Entropy Production in a Fractal System with Diffusive Dynamics.

We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.

Scaling properties of d-dimensional complex networks.

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 the scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r_{ij}^{-α_{A}} (α_{A}≥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 α_{A}. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable α_{A}/d. These observations confirm the exist- ence of three regimes. The first one occurs in the interval α_{A}/d∈[0,1]; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a q exponential with q constant and above unity. The critical value α_{A}/d=1 that emerges in many of these properties is replaced by α_{A}/d=1/2 for the ß 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 α_{A}/d increasing from its critical value to a characteristic value α_{A}/d≃5. Finally, the third regime is Boltzmannian-like (with q≃1) and corresponds to short-range interactions.

Spreading gossip in social networks.

We study a simple model of information propagation in social networks, where two quantities are introduced: the spread factor, which measures the average maximal reachability of the neighbors of a given node that interchange information among each other, and the spreading time needed for the information to reach such a fraction of nodes. When the information refers to a particular node at which both quantities are measured, the model can be taken as a model for gossip propagation. In this context, we apply the model to real empirical networks of social acquaintances and compare the underlying spreading dynamics with different types of scale-free and small-world networks. We find that the number of friendship connections strongly influences the probability of being gossiped. Finally, we discuss how the spread factor is able to be applied to other situations.

Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs.

We introduce a new family of networks, the Apollonian networks, that are simultaneously scale-free, small-world, Euclidean, space filling, and with matching graphs. These networks describe force chains in polydisperse granular packings and could also be applied to the geometry of fully fragmented porous media, hierarchical road systems, and area-covering electrical supply networks. Some of the properties of these networks, namely, the connectivity exponent, the clustering coefficient, and the shortest path are calculated and found to be particularly rich. The percolation, the electrical conduction, and the Ising models on such networks are also studied and found to be quite peculiar. Consequences for applications are also discussed.