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.     

Vortex motion on surfaces of small curvature

We consider a single Abelian Higgs vortex on a surface Σ$\Sigma$ whose Gaussian curvature K$K$ is small relative to the size of the vortex, and analyse vortex motion by using geodesics on the moduli space of static solutions. The moduli space is Σ$\Sigma$ with a modified metric, and we propose that this metric has a universal expansion, in terms of K$K$ and its derivatives, around the initial metric on Σ$\Sigma$. Using an integral expression for the Kähler potential on the moduli space, we calculate the leading coefficients of this expansion numerically, and find some evidence for their universality. The expansion agrees to first order with the metric resulting from the Ricci flow starting from the initial metric on Σ$\Sigma$, but differs at higher order. We compare the vortex motion with the motion of a point particle along geodesics of Σ$\Sigma$. Relative to a particle geodesic, the vortex experiences an additional force, which to leading order is proportional to the gradient of K$K$. This force is analogous to the self-force on bodies of finite size that occurs in gravitational motion.     

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.     

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.     

Effects of uniform rotational flow on predator–prey system

Rotational flow is often observed in lotic ecosystems, such as streams and rivers. For example, when an obstacle interrupts water flowing in a stream, energy dissipation and momentum transfer can result in the formation of rotational flow, or a vortex. In this study, I examined how rotational flow affects a predator–prey system by constructing a spatially explicit lattice model consisting of predators, prey, and plants. A predation relationship existed between the species. The species densities in the model were given as S$S$ (for predator), P$P$ (for prey), and G$G$ (for plant). A predator (prey) had a probability of giving birth to an offspring when it ate prey (plant). When a predator or prey was first introduced, or born, its health state was assigned an initial value of 20 that subsequently decreased by one with every time step. The predator (prey) was removed from the system when the health state decreased to less than zero. The degree of flow rotation was characterized by the variable, R$R$. A higher R$R$ indicates a higher tendency that predators and prey move along circular paths. Plants were not affected by the flow because they were assumed to be attached to the streambed. Results showed that R$R$ positively affected both predator and prey survival, while its effect on plants was negligible. Flow rotation facilitated disturbances in individuals’ movements, which consequently strengthens the predator and prey relationship and prevents death from starvation. An increase in S$S$ accelerated the extinction of predators and prey.     

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.     

Hermitian–Einstein connections on polystable orthogonal and symplectic parabolic Higgs bundles

Let X$X$ be a smooth complex projective curve and S⊂X$S\subset X$ a finite subset. We show that an orthogonal or symplectic parabolic Higgs bundle on X$X$ with parabolic structure over S$S$ admits a Hermitian–Einstein connection if and only if it is polystable.     

Post-breakthrough scaling in reservoir field simulation

We study the oil displacement and production behavior in an isothermal thin layered reservoir model subjected to water flooding. We use the CMG’s (Computer Modelling Group  ) numerical simulators to solve mass balance equations. The influences of the viscosity ratio (m≡_μoil/_μwater$m\equiv {\mu }_{oil}/{\mu }_{water}$) and the inter-well (injector-producer) distance r$r$ on the oil production rate C(t)$C\left(t\right)$ and the breakthrough time _tbr$t_{br}$ are investigated. Two types of reservoir configuration are used, namely one with random porosities and another with a percolation cluster structure. We observe that the breakthrough time follows a power-law of m$m$ and r$r$, _tbr∝r^αm^β$t_{br}\propto r^{\alpha }{m}^{\beta }$, with α=1.8$\alpha =1.8$ and β=−0.25$\beta =-0.25$ for the random porosity type, and α=1.0$\alpha =1.0$ and β=−0.2$\beta =-0.2$ for the percolation cluster type. Moreover, our results indicate that the oil production rate is a power law of time. In the percolation cluster type of reservoir, we observe that P(t)∝t^γ$P\left(t\right)\propto t^{\gamma }$, with γ=−1.81$\gamma =-1.81$, where P(t)$P\left(t\right)$ is the time derivative of C(t)$C\left(t\right)$. The curves related to different values of m$m$ and r$r$ may be collapsed suggesting a universal behavior for the oil production rate.     

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.     

A Monte Carlo study of ant traffic in a uni-directional trail

In this study, we aim to investigate ant traffic in the uni-directional ant trail. We consider two types of ants moving in the trail: one of which smells well and the other does not. The theoretical base of the study is similar to that of the Nagel–Schreckenberg (NaSch) model, but we do not use the exclusion rule, the asymmetrical exclusion rule is employed instead. Ants are placed on the trail as mixed. By keeping the number of ‘poor-smelling ants’ constant, the traffic in the trail is studied as a function of the number of “good-smelling” ants and the evaporation rate probability of pheromone f$f$. The fundamental physical quantities, i.e., mean speed V$V$ and flux F$F$, interestingly show non-monotonic density dependence for some values of f$f$ at some densities.     

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

Intertwining operators for Sklyanin algebra and elliptic hypergeometric series

Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins ?$?$ and −?−1$-?-1$ are constructed using the technique of intertwining vectors for elliptic L$L$-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W$W$-operators) serve as building blocks for the elliptic R$R$-matrix which intertwines tensor product of two L$L$-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang–Baxter equation for this R$R$-matrix follows from simpler equations of the star–triangle type for the W$W$-operators. A natural graphic representation of the objects and equations involved in the construction is used.     

The spatial distribution of clusters and the formation of mixed languages in bilingual competition

We use cellular automata simulation methods to study the competition between two languages (language A$A$ and B$B$). We assume each of the two languages consists of F$F$ independent features and define an individual as two F$F$-length “identity level” integer strings. The value of each integer of the strings indicates whether the individual is willing or unwilling to express a certain feature and how his willingness or unwillingness is. In our model, individuals who speak either language A$A$ or B$B$ are randomly placed on a square lattice initially and programmed to evolve under the communication and interaction methods. First, we consider the situation that the competition occurs between two languages with only one feature. We find the differences between short-time coexistence processes and long-time coexistence processes and discuss how the spatial distribution of languages evolves. We observe that periodic line-shaped boundaries take shape in some simulations and lead to long-time coexistence between the two languages. Then we study the multi-feature cases when F=10$F=10$ and find the correlation among the evolution of different features decreases with time. We also observe the extinction of two primitive languages and formation of mixed languages.     

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.     

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P$P$, there are two approaches to its geometric quantization: one involves a circle bundle Q$Q$ over P$P$ endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P$P$. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P$P$ is obtained from the Lie groupoid of Q$Q$ via an S¹$S^1$ reduction that preserves both the Lie groupoid and the geometric structures.     

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

In this paper we revisit the Bialynicki-Birula and Mycielski uncertainty principle and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both Shannon and Rényi cases, and for a given dimension n$n$, the only case of equality occurs for Gaussian random vectors. We show that as n$n$ grows, however, the bound is also asymptotically attained in the cases of n$n$-dimensional Student-t$t$ and Student-r$r$ distributions. A complete analytical study is performed in a special case of a Student-t$t$ distribution. We also show numerically that this effect exists for the particular case of a n$n$-dimensional Cauchy variable, whatever the Rényi entropy considered, extending the results of Abe and illustrating the analytical asymptotic study of the Student-t$t$ case. In the Student-r$r$ case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a “Gaussianization” of the vector when the dimension increases.     

Magnetic field effects in the Casimir force between two parallel anti-ferromagnetic slabs

We study the Casimir force F$F$ between two parallel anti-ferromagnetic slabs taking into account an external magnetic field in the Voigt configuration. Using a frequency and magnetic field dependent magnetic permeability tensor and a frequency independent dielectric permittivity, to describe the slabs, we calculate the Casimir force using non-normal incidence reflectivity of the electromagnetic waves in the free space between the slabs. We determine the Casimir force by performing two-dimensional calculations. F$F$ is investigated as a function of the layer thickness d$d$, the vacuum gap width L$L$ between slabs, and the external magnetic field strength H$H$. Features of F$F$ as function of the external field include the presence of sharp dips and peaks, which appear in the vicinity of the resonance frequency, and are consequences of the interaction of the external magnetic field with the electron spin. In addition, an external field may diminish F$F$, which is an important effect not found in any other system.