Visibility graph analysis of wind speed records measured in central Argentina

Hourly means of wind speed time series recorded at two wind stations in central Argentina (one inland and the other coastal) are analyzed by means of the visibility graph method. The degree distribution of both series was calculated along with that of their shuffled, Gaussian- and uniform-distributed surrogates. The original series as well as their surrogates show an apparent exponential degree distribution. The v$v$–k$k$ (value–degree) plot of the original wind series indicates that the higher values of the series are not necessarily “hubs” for the series, while that of the surrogates are characterized by a reduced connectivity degree of the higher hubs.     

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.     

Particle diagrams and statistics of many-body random potentials

We present a method using Feynman-like diagrams to calculate the statistical properties of random many-body potentials. This method provides a promising alternative to existing techniques typically applied to this class of problems, such as the method of supersymmetry and the eigenvector expansion technique pioneered in Benet et al. (2001). We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with m$m$ bosons or fermions that interact through a random k$k$-body Hermitian potential (k≤m$k\le m$); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble   (eGUE) (Mon and French, 1975). Our results apply in the limit where the number l$l$ of available single-particle states is taken to infinity. A key advantage of the method is that it provides an efficient way to identify only those expressions which will stay relevant in this limit. It also provides a general argument for why these terms have to be the same for bosons and fermions. The moments are obtained as sums over ratios of binomial expressions, with a transition from moments associated to a semi-circular level density for m<2k$m<2k$ to Gaussian moments in the dilute limit k?m?l$k?m?l$. Regarding the form of this transition, we see that as m$m$ is increased, more and more diagrams become relevant, with new contributions starting from each of the points m=2k,3k,…,nk$m=2k,3k,\dots ,nk$ for the 2n$2n$th moment.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Ranking the spreading influence in complex networks

Identifying the node spreading influence in networks is an important task to optimally use the network structure and ensure the more efficient spreading in information. In this paper, by taking into account the shortest distance between a target node and the node set with the highest k$k$-core value, we present an improved method to generate the ranking list to evaluate the node spreading influence. Comparing with the epidemic process results for four real networks and the Barabási–Albert network, the parameterless method could identify the node spreading influence more accurately than the ones generated by the degree k$k$, closeness centrality, k$k$-shell and mixed degree decomposition methods. This work would be helpful for deeply understanding the node importance of a network.     

Kinetics of node splitting in evolving complex networks

We introduce a collection of complex networks generated by a combination of preferential attachment and a previously unexamined process of “splitting” nodes of degree k$k$ into k$k$ nodes of degree 1$1$. Four networks are considered, each evolves at each time step by either preferential attachment, with probability p$p$, or splitting with probability 1−p$1-p$. Two methods of attachment are considered; first, attachment of an edge between a newly created node and an existing node in the network, and secondly by attachment of an edge between two existing nodes. Splitting is also considered in two separate ways; first by selecting each node with equal probability and secondly, selecting the node with probability proportional to its degree. Exact solutions for the degree distributions are found and scale-free structure is exhibited in those networks where the candidates for splitting are chosen with uniform probability, those that are chosen preferentially are distributed with a power law with exponential cut-off.     

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied — in particular, the number of nodal domains, ν$\nu$ of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrödinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, ν$\nu$ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of mmodkn$mmodkn$, given a particular k$k$, for a set of quantum numbers, m,n$m,n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.     

Percolation processes in monomer-polyatomic mixtures

In this paper, the percolation of mixtures of monomers and polyatomic species (k$k$-mers) on a square lattice is studied. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattices are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in order to predict its evolution for larger k$k$-mer sizes.     

A Monte Carlo model for networks between professionals and society

We propose a network model with a fixed number of nodes and links and with a dynamic which favors links between nodes differing in connectivity. We observe a phase transition and parameter regimes with degree distributions following power laws, P(k)∼k^-γ$P(k)\sim k^{-\gamma }$, with γ$\gamma$ ranging from 0.2$0.2$ to 0.5$0.5$, small-world properties, with a network diameter following D(N)∼logN$D(N)\sim \log N$ and relative high clustering, following C(N)∼1/N$C(N)\sim 1/N$ and C(k)∼k^-α$C(k)\sim k^{-\alpha }$, with α$\alpha$ close to 3. We compare our results with data from real-world protein interaction networks.     

A generalized Beraha conjecture for non-planar graphs

We study the partition function _ZG(nk,k)(Q,v)$Z_{G(nk,k)}(Q,v)$ of the Q  -state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k)$G(nk,k)$. We study its zeros in the plane (Q,v)$(Q,v)$ for 1?k?7$1?k?7$. We also consider two specializations of _ZG(nk,k)$Z_{G(nk,k)}$, namely the chromatic polynomial _PG(nk,k)(Q)$P_{G(nk,k)}(Q)$ (corresponding to v=−1$v=-1$), and the flow polynomial _ΦG(nk,k)(Q)${\Phi }_{G(nk,k)}(Q)$ (corresponding to v=−Q$v=-Q$). In these two cases, we study their zeros in the complex Q  -plane for 1?k?7$1?k?7$. We pay special attention to the accumulation loci of the corresponding zeros when n→∞$n\to \infty$. We observe that the Berker–Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.     

Eigenvalue amplitudes of the Potts model on a torus

We consider the Q-state Potts model in the random-cluster formulation, defined on finite   two-dimensional lattices of size L×N$L\times N$ with toroidal boundary conditions. Due to the non-locality of the clusters, the partition function Z(L,N)$Z(L,N)$ cannot be written simply as a trace of the transfer matrix TL${\mathrm{T}}_L$. Using a combinatorial method, we establish the decomposition Z(L,N)=_∑l,Dkb(l,Dk)K_l,Dk$Z(L,N)={\sum }_{l,D_k}{b}^{(l,D_k)}{K}_{l,D_k}$, where the characters K_l,Dk=i_∑N(λi)$K_{l,D_k}={\sum }_i{({\lambda }_i)}^N$ are simple traces. In this decomposition, the amplitudes b(l,Dk)$b^{(l,D_k)}$ of the eigenvalues λi${\lambda }_i$ of TL${\mathrm{T}}_L$ are labelled by the number l=0,1,…,L$l=0,1,\dots ,L$ of clusters which are non-contractible with respect to the transfer (N  ) direction, and a representation Dk$D_k$ of the cyclic group Cl$C_l$. We obtain rigorously a general expression for b(l,Dk)$b^{(l,D_k)}$ in terms of the characters of Cl$C_l$, and, using number theoretic results, show that it coincides with an expression previously obtained in the continuum limit by Read and Saleur.     

Time-varying human mobility patterns with metapopulation epidemic dynamics

In this paper, explicitly considering the influences of an epidemic outbreak on human travel, a time-varying human mobility pattern is introduced to model the time variation of global human travel. The impacts of the pattern on epidemic dynamics in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals, are investigated by using a mean-field approach. The results show that the pattern does not alter the epidemic threshold, but can slightly lower the final average density of infected individuals as a whole. More importantly, we also find that the pattern produces different impacts on nodes with different degree, and that there exists a critical degree _kc$k_c$. For nodes with degree smaller than _kc$k_c$, the pattern produces a positive impact on epidemic mitigation; conversely, for nodes with degree larger than _kc$k_c$, the pattern produces a negative impact on epidemic mitigation.