Abstract
Deep connections are known to exist between scalefree networks and nonGibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form , where the qexponential form optimizes the nonadditive entropy S_{q} (which, for q → 1, recovers the BoltzmannGibbs entropy). We introduce and study here ddimensional geographicallylocated networks which grow with preferential attachment involving Euclidean distances through . Revealing the connection with qstatistics, we numerically verify (for d = 1, 2, 3 and 4) that the qexponential degree distributions exhibit, for both q and k, universal dependences on the ratio α_{A}/d. Moreover, the q = 1 limit is rapidly achieved by increasing α_{A}/d to infinity.
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Introduction
Networks emerge spontaneously in many natural, artificial and social systems. Their study is potentially important for physics, biology, economics, social sciences, among other areas. For example, many empirical studies have identified peculiar properties in very different networks such as the Internet and online social networks (e.g., Facebook), citations networks, neurons networks^{1,2,3}, to quote but a few. An ubiquitous class of such networks is constituted by the scalefree ones (more precisely, asymptotically scalefree). As we shall soon verify, these networks can be seen as a particular application of nonextensive statistical mechanics, based on the nonadditive entropy , where BG stands for BoltzmannGibbs)^{4,5,6}. This current generalization of the BG entropy and corresponding statistical mechanics has been widely successful in clarifying the foundations of thermal statistics as well as for applications in complex systems in highenergy collisions at LHC/CERN (CMS, ALICE, ATLAS and LHCb detectors) and at RHIC/Brookhaven (PHENIX detector)^{7,8,9,10,11,12,13,14,15,16}, cold atoms^{17}, dusty plasmas^{18}, spinglasses^{19}, trapped ions^{20}, astrophysical plasma^{21,22}, biological systems^{23}, typeII superconductors^{24}, granular matter^{25}, the Kuramoto model at the edge of chaos^{26}, lowdimensional maps, for instance the (areapreserving) standard map^{27} (see Bibliography in http://tsallis.cat.cbpf.br/biblio.htm). Many other physical situations are described which are analogous, such as longrangeinteracting Hamiltonians, for example, gravitational problems like globular clusters, spins systems, like the Ising, XY and Heisenberg longrange models. We may also point out randomwalk anomalous diffusion where the jumps obey a power low probability distribution function. Some (naturally not all) of the properties of longrangeinteracting systems may be described as forming complex network where the sites are linked according to powerlaw preferential attachment. In the present work we address a wide class of this kind of problems focusing on some basic universality relations.
The deep relationship between scalefree networks and qstatistics started being explored in 2005^{28,29,30}, and is presently very active^{31,32,33,34,35}. The basic connection comes (along the lines of the BG canonical ensemble) from the fact that, if we optimize the functional with the constraint or analogous (k being the degree of a generic site, i.e., the number of links connected to a given site; P(k) denotes the degree or connectivity distribution), we straightforwardly obtain , which turns out to be the generic degree distribution for virtually all kinds of scalefree networks. The qexponential function is defined as . We verify that, for q > 1 and k → ∞, P(k) ~ 1/k^{γ} with γ ≡ 1/(q − 1). The classical result γ = 3^{36} corresponds to q = 4/3.
In the present work we address the question of how universal such results might be, and more specifically, how P(k) varies with the dimension d of the system?
Our growing model starts with one site at the origin. We then stochastically locate a second site (and then a third, a fourth, and so on up to N) through the ddimensional isotropic distribution
where r ≥ 1 is the Euclidean distance from the newly arrived site to the center of mass of the preexisting system (in one dimension, r = x; in two dimensions, ; in three dimensions , and so on); we assume angular isotropy; p(r) is zero for 0 ≤ r < 1; the subindex G stands for growth. We consider α_{G} > 0 so that the distribution P(r) is normalizable; indeed, , which is finite for α_{G} > 0, and diverges otherwise. See Fig. 1.
Every new site which arrives is then attached to one and only one site of the preexisting cluster. The choice of the site to be linked with is done through the following preferential attachment probability:
where k_{i} is the connectivity of the ith preexisting site (i.e., the number of sites that are already attached to site i), and r_{ij} is the Euclidean distance from site i to the newly arrived site j; subindex A stands for attachment.
For α_{A} approaching zero and arbitrary d, the physical distances gradually loose relevance and, at the limit α_{A} = 0, all distances becomes irrelevant in what concerns the connectivity distribution, and we therefore recover the BarabásiAlbert (BA) model^{36}, which has topology but no metrics. The BA model was extended^{37} in such a way that it would be able to yield an exponent γ such that 2 < γ < 3, thus making the model more realistic. In this work they showed a topological phase transition which range from scalefree networks to exponential networks through three control parameters (addition of links, redirection of edges, and addition of new sites). In the present paper we show that, for arbitrary dimensionality, γ can be controlled in a kind of simpler manner, namely by metric changes through only one control parameter (namely the ratio α_{A}/d) in the structure of the network. Notice, however, that the BA generalized model is not a particular case of our model, and neither the other way around.
Largescale simulations have been performed for the (d = 1, 2, 3, 4) models for fixed (α_{G}, α_{A}), and we have verified in all cases that the degree distribution P(k) is completely independent from α_{G}: see Fig. 2. Using this fact, we have arbitrarily fixed α_{G} = 2, and have numerically studied the influence of (d, α_{A}) on P(k): see Figs 3 and 4. In all cases, the qexponential fittings with q > 1 and κ > 0 have been remarkably good. To test the goodness of fit, we performed KolmogorovSmirnov test^{38} (see Table 1). To deal with the problem that the data are very sparse in the tail, we excluded data points with sample probability less than 10^{−6}. The best fitting values for (q, κ) are indicated in Fig. 5. From normalization of P(k), P(0) can be expressed as a straightforward function of (q, κ).
Our most remarkable results are presented in Fig. 6, namely the fact that both the index q and the characteristic degree (or “effective temperature”) κ do not depend from (α_{A}, d) in an independent manner but only from the ratio α_{A}/d. This nontrivial fact puts the growing ddimensional geographically located models that have been introduced here for scalefree networks, on similar footing as longrangeinteracting manybody classical Hamiltonian systems such as the inertial XY planar rotators^{39,40,41,42} (possibly the generic inertial nvector rotators as well^{43,44}) and FermiPastaUlam oscillators, assuming that the strength of the twobody interaction decreases with distance as 1/(distance)^{α}. Moreover, as first pointed out generically by Gibbs himself^{45}, we have the facts that the BG canonical partition function of these classical systems anomalously diverges with size for 0 ≤ α/d ≤ 1 (longrange interactions, e.g., gravitational and dipolemonopole interactions) and converges for α/d > 1 (shortrange interactions, e.g., LennardJones interaction), and the internal energy per particle is, in the thermodynamical limit, constant for shortrange interactions whereas it diverges like N^{1−α/d} for longrange interactions, N being the total number of particles.
If all these meaningful scalings are put together, we obtain a highly plausible scenario for the respective domains of validity of the BoltzmannGibbs (additive) entropy and associated statistical mechanics, and that of the nonadditive entropies S_{q} (with q ≠ 1) and associated statistical mechanics.
Finally, we notice in Fig. 6 that both q and κ approach quickly their BG limits (q = 1) for α_{A}/d → ∞. Moreover, the same exponential e^{1−α/d} appears in both heuristic expressions for q and κ. Consequently, the following linear relation can be straightforwardly established:
In fact, this simple relation is numerically quite well satisfied as can be seen in Fig. 7. Its existence reveals an interesting peculiarity of the nature of qstatistics. If in the celebrated BG factor e^{−energy/kT}, corresponding to q = 1, we are free to consider an arbitrary value for T, how come in the present problem, κ is not a free parameter but has instead a fixed value for each specific model that we are focusing on? This is precisely what occurs in the highenergy applications of qstatistics, e.g., in quarkgluon soup^{46} where q = 1.114 and T = 135.2 Mev, as well as in all the LHC/CERN and RHIC/Brookhaven experiments^{7}. Another example which is reminiscent of this type of behavior is the sensitivity to the initial conditions at the edge of chaos (Feigenbaum point) of the logistic map; indeed, the inverse qgeneralized Lyapunov exponent satisfies the linear relation 1/λ_{q} = 1 − q^{47,48}. The cause of this interesting and ubiquitous feature comes from the fact that qstatistics typically emerges at criticallike regimes and is deeply related to an hierarchical occupation of phase space (or Hilbert space or Fock space), which in turn points towards asymptotic powerlaws (see also^{49}). In other words, κ plays a role analogous to a critical temperature, which is of course not a free parameter but is instead fixed by the specific model.
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How to cite this article: Brito, S. et al. Role of dimensionality in complex networks. Sci. Rep. 6, 27992; doi: 10.1038/srep27992 (2016).
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Acknowledgements
We have benefitted from fruitful discussions with D. Bagchi, E.M.F. Curado, F.D. Nobre, P. Rapcan and G. Sicuro. We also appreciate the suggestions of an anonymous referee which helped us to improve this work. We gratefully acknowledge partial financial support from CNPq and Faperj (Brazilian agencies) and from the John Templeton FoundationUSA.
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C.T. conceived the research, analyzed the results, contributed to the manuscript text and revised it. S.B. developed and carried out the numerics, prepared figures, analyzed the results, contributed to the manuscript text and revised it. L.R.d.S. offered constructive suggestions, analyzed the results, contributed to the manuscript text and revised it.
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Brito, S., da Silva, L. & Tsallis, C. Role of dimensionality in complex networks. Sci Rep 6, 27992 (2016). https://doi.org/10.1038/srep27992
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DOI: https://doi.org/10.1038/srep27992
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