l rand N k p C rand = = k 1 N k k 1 N rand ) p 1 ( p C ) k ( P Expected degree distribution: Path length and order in real networks k log N log l rand = N k C rand = Real networks have short distances like random graphs yet show signs of local order.
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2 Small-world networks Watts-Strogatz model - D. Watts, S. Strogatz, Nature 393, 440 (1998) • lattice with K neighbors id i t h Real networks resemble both regular lattices and random graphs – perhaps they are in between. •rewire edges with probability p , K 2 N l = ) 1 K ( 4 ) 2 K ( 3 C = , K log N log l N K C Is there a regime with small l and large C? Transition from a lattice to a small world lattice small world random There is a broad interval of p over which but ) 0 ( ) ( C p C ) 1 ( ) ( l p l The onset of the small-world behavior depends on the system size ) pKN ( f K N ) p , N ( l d / 1 d is the dimension of the lattice = ) u ( f 1 u const << if 1 u u / u ln >> if lattice - like random graph - like 3 ) p 1 )( 0 ( C ) p ( C = These results cannot be directly compared to most real networks because the rewiring probability p is not known. random graph like The transition point depends on the rewiring probability, the size of the network and the average degree. Degree distribution of a small-world network K k = Rewiring does not change the average degree, but modifies the degree distribution. P(k) depends on the rewiring parameter p, but is always centered around <k>. Degree distribution similar to that of a random graph, with exponentially small probability for very highly connected nodes .
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3 Ex. 1 A variant of the Watts-Strogatz model adds random edges to a regular lattice. Start with a 1D lattice where every node has degree K . For each existing edge of a node, add an edge with a probability p . The endpoint of the edge is selected randomly from all other
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c09_netw_mod_handout - Network models Properties common to...

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