Gradient flow-based modularity maximization for community detection in multiplex networks.

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

Multiplex community detection in complex networks using an evolutionary approach

The approximations of the nonlinear heat transport problem are based on the finite volume (FM) and averaging (AM) methods [1,2]. This procedures allows reduce the nonlinear 2‐D problem for partial differential equation (PDE) to a initial‐value problem for a system of 2 nonlinear ordinary differential equations(ODE) of first order in the time t or to a initial‐value problem for one nonlinear ODE of first order with two nonlinear algebraic equations.

Complex networks arise in various fields, such as biology, sociology and communication, to model interactions among entities. Entities in many real-world systems exhibit different types of interactions, which requires modeling these type of systems properly. Multiplex networks are used to model these systems, as they can reflect the nodes’ pair-wise interactions as multiple distinct types of links across layers. Community detection is a widely studied application in network analysis as it provides insights into the structure and organization of the network. Even though multiple algorithms have been developed in the community detection field, many of them have a limited performance in the presence of noise. In this article, we develop a novel algorithm that combines tensor low-rank representation, spectral clustering and distance regularization to improve the accuracy in discovering communities in multiplex networks. The low-rank representation leads to reducing the noise and errors existing in the network and the optimization of an accurate consensus set of eigenvectors that reveals the communities in the network. Moreover, the proposed approach balances the agreement between the eigenvectors of each layer, i.e., individual subspaces, and the consensus set of eigenvectors, i.e., common subspaces, by minimizing the projection distance between them. The common and individual subspaces are computed efficiently through Tucker decomposition and modified spectral clustering, respectively. Finally, multiple experiments are conducted on real and simulated networks to evaluate the proposed approach and compare it to state-of-the-art algorithms. The proposed approach shows its robustness and efficiency in discovering the communities in multiplex networks.

In recent years, community detection has received increasing interest. In network analysis, community detection refers to the identification of tightly connected subsets of nodes, which are called “communities” or “groups”, in the network. Non-negative matrix factorization models are often used to solve the problem. Orthogonal non-negative matrix tri-factorization (ONMTF) exhibits significant potential as an approach for community detection within multiplex networks. This paper explores the application of ONMTF in multiplex networks, aiming to detect both shared and exclusive communities simultaneously. The model decomposes each layer within the multiplex network into two low-rank matrices. One matrix corresponds to shared communities across all layers, and the other to unique communities within each layer. Additionally, graph regularization and the diversity of private communities are taken into account in the algorithm. The Hilbert Schmidt Independence Criterion (HSIC) is used to constrain the independence of private communities. The results prove that ONMTF effectively addresses community detection in multiplex networks. It also offers strong interpretability and feature extraction capabilities. Therefore, it is an advanced method for community detection in multiplex networks.

Non-linear vibration and instability of multi-phase composite plate subjected to non-uniform in-plane parametric excitation: Semi-analytical investigation

Community detection in social networks is a promising research area. However, network exploration with multiple types of relationships remains an underexplored yet crucial domain. To address the challenge of community detection in multiplex networks, we propose SFCSA, a smart flattening-based Cuckoo Search Algorithm, designed to address the complexities of such networks effectively. SFCSA unfolds in three integral steps. Firstly, we employ DeepWalk and node2vec algorithms to extract node embeddings, which are merged to construct the multiplex network node embeddings, minimizing information loss. Secondly, a flattened network is built by merging the network layers and adopting the Pearson correlation coefficient as a weight to capture node similarity. Thirdly, we employ a community detection approach that unveils the community structure through a cuckoo search algorithm applied to the multiplex node embeddings to extract the best set of cluster centres by maximizing the modularity function. Experimental evaluations across synthetic and real-world multiplex networks attest to the efficiency and effectiveness of our approach, outperforming state-of-the-art algorithms.

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

We introduce a partial differential equation whose structure, in many respects, mimics that of the second Painlevé equation. For this equation we give auto-Bäcklund transformations of both ordinary differential equation and partial differential equation types. Our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto-Bäcklund transformations and the integrability of ordinary and partial differential equations.

PurposeDiscrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.Design/methodology/approachThis paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.FindingsAn alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.Research limitations/implicationsThe proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.Practical implicationsThis paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.Originality/valueThe presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

Community detection in network data aims at grouping similar nodes sharing certain characteristics together. Most existing methods focus on detecting communities in undirected networks, where similarity between nodes is measured by their node features and whether they are connected. In this article, we propose a novel method to conduct network embedding and community detection simultaneously in a directed network. The network embedding model introduces two sets of vectors to represent the out- and in-nodes separately, and thus allows the same nodes belong to different out- and in-communities. The community detection formulation equips the negative log-likelihood with a novel regularization term to encourage community structure among the nodes representations, and thus achieves better performance by jointly estimating the nodes embeddings and their community structures. To tackle the resultant optimization task, an efficient alternative updating scheme is developed. More importantly, the asymptotic properties of the proposed method are established in terms of both network embedding and community detection, which are also supported by numerical experiments on some simulated and real examples.

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.

Nonlinear Stabilization in Infinite Dimension

Supervised community detection in multiplex networks based on layers convex flattening and modularity optimization

Magneto-bioconvection flow of Casson nanofluid configured by a rotating disk in the presence of gyrotatic microorganisms and Joule heating