The Tsallis–Laplace transform

We introduce here the q$q$-Laplace transform as a new weapon in Tsallis’ arsenal, discussing its main properties and analyzing some examples. The q$q$-Gaussian instance receives special consideration. Also, we derive the q$q$-partition function from the q$q$-Laplace transform.     

Some properties of generalized Fisher information in the context of nonextensive thermostatistics

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized q$q$-Gaussian distributions and the q$q$-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed q$q$-entropy is attained, in both cases, by a generalized q$q$-Gaussian distribution. This complements the fact that the q$q$-Gaussians maximize the q$q$-entropies subject to a moment constraint, and yields new variational characterizations of the generalizedq$q$-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the q$q$-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for q≥1$q\ge 1$. Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.     

Skeleton of weighted social network

In the literature of social networks, understanding topological structure is an important scientific issue. In this paper, we construct a network from mobile phone call records and use the cumulative number of calls as a measure of the weight of a social tie. We extract skeletons from the weighted social network on the basis of the weights of ties, and we study their properties. We find that strong ties can support the skeleton in the network by studying the percolation characters. We explore the centrality of w$w$-skeletons based on the correlation between some centrality measures and the skeleton index w$w$ of a vertex, and we find that the average centrality of a w$w$-skeleton increases as w$w$ increases. We also study the cumulative degree distribution of the successive w$w$-skeletons and find that as w$w$ increases, the w$w$-skeleton tends to become more self-similar. Furthermore, fractal characteristics appear in higher w$w$-skeletons. We also explore the global information diffusion efficiency of w$w$-skeletons using simulations, from which we can see that the ties in the high w$w$-skeletons play important roles in information diffusion. Identifying such a simple structure of a w$w$-skeleton is a step forward toward understanding and representing the topological structure of weighted social networks.     

Dynamic allocation of investments promotes cooperation in spatial public goods game

In this paper, we introduce a dynamic allocation investment mechanism into the spatial public goods game. In the model, if a cooperator i$i$ collects more payoffs from one group centered on the player j$j$ in the last round, he will put more investments to the group in the next round. The intensity of investment is controlled by a tunable parameter α$\alpha$. Simulation results show that the emergence of cooperation is remarkably promoted, and a larger α$\alpha$ induces a higher cooperation level. These interesting phenomena are then extensively explained by pairwise investments, average payoffs and evolution snapshots.     

Nonextensivity measure for earthquake networks

Studying earthquakes and the associated geodynamic processes based on the complex network theory enables us to learn about the universal features of the earthquake phenomenon. In addition, we can determine new indices for identification of regions geophysically. It was found that earthquake networks are scale free and its degree distribution obeys the power law. Here we claim that the q$q$-exponential function is better than power law model for fitting the degree distribution. We also study the behavior of q$q$ parameter (nonextensivity measure) with respect to resolution. It was previously asserted in Eur. Phys. J. B (2012) 85: 23; that the topological characteristics of earthquake networks are dependent on each other for large values of the resolution. A peak in the plot of q$q$ against resolution determines the beginning of the assertion range.     

Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M$M$ induced by complex structures on the generalized tangent bundle of M$M$.     

Complex scale-free networks with tunable power-law exponent and clustering

We introduce a network evolution process motivated by the network of citations in the scientific literature. In each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network. This set includes m$m$ “ambassador” nodes and l$l$ of each ambassador’s descendants where m$m$ and l$l$ are random variables selected from any choice of distributions _pl$p_l$ and _qm$q_m$. The process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite. We show that the degree distributions of the networks generated after a large number of iterations are scale-free and derive an expression for the power-law exponent. In a particular case of the model where the number of ambassadors is always the constant m$m$ and the number of selected descendants from each ambassador is the constant l$l$, the power-law exponent is (2l+1)/l$(2l+1)/l$. For this example we derive expressions for the degree distribution and clustering coefficient in terms of l$l$ and m$m$. We conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale-free distributions that have been studied empirically.     

Evolution of public cooperation with weighted and conditional strategies

In real systems, generous individuals who prefer to contribute can play more important roles on the public than selfish individuals. Inspired by this, a spatial public goods game with weighted and conditional strategies is proposed in this paper. In our model, a player’s contribution behavior is mainly determined by the less cautious neighbors rather than those more cautious ones. The weight effect is adjusted by a non-negative parameter α$\alpha$. Our results show that the number of strategies increases with the weight parameter α$\alpha$ for the low multiplication factor r$r$. On the contrary, for the high multiplication factor r$r$, the less cautious cooperators are more likely to appear in the system.     

Tsallis’ maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schrödinger equation

Tsallis maximum entropy distributions provide useful tools for the study of a wide range of scenarios in mathematics, physics, and other fields. Here we apply a Tsallis maximum entropy ansatz, the q$q$-Gaussian, to obtain time dependent wave-packet solutions to a nonlinear Schrödinger equation recently advanced by Nobre, Rego-Monteiro and Tsallis (NRT) [F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106 (2011) 140601]. The NRT nonlinear equation admits plane wave-like solutions (q$q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the q$q$-generalized thermostatistical formalism, is characterized by a parameter q$q$ and in the limit q→1$q\to 1$ reduces to the standard, linear Schrödinger equation. The q$q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known q$q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the q→1$q\to 1$ limit the q$q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrödinger equation. In the present work we also show that there are other families of nonlinear Schrödinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.     

Exploring associations between micro-level models of innovation diffusion and emerging macro-level adoption patterns

A micro-level agent-based model of innovation diffusion was developed that explicitly combines (a) an individual’s perception of the advantages or relative utility derived from adoption, and (b) social influence from members of the individual’s social network. The micro-model was used to simulate macro-level diffusion patterns emerging from different configurations of micro-model parameters. Micro-level simulation results matched very closely the adoption patterns predicted by the widely-used Bass macro-level model (Bass, 1969 [1]). For a portion of the p−q$p-q$ domain, results from micro-simulations were consistent with aggregate-level adoption patterns reported in the literature. Induced Bass macro-level parameters p$p$ and q$q$ responded to changes in micro-parameters: (1) p$p$ increased with the number of innovators and with the rate at which innovators are introduced; (2) q$q$ increased with the probability of rewiring in small-world networks, as the characteristic path length decreases; and (3) an increase in the overall perceived utility of an innovation caused a corresponding increase in induced p$p$ and q$q$ values. Understanding micro to macro linkages can inform the design and assessment of marketing interventions on micro-variables–or processes related to them–to enhance adoption of future products or technologies.     

Opinion evolution on a BA scaling network

In this paper, the dynamics of opinion formation is investigated based on a BA (Barabási–Albert) scale-free network, using a majority–minority rule governed by parameter q$q$. As the value of q$q$ is smoothly varied, a phase transition occurs between an ordered phase and a disordered one. By performing extensive Monte Carlo simulations, we show that the phase transition is dependent on the system size, as well as on m$m$, the number of edges added at each time step during the growth of the BA scaling network. Additionally, some theoretical analysis is given based on mean-field theory, by neglecting fluctuations and correlations. It is observed that the theoretical results coincide with results from simulations, especially for very large m$m$.     

Generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy

We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order q$q$, we first provide a generalized monogamy inequality of multi-qubit entanglement for q=2$q=2$ or 3$3$. This generalization encapsulates the multi-qubit CKW-type inequality as a special case. We further provide a generalized polygamy inequality of multi-qubit entanglement in terms of Tsallis-q$q$ entropy for 1≤q≤2$1\le q\le 2$ or 3≤q≤4$3\le q\le 4$, which also contains the multi-qubit polygamy inequality as a special case.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

The nonextensive parameter for nonequilibrium electron gas in an electromagnetic field

The nonextensive parameter for nonequilibrium electron gas of the plasma in an electromagnetic field is studied. We exactly obtained an expression of the q$q$-parameter based on Boltzmann kinetic theories for plasmas, where Coulombian interactions and Lorentz forces play dominant roles. We show that the q$q$-parameter different from unity is related by an equation to temperature gradient, electric field strength, magnetic induction as well as overall bulk velocity of the gas. The effect of the magnetic field on the q$q$-parameter depends on the overall bulk velocity. Thus the q$q$-parameter for the electron gas in an electromagnetic field represents the nonequilibrium nature or nonisothermal configurations of the plasma with electromagnetic interactions.     

Implication of Tsallis entropy in the Thomas–Fermi model for self-gravitating fermions

The Thomas–Fermi approach for self-gravitating fermions is revisited within the theoretical framework of the q$q$-statistics  . Starting from the q$q$-deformation of the Fermi–Dirac distribution function, a generalized Thomas–Fermi equation is derived. It is shown that the Tsallis entropy   preserves a scaling property of this equation. The q$q$-statistical   approach to Jeans’ instability in a system of self-gravitating fermions is also addressed. The dependence of the Jeans’ wavenumber (or the Jeans length) on the parameter q$q$ is traced. It is found that the q$q$-statistics makes the Fermionic system unstable at scales shorter than the standard Jeans length.     

An efficient community detection method based on rank centrality

Community detection is a very important problem in social network analysis. Classical clustering approach, K$K$-means, has been shown to be very efficient to detect communities in networks. However, K$K$-means is quite sensitive to the initial centroids or seeds, especially when it is used to detect communities. To solve this problem, in this study, we propose an efficient algorithm K$K$-rank, which selects the top-K$K$ nodes with the highest rank centrality as the initial seeds, and updates these seeds by using an iterative technique like K$K$-means. Then we extend K$K$-rank to partition directed, weighted networks, and to detect overlapping communities. The empirical study on synthetic and real networks show that K$K$-rank is robust and better than the state-of-the-art algorithms including K$K$-means, BGLL, LPA, infomap and OSLOM.     

Nonextensivity and entropy of astrophysical sources

We study the X-ray intensities of 142 light curves of cataclysmic variables, galaxies, pulsars, supernova remnants and other X-ray sources present in the public data collected by the instrument All Sky Monitor on board the satellite Rossi X-ray Timing Explorer. We show that the X-ray light curves coming from astrophysical systems obey Tsallis’s q$q$-Gaussian distribution as probability density. This fact strongly suggests that these astrophysical systems behave in a non-extensive manner. Furthermore, the q$q$ entropic indices for these systems were obtained and they provide an indication of the nonextensivity degree of each of these astrophysical systems. The q$q$-value increases for systems if the Tsallis entropy decreases.     

Lobby index as a network centrality measure

We study the lobby index (l$l$-index for short) as a local node centrality measure for complex networks. The l$l$-index is compared with degree (a local measure), betweenness and Eigenvector centralities (two global measures) in the case of a biological network (Yeast interaction protein–protein network) and a linguistic network (Moby Thesaurus   II). In both networks, the l$l$-index has a poor correlation with betweenness but correlates with degree and Eigenvector centralities. Although being local, the l$l$-index carries more information about its neighbors than degree centrality. Also, it requires much less time to compute when compared with Eigenvector centrality. Results show that the l$l$-index produces better results than degree and Eigenvector centrality for ranking purposes.     

If Gauss–Bonnet interaction plays the role of dark energy

A cosmological model has been constructed with Gauss–Bonnet-scalar interaction, where the Universe starts with exponential expansion but encounters infinite deceleration, q→∞$q\to \infty$ and infinite equation of state parameter, w→∞$w\to \infty$. During evolution it subsequently passes through the stiff fluid era, q=2$q=2$, w=1$w=1$, the radiation dominated era, q=1$q=1$, w=1/3$w=1/3$ and the matter dominated era, q=1/2$q=1/2$, w=0$w=0$. Finally, deceleration halts, q=0$q=0$, w=−1/3$w=-1/3$, and it then encounters a transition to the accelerating phase. Asymptotically the Universe reaches yet another inflationary phase q→−1$q\to -1$, w→−1$w\to -1$. Such evolution is independent of the form of the potential and the sign of the kinetic energy term, i.e., even a non-canonical kinetic energy is unable to phantomize (w<−1)$(w<-1)$ the model.