Large Scale Structure and Dynamics of Complex Networks: From by Guido Caldarelli

By Guido Caldarelli

This booklet is the end result of 3 years of analysis attempt on a multidisciplinary undertaking within which physicists, mathematicians, computing device scientists and social scientists labored jointly to reach at a unifying photograph of advanced networks. The contributed chapters shape a reference for a number of the difficulties in facts research visualization and modeling of advanced networks.

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An edge is then present between i and j if zij < pK(xi , xj ). The distribution q(x), the probability p and the connection kernel K(x, y) are the shaping ingredients of the networks. A symmetric choice of the kernel, K(x, y) = K(y, x) produces non-directed networks, and in this case the choice K(x, y) =const (const= 1 without loss of generality) produces simple ER networks with a governing probability p. The degree distribution of the network can be computed by finding the relation between the degree k of a vertex and its variable x.

7 June 6, 2007 9:49 46 WSPC/Trim Size: 9in x 6in for Review Volume book M. A. Serrano, M. Bogu˜ n´ a, R. Pastor-Satorras and A. Vespignani Although overall scalar measures are helpful as a first indication of clustering, it is always more informative to work with quantities which explicitly depend on the degree. As in the case of two vertices correlations, an uniparametric function c¯(k)50 can also be computed. In practice, the degreedependent local clustering c¯(k) is calculated as the clustering coefficient averaged for each degree class k.

Yet the good news are that the γ exponent of the Barab´ asi-Albert model is actually sensitive to the details of the preferential attachment rule if α = 1. If for example f (k) = a + k, with a a constant, the resulting exponent is γ = 3 + a/m, showing that any exponent γ > 2 can actually be obtained. The Barab´ asi-Albert model endowed with a linear preferential attachment rule defines therefore a class of scale-free networks of tunable degree distribution exponent. One last comment is necessary for the case α > 1.

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