Role of dimensionality in complex networks

Abstract

Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form , where the q-exponential form optimizes the nonadditive entropy Sq (which, for q → 1, recovers the Boltzmann-Gibbs entropy). We introduce and study here d-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through . Revealing the connection with q-statistics, we numerically verify (for d = 1, 2, 3 and 4) that the q-exponential degree distributions exhibit, for both q and k, universal dependences on the ratio αA/d. Moreover, the q = 1 limit is rapidly achieved by increasing αA/d to infinity.

Scaling properties of d -dimensional complex networks

“…In 2016, we studied a d-dimensional network model where the interactions are short-or long-ranged depending on the choice of the parameter α A ≥ 0. The results that were obtained reinforced the connection between nonextensive statistical mechanics and the networks theory [13]. In that work, we found some quantities which present a universal behaviour with respect to the particular variable α A /d and observed the existence of three regimes.…”

“…where r ≥ 1 is the Euclidean distance from the newly arrived site to the center of mass of the pre-existing system. For more details see [13]. This network is characterized by three parameters α A , α G and d, where α A controls the importance of the distance in the preferential attachment rule, α G is associated with the geographical distribution of the sites, and d is the dimension of the system.…”

“…We have analyzed the exponent β, which is associated with the time evolution of the connectivity of sites, the average shortest path l , the degree distribution entropy S q and the average clustering coefficient C . Along the lines of [13], in order to analyse these properties, we choose the typical value α G = 2 and vary the parameters (α A , d).…”

Nonlinear Wave Equations Related to Nonextensive Thermostatistics

“…The so-called q-exponential function, arising when considering constrained optimization problems based on the entropic measure S q , plays a central role within the nonextensive thermostatistical formalism [1,2,24]. This function is defined as…”

“…These functions reduce to the standard exponential and Gaussian ones in the q → 1 limit. Just to list a few recent examples of applications of the nonextensive thermostatistical formalism, we can mention applications to self-gravitating systems [23], to the dynamics of vortices in type II superconductors [19], and to complex networks [24]. The remarkably diverse scenarios within which q-exponentials and q-Gaussians appear to be relevant suggests that the dynamical mechanisms giving rise to these distributions are not unique.…”