Diffusion Processes on Small-World Networks with Distance-Dependent Random-Links

We considered diffusion-driven processes on small-world networks with distance-dependent random links. The study of diffusion on such networks is motivated by transport on randomly folded polymer chains, synchronization problems in task-completion networks, and gradient driven transport on networks. Changing the parameters of the distance-dependence, we found a rich phase diagram, with different transient and recurrent phases in the context of random walks on networks. We performed the calculations in two limiting cases: in the annealed case, where the rearrangement of the random links is fast, and in the quenched case, where the link rearrangement is slow compared to the motion of the random walker or the surface. It has been well-established that in a large class of interacting systems, adding an arbitrarily small density of, possibly long-range, quenched random links to a regular lattice interaction topology, will give rise to mean-field (or annealed) like behavior. In some cases, however, mean-field scaling breaks down, such as in diffusion or in the Edwards-Wilkinson process in "low-dimensional" small-world networks. This break-down can be understood by treating the random links perturbatively, where the mean-field (or annealed) prediction appears as the lowest-order term of a naive perturbation expansion. The asymptotic analytic results are also confirmed numerically by employing exact numerical diagonalization of the network Laplacian. Further, we construct a finite-size scaling framework for the relevant observables, capturing the cross-over behaviors in finite networks. This work provides a detailed account of the self-consistent-perturbative and renormalization approaches briefly introduced in two earlier short reports.Comment: 36 pages, 27 figures. Minor revisions in response to the referee's comments. Furthermore, some typos were fixed and new references were adde

Stellar-Evolution Limits on Axion Properties

If axions exist, they are copiously produced in hot and dense plasmas, carrying away energy directly from the interior of stars. Various astronomical observables constrain the operation of such anomalous stellar energy-loss channels and thus provide restrictive limits on the axion interactions with photons, nucleons, and electrons. In typical axion models a limit m_a < 0.01 eV is implied. The main arguments leading to this result are explained, including more recent work on the important supernova 1987A constraint.Comment: 11 pages, 12 eps figs, to be published in Proc. AXION WORKSHOP, Gainesville, Florida, 13-15 March 1998, ed. by P.Sikivi

Energy-based comparison between the Fourier--Galerkin method and the finite element method

The Fourier-Galerkin method (in short FFTH) has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom. Because the method incorporates the fast Fourier transform (FFT) in the linear solver, it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method (FEM). Here, we systematically compare these two methods using the energetic norm of local fields, which has the clear physical interpretation as being the error in the homogenised properties. This enables the comparison of memory and computational requirements at the same level of approximation accuracy. We show that the methods' effectiveness relies on the smoothness (regularity) of the solution and thus on the material coefficients. Thanks to its approximation properties, FEM outperforms FFTH for problems with jumps in material coefficients, while ambivalent results are observed for the case that the material coefficients vary continuously in space. FFTH profits from a good conditioning of the linear system, independent of the number of degrees of freedom, but generally needs more degrees of freedom to reach the same approximation accuracy. More studies are needed for other FFT-based schemes, non-linear problems, and dual problems (which require special treatment in FEM but not in FFTH).Comment: 24 pages, 10 figures, 2 table

A Review of Graph Neural Networks and Their Applications in Power Systems

Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data in these tasks is typically represented in Euclidean domains. Nevertheless, there is an increasing number of applications in power systems, where data are collected from non-Euclidean domains and represented as graph-structured data with high dimensional features and interdependency among nodes. The complexity of graph-structured data has brought significant challenges to the existing deep neural networks defined in Euclidean domains. Recently, many publications generalizing deep neural networks for graph-structured data in power systems have emerged. In this paper, a comprehensive overview of graph neural networks (GNNs) in power systems is proposed. Specifically, several classical paradigms of GNNs structures (e.g., graph convolutional networks) are summarized, and key applications in power systems, such as fault scenario application, time series prediction, power flow calculation, and data generation are reviewed in detail. Furthermore, main issues and some research trends about the applications of GNNs in power systems are discussed

Non-equilibrium diagrammatic approach to strongly interacting photons

We develop a non-equilibrium field-theoretical approach based on a systematic diagrammatic expansion for strongly interacting photons in optically dense atomic media. We consider the case where the characteristic photon-propagation range $L_P$ is much larger than the interatomic spacing $a$ and where the density of atomic excitations is low enough to neglect saturation effects. In the highly polarizable medium the photons experience nonlinearities through the interactions they inherit from the atoms. If the atom-atom interaction range $L_E$ is also large compared to $a$, we show that the subclass of diagrams describing scattering processes with momentum transfer between photons is suppressed by a factor $a/L_E$. We are then able to perform a self-consistent resummation of a specific (Hartree-like) diagram subclass and obtain quantitative results in the highly non-perturbative regime of large single-atom cooperativity. Here we find important, conceptually new collective phenomena emerging due to the dissipative nature of the interactions, which even give rise to novel phase transitions. The robustness of these is investigated by inclusion of the leading corrections in $a/L_E$. We consider specific applications to photons propagating under EIT conditions along waveguides near atomic arrays as well as within Rydberg ensembles.Comment: 72 pages, 36 figure

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group