Role of dimensionality in preferential attachment growth in the Bianconi–Barabási model

Abstract

Scale-free networks are quite popular nowadays since many systems are well represented by such structures. In order to study these systems, several models were proposed. However, most of them do not take into account the nodeto-node Euclidean distance, i.e. the geographical distance. In real networks, the distance between sites can be very relevant, e.g. those cases where it is intended to minimize costs. Within this scenario we studied the role of dimensionality d in the Bianconi-Barabási model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the, where η i characterizes the fitness of the ith site and is randomly chosen within the (0, 1] interval. We verified that the degree distribution P (k) for dimensions d = 1, 2, 3, 4 are well fitted by P (k) ∝ e −k/κ q, where e −k/κ q is the q-exponential function naturally emerging

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

“…The present numerical analysis is consistent with the theoretical prevision of a change of behavior at , in agreement with recent results obtained in other long-range-interaction models, such as a generalized Fermi-Pasta-Ulam model [ 35 , 36 , 37 , 38 , 39 ], complex networks with preferential attachment growth [ 40 , 41 ], and a system of particles under overdamped motion interacting repulsively [ 42 ]. In future works we intend to cover the entire region in order to clarify further this point, and particularly, to investigate why some of the Heisenberg results found herein sensibly differ from those available in the literature for the XY and Fermi-Pasta-Ulam models.…”

“…The same departure from the BG predictions is observed for the time-averaged energy distributions; in this case, instead of the usual BG exponential, q -exponential probability distributions are obtained, with [ 34 ]. Similar universality features have been recently obtained in quite different contexts, such as a generalized Fermi-Pasta-Ulam model with long-range interactions [ 35 , 36 , 37 , 38 , 39 ], complex networks with preferential attachment growth [ 40 , 41 ], and a system of particles under overdamped motion interacting repulsively with power-law interactions [ 42 ].…”

Validity and failure of the Boltzmann weight

“…and the q-Gaussian exp q [−βx 2 ] appear naturally by extremizing S q under appropriate constraints [27][28][29]. The generalized thermostatistics based on S q frequently applies when assumptions underlying the BG thermostatistics are not fulfilled (like, e.g., mixing and ergodicity) [14,15,[18][19][20][29][30][31][32][33][34].…”

Analytic approaches of the anomalous diffusion: A review

“…We show that these equations gain a foundation when they are involved with the proposal of non-extensive statistical mechanics [31]. Nowadays, the Tsallis statistic [31] and nonlinear diffusion have assumed important roles in the application of more subtle problems in thermodynamics, such as black holes [21,143], generalised forms of H-theorem [19,20,144], financial market [145,146], and many other systems [94,147,148]. Here, it is worth mentioning to the reader that from the point of view of the H theorem [11] the porous media equations may imply the Tsallis entropy.…”