Nonextensive Statistical Mechanics

7.8 Scale-Free Networks 289Fig. 7.94 The α-dependence of q ≡ q sen . See details in [846].mixed connections. Networks can be topological in nature, in the sense that we areallowed to arbitrarily deform them as long as we do not modify the connectionsbetween nodes and links. But they can also be metrical, in the sense that they mayhave a “geography” with a concept of distance, which can sensibly influence a varietyor properties. Bravais lattices can be thought as networks which are invariantthrough discrete translations. Through the concept of unitary crystalline cell, wecan attribute to them a nonzero Lebesgue measure. Hierarchical networks typicallyare scale-invariant, and can be characterized through a Hausdorff or fractal dimension.More complex networks can exhibit a multifractal structure, and can thus becharacterized by a f (α) function [212]. In what follows we focus on the so-calledscale-free networks. Indeed, they play an interesting role as systems that can be (atleast for some of their properties) addressed by the entropy S q and nonextensivestatistical mechanics.These networks are of the hierarchical kind, made of hubs, sub-hubs, sub-subhubs,and their links, the whole constituting a connected structure which exhibits(strict or statistical) invariance under dilation. Their basic characterization is donethrough the degree distribution p(k), defined as the probability of a node having klinks (k = 1, 2,...). It happens that many of them exhibit a power-law dependencein k for large values of k. And many among those, precisely have the formp(k) = p(0) e −k/κq (κ >0) , (7.31)where p(0) is a normalizing factor. This form is known to extremize S q with simpleconstraints (see Section (3.5)). It appears frequently in the literature asp(k) ∝which is identical to Eq. (7.31) through the transformation1(k 0 + k) μ , (7.32)