Roberto Fernandes Silva Andrade

The network of concepts in written texts

Abstract.Complex network theory is used to investigate the structure of meaningful concepts in written texts of individual authors. Networks have been constructed after a two phase filtering, where words with less meaning contents are eliminated and all remaining words are set to their canonical form, without any number, gender or time flexion. Each sentence in the text is added to the network as a clique. A large number of written texts have been scrutinised, and it is found that texts have small-world as well as scale-free structures. The growth process of these networks has also been investigated, and a universal evolution of network quantifiers have been found among the set of texts written by distinct authors. Further analyses, based on shuffling procedures taken either on the texts or on the constructed networks, provide hints on the role played by the word frequency and sentence length distributions to the network structure.

Ising chain in the generalized Boltzmann-Gibbs statistics

Abstract The partition function and specific heat of the Ising chain are evaluated in a generalization of the Boltzmann-Gibbs statistics. The extensivity and the influence of the ground state energy in the thermodynamical properties are discussed.

Magnetic models on Apollonian networks.

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering a Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices. Numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model. Different choices for the coupling constants between only nearest neighbors along the lattice are taken into account. For both ferromagnetic and antiferromagnetic constants, long-range magnetic ordering is obtained. With exception of a size-dependent effective critical behavior of the correlation length, no evidence of asymptotic criticality was detected.

Modularity map of the network of human cell differentiation

Cell differentiation in multicellular organisms is a complex process whose mechanism can be understood by a reductionist approach, in which the individual processes that control the generation of different cell types are identified. Alternatively, a large-scale approach in search of different organizational features of the growth stages promises to reveal its modular global structure with the goal of discovering previously unknown relations between cell types. Here, we sort and analyze a large set of scattered data to construct the network of human cell differentiation (NHCD) based on cell types (nodes) and differentiation steps (links) from the fertilized egg to a developed human. We discover a dynamical law of critical branching that reveals a self-similar regularity in the modular organization of the network, and allows us to observe the network at different scales. The emerging picture clearly identifies clusters of cell types following a hierarchical organization, ranging from sub-modules to super-modules of specialized tissues and organs on varying scales. This discovery will allow one to treat the development of a particular cell function in the context of the complex network of human development as a whole. Our results point to an integrated large-scale view of the network of cell types systematically revealing ties between previously unrelated domains in organ functions.

An Abelian model for rainfall

Statistical analyses of long-term records of daily rain suggest that rain phenomena might be a manifestation of self-organized criticality. In this work the essential mechanisms of rain phenomena, the growth of droplets inside a cloud and the subsequent rainfall, are described by an Abelian sandpile model of self-organized criticality. Several simulations support the existence of scale invariance. The introduction of variations of the basic model, to provide a better description of the phenomena, does not alter the critical behavior.

Ising model on the Apollonian network with node-dependent interactions.

This work considers an Ising model on the Apollonian network, where the exchange constant J(i,j) approximately 1/(k(i)k(j))(mu) between two neighboring spins (i,j) is a function of the degree k of both spins. Using the exact geometrical construction rule for the network, the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit. The results can be compared to the predictions of a general framework for spin models on scale-free networks, where the node distribution P(k) approximately k(-gamma) , with node-dependent interacting constants. We observe that, by increasing mu , the critical behavior of the model changes from a phase transition at T=infinity for a uniform system (mu=0) to a T=0 phase transition when mu=1 : in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole mu > or = 0 interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.

What are the best concentric descriptors for complex networks

This work reviews several concentric measurements of the topology of complex networks and then applies feature selection concepts and methods in order to quantify the relative importance of each measurement with respect to the discrimination between four representative theoretical network models, namely Erdos–Renyi, Barabasi–Albert, Watts–Strogatz, as well as a geographical type of network. Progressive randomizations of the geographical model have also been considered. The obtained results confirmed that the four models can be well-separated by using a combination of measurements. In addition, the relative contribution of each considered feature for the overall discrimination of the models was quantified in terms of the respective weights in the canonical projection into two-dimensions, with the traditional clustering coefficient, concentric clustering coefficient and neighborhood clustering coefficient being particularly effective. Interestingly, the average shortest path length and concentric node degrees contributed little for the separation of the four network models.

Neighborhood properties of complex networks.

A concept of neighborhood in complex networks is addressed based on the criterion of the minimal number of steps to reach other vertices. This amounts to, starting from a given network R1, generating a family of networks Rl, l = 2, 3,... such that, the vertices that are l steps apart in the original R1, are only 1 step apart in Rl. The higher order networks are generated using Boolean operations among the adjacency matrices Ml that represent Rl. The families originated by the well known linear and the Erdös-Renyi networks are found to be invariant, in the sense that the spectra of Ml are the same, up to finite size effects. A further family originated from small world network is identified.

Mandala Networks: ultra-small-world and highly sparse graphs

The increasing demands in security and reliability of infrastructures call for the optimal design of their embedded complex networks topologies. The following question then arises: what is the optimal layout to fulfill best all the demands? Here we present a general solution for this problem with scale-free networks, like the Internet and airline networks. Precisely, we disclose a way to systematically construct networks which are robust against random failures. Furthermore, as the size of the network increases, its shortest path becomes asymptotically invariant and the density of links goes to zero, making it ultra-small world and highly sparse, respectively. The first property is ideal for communication and navigation purposes, while the second is interesting economically. Finally, we show that some simple changes on the original network formulation can lead to an improved topology against malicious attacks.

Mother wavelet functions generalized through q-exponentials

We generalize some widely used mother wavelets by means of the q-exponential function exq ? [1 + (1 ? q)x]1/(1?q) that emerges from nonextensive statistical mechanics. In particular, we define extended versions of the Mexican hat and the Morlet wavelets. We also introduce new wavelets that are q-generalizations of the trigonometric functions. All cases reduce to the usual ones as q ? 1. Within nonextensive statistical mechanics, departures from unity of the entropic index q are expected in the presence of long-range interactions, long-term memory, multi-fractal structures, among others. Consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q. We exemplify with the wavelet transform modulus-maxima method for mono- and multi-fractal self-affine signals.