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
The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r −α A ij (αA ≥ 0). We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient, for d = 1, 2, 3, 4, and typical values of αA. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable αA/d. These observations confirm the existence of three regimes. The first one occurs in the interval αA/d ∈ [0, 1]; it is non-Boltzmannian with verylong-range interactions in the sense that the degree distribution is a q-exponential with q constant and above unity. The critical value αA/d = 1 that emerges in many of these properties is replaced by αA/d = 1/2 for the β-exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately long-range interactions, and reflects in an index q monotonically decreasing with αA/d increasing from its critical value to a characteristic value αA/d ≃ 5. Finally, the third regime is Boltzmannian-like (with q ≃ 1), and corresponds to short-range interactions. *
We investigated the impact of Kaniadakis statistics on thermodynamic properties for a square magnetic grid. We used the Ising model. We reported numerical results for a two-dimensional magnetic network in a thermal bath. We calculated the probabilities of transition between states using-statistics, the Metropolis dynamics for stochastic processes of the finite-sized magnetic network, considering that the system interacts with a thermal reservoir. We investigated the behavior of various thermodynamic properties. We observed typical measurements of magnetization, energy and specific heat. Increasing the parameter reduces the critical temperature. We observed by the measurements of the fourth order Binder cumulative of magnetization, that for different network sizes and different values of the parameter , the system transition temperature magnetic decreases as κ increases.
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