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...thers, the factor 3 accounting to the fact that each triangle contributes to 3 connected triples. This definition corresponds to the concept of ”fraction of transitive triples” used in sociology (see =-=Wasserman and Faust 1994-=-). To calculate C ′ (p) for the WS model, let us start with a regular lattice with a clustering coefficient C(0). For p > 0 two neighbors of a node i that were connected at p = 0 are still neighbors o...

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...ii and volume. The correlation length ξ is a measure of the mean cluster radius, and we know that ξ scales with the cutoff cluster size sξ as ξ ∝ s 1/νσ ξ . Thus the finite clusters are fractals (see =-=Mandelbrot 1982-=-), because their size does not scale as their radius to the dth power, but as s(r) ∼ r df , (42) where df = 1/σν. It can be also shown that at the percolation threshold the infinite cluster is still a...

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...ether there is a critical probability which marks the appearance of arbitrary subgraphs consisting of k nodes and l edges. In random graph theory there is a rigorously proven answer to this question (=-=Bollobás 1985-=-). Consider a random graph G = GN,p. In addition, consider a small graph F consisting of k nodes and l edges. In principle, the random graph G can contain several such subgraphs F. Our first goal is t...

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...000) and a Bose gas (Bianconi and Barabási 2000b). 1. The Simon model Aiming to account for the wide range of empirical distributions following a power-law, such as the frequency of word occurrences (=-=Zipf 1949-=-), the number of articles published by scientists (Lotka 1926), the city populations (Zipf 1949) or incomes (Pareto 1898), Simon (1955) proposed a class of stochastic models that result in a powerlaw ...

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...scale showing the linear behavior l(N, p) ∼ N/(4K) at small N/N ∗ . After Barrat and Weigt (2000). Numerical simulations and analytical arguments (Barthélémy and Amaral 1999a, Barrat 1999, Newman and =-=Watts 1999-=-a, Bartélémy and Amaral 1999b, Argollo de Menezes et al. 2000, Barrat and Weigt 2000) concluded that the crossover length N ∗ scales with p as N ∗ ∼ p −τ , where τ = 1/d and d is the dimension of the ...

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...er of the distance pc−p from the percolation threshold. This behavior is an example of critical phenomena: an order parameter goes to zero following a power-law in the vicinity of the critical point (=-=Stanley 1971-=-, Ma 1976). Cluster size distribution: The probability of having a cluster of size s is (Durett 1985) Pp(|C| = s) = 1 s Cs−1 2s ps−1 (1 − p) s+1 . (34) Here the number of edges surrounding the s nodes...

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...ance pc−p from the percolation threshold. This behavior is an example of critical phenomena: an order parameter goes to zero following a power-law in the vicinity of the critical point (Stanley 1971, =-=Ma 1976-=-). Cluster size distribution: The probability of having a cluster of size s is (Durett 1985) Pp(|C| = s) = 1 s Cs−1 2s ps−1 (1 − p) s+1 . (34) Here the number of edges surrounding the s nodes is 2s, f...

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...2πNp(1−p) if |λ| < 2 √ Np(1 − p) . (21) 0 otherwise Known as Wigner’s (see Wigner 1955, 1957, 1958), or the semicircle law, (21) has many applications in quantum, statistical and solid state physics (=-=Mehta 1991-=-, Crisanti et al. 1993, Guhr et al. 1998). The largest (principal) eigenvalue, λ1, is isolated from the bulk of the spectrum, and it increases with the network size as pN. When z > 1 the spectral dens...

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...onnected. B. Generating function formalism A general approach to random graphs with given degree distribution was developed by Newman, Strogatz and Watts (2000) using a generating function formalism (=-=Wilf 1990-=-). The generating function of the degree distribution, G0(x) = ∞∑ P(k)x k , (46) k=0 encapsulates all the information contained in P(k), since P(k) = 1 d k! kG0 dxk ∣ . (47) ∣ x=0 An important quantit...

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...∼ (k + k0) −γ e −(k+k0)/kc with k0 = 1, kc = 20 and γ = 2.4 (Jeong et al. 2001). 7. Ecological networks Food webs are used regularly by ecologists to quantify the interaction between various species (=-=Pimm 1991-=-). In a food web the nodes are species and the edges represent predator-prey relationships between them. In a recent study, Williams et al. (2000) investigated the topology of the seven most documente...

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...clusters but above pc a giant cluster spans the entire network. This phenomenon is markedly similar to a percolation transition, a topic much studied both in mathematics and in statistical mechanics (=-=Stauffer and Aharony 1992-=-, Bunde and Havlin 1994, 1996, Grimmett 1999, ben Avraham and Havlin 2000). Indeed, the percolation transition and the emergence of the giant cluster are the same phenomenon expressed in different lan...

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...00), which is shaped by evolution over many generations. The network structure plays a crucial role in determining the spread of ideas, innovations or computer viruses (Coleman, Menzel and Katz 1957, =-=Valente 1995-=-). In this light, spreading and diffusion has been studied both on regular (Kauffman 1992, Keeling 1999), random (Solomonoff, Rapoport 1951, Rapoport 1957, Weigt and Hartmann 2001), small-world (Abram...

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...the topological aspects of robustness, caused by edge and/or node removal. The first results regarding network reliability under edge removal came from random graph theory (Moore and Shannon 1956a,b, =-=Margulis 1974-=- and Bollobás 1985). Consider an arbitrary connected graph HN of N nodes, and assume that a p fraction of the edges have been removed. What is the probability that the resulting subgraph is connected,...

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... tail with γsyn = 2.8. These results indicate that in many respects the language also forms a complex network with organizing principles not so different from the examples discussed earlier (see also =-=Steyvers and Tennenbaum 2001-=-). 11. Power and neural networks The power grid of the western United States is described by a complex network whose nodes are generators, transformers and substations, and the edges are high-voltage ...

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...ding actors have acted in a movie together. This is a continuously expanding network, with 225, 226 nodes in 1998 (Watts, Strogatz 1998) which grew to 449, 913 nodes by May 2000 (Newman, Strogatz and =-=Watts 2000-=-). The average path length of the actor network is close to that of a random graph with the same size and average degree, 3.65 compared to 2.9, but its clustering coefficient is more than 100 times hi...