We introduce a two-dimensional growth model where every new site is located, at a distance r from the barycenter of the pre-existing graph, according to the probability law 1/r 2+α G (αG ≥ 0), and is attached to (only) one pre-existing site with a probability ∝ ki/r α A i (αA ≥ 0; ki is the number of links of the i th site of the pre-existing graph, and ri its distance to the new site). Then we numerically determine that the probability distribution for a site to have k links is asymptotically given, for all values of αG, by P (k) ∝ e −k/κ q , where eis the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for αA not too large) by q = 1 + (1/3)e −0.526 α A , and the characteristic number of links by κ ≃ 0.1 + 0.08 αA. The αA = 0 particular case belongs to the same universality class to which the Barabasi-Albert model belongs. In addition to this, we have numerically studied the rate at which the average number of links ki increases with the scaled time t/i; asymptotically, ki ∝ (t/i) β , the exponent being close to β = 1 2(1 − αA) for 0 ≤ αA ≤ 1, and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs Γ-space for Hamiltonian systems) a scale-free network.Among the subjects that are being studied intensively nowadays in statistical physics, there are two, namely nonextensive statistical mechanics (see [1] for a review) and networks [2], in particular scale-free networks [3], which receive special attention in connection with complex systems [4,5,6,7]. Could these two topics be intimately related? This would not be so surprising after all, given the fact that both research lines frequently address similar types of natural and artificial systems, in physics, economics, chemistry, biology, linguistics, social sciences and others. In fact, such a connection has already been conjectured in several occasions, e.g., [1] (Preface and Chapter 1) and [8]. In the present paper we propose a growth model, on which we exhibit and quantitatively explore this connection.Let us consider a continuous plane. We shall construct a single connected network of sites (or nodes or vertices) and links (or bonds or edges) by gradually (sequentially) making it grow. We first fix one site (i = 1) at some arbitrary origin of the plane. The second site (i = 2) is randomly and isotropically chosen at a distance r distributed according to the probability law P G (r) ∝ 1/r 2+αG (α G ≥ 0; G stands for growth). This second site is then linked to the first one. To locate the third site (i = 3) we move the origin to the barycenter of the two first sites, and apply again the distribution P G (r) from this new origin. This third site is now going to be linked to only one of the pre-existing two sites. To do this, we use an attachement probability p A ∝ k i /r αA i (α A ≥ 0; A stands for attachment), where r i is the distance of the newly arrived site to the i th site of the pre-exi...

We investigate the optical transmission spectra of quasi-periodic dielectric multilayer slabs arranged in a fashion that exhibits what has been called deterministic disorders. They can be of the so-called substitutional sequences type, and are characterized by the nature of their Fourier spectrum, which can be dense pure point (e.g. a Fibonacci sequence) or singular continuous (e.g. Thue-Morse and double-period sequences). The transmission coefficients are conveniently derived by using a theoretical model based on the transfer-matrix approach. A comparison between the oblique-incidence optical transmission spectrum and the normal-incidence one shows quite a different transmission behaviours over a particular range of frequency.

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