Search

Recently, there has been growing interest in deep spectral methods for image localization and segmentation, influenced by traditional spectral segmentation approaches. These methods reframe the image decomposition process as a graph partitioning task by extracting features using self-supervised learning and utilizing the Laplacian of the affinity matrix to obtain eigensegments. However, instance segmentation has received less attention than other tasks within the context of deep spectral methods. This paper addresses that not all channels of the feature map extracted from a self-supervised backbone contain sufficient information for instance segmentation purposes. Some channels are noisy and hinder the accuracy of the task. To overcome this issue, this paper proposes two channel reduction modules, Noise Channel Reduction (NCR) and Deviation-based Channel Reduction (DCR). The NCR retains channels with lower entropy, as they are less likely to be noisy, while DCR prunes channels with low standard deviation, as they lack sufficient information for effective instance segmentation. Furthermore, the paper demonstrates that the dot product, commonly used in deep spectral methods, is not suitable for instance segmentation due to its sensitivity to feature map values, potentially leading to incorrect instance segments. A novel similarity metric called Bray-curtis over Chebyshev (BoC) is proposed to address this issue. This metric considers the distribution of features in addition to their values, providing a more robust similarity measure for instance segmentation. Quantitative and qualitative results on the Youtube-VIS 2019 and OVIS datasets highlight the improvements achieved by the proposed channel reduction methods and using BoC instead of the conventional dot product for creating the affinity matrix. These improvements regarding mean Intersection over Union (mIoU) and extracted instance segments are observed, demonstrating enhanced instance segmentation performance. The code is available on: https://github.com/farnooshar/SpecUnIIS. [ABSTRACT FROM AUTHOR]

This paper delves into three research directions, leveraging the Lovász ϑ-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that n≥14, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on n vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász ϑ-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the ϑ-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the ϑ-function variant by Schrijver does not possess the property of the Lovász ϑ-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory. [ABSTRACT FROM AUTHOR]

With the increased complexity of power systems stemming from the connection of high-proportion renewable energy sources, coupled with the escalating volatility and uncertainty, the key transmission sections that serve as indicators of the power grid's security status are also subject to frequent changes, posing challenges to grid monitoring. The search method for key transmission sections based on an improved spectral clustering algorithm is proposed in this paper. A branch weight model, considering the impact of node voltage and power flow factors, is initially established to comprehensively reflect the electrical connectivity between nodes. Subsequently, a weighted graph model is constructed based on spectral graph theory, and an improved spectral clustering algorithm is employed to partition the power grid. Finally, a safety risk indicator is utilized to identify whether the partitioned sections are key transmission sections. Results from case studies on the IEEE39-node system and actual power grid examples demonstrate that the proposed method accurately and effectively searches for all key transmission sections of the system and identifies their security risks. The application in real power grid scenarios validates its ability to screen out some previously unrecognized key transmission sections. [ABSTRACT FROM AUTHOR]

Stochastic complex networks with multi-weights which were driven by Brownian motion were widely investigated by many researchers. However, Brownian motion is not suitable for the modeling of engineering issues by reason of its variance, which is infinite at any time. So, in this paper, a novel kind of stochastic complex network with multi-weights driven by second-order process is developed. To disclose how the weights and second-order process affect the dynamical properties of stochastic complex networks with multi-weights driven by the second-order process, we discuss exponential stability of the system. Two types of sufficient criteria are provided to ascertain exponential stability of the system on the basis of Kirchhoff's matrix tree theorem and the Lyapunov method. Finally, some numerical examples are given to verify the correctness and validity of our results. [ABSTRACT FROM AUTHOR]

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of "information" across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with the k -path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization. [ABSTRACT FROM AUTHOR]

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n ≤ 10 . We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. [ABSTRACT FROM AUTHOR]

In dual traction power supply systems, the overhead catenary system operates in different power supply modes. It passes across the AC section, DC section, and a neutral part, which influences the features and properties of the feedback current and aggregates its effects on stray current and rail potential. This research paper presents an integrated model of an AC and DC traction power system (TPS), along with a unified chain equivalent circuit model, the ground wire through the AC section, and the insulation joints fixed in the AC–DC neutral section under different operating positions. To make a unified impedance and admittance matrix for traction network, a virtual conductor wire has been added in the DC section to make the balance order of the TPS. The influence of reflex system configuration on rail potential, stray current, and distribution of rail potential at AC and DC stations has also been calculated. The simulation results demonstrate that establishing the insulation joints in the AC–DC section and using ground wires in the AC part can significantly decrease the stray current and rail potential. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices. Denote by L+ (G) = D(G)-1/2 Q(G) D(G)-1/2 the normalized signless Laplacian matrix of graph G, where Q(G) and D(G) are the signless Laplacian and diagonal degree matrices of G, respectively. The eigenvalues of matrix L+ (G), 2 = γ1+ ≥ γ2+ > … ≥ γn+ ≥ 0, are normalized signless Laplacian eigenvalues of G. In this paper, we introduce the normalized signless Laplacian resolvent energy of G as ERNS (G) = ∑i=1n 1/3-γ4+. We also obtain some lower and upper bounds for ERNS (G) as well as its relationships with other energies and signless Kemeny's constant. [ABSTRACT FROM AUTHOR]

It is widely understood that quantum computing — quantum gates upon qubits — is the general case, encompassing computing by classical means, viz. Boolean logic upon classical bits. It also seems reasonable that Quantum Software should encompass Classical Software. However, to accept such a statement regarding software, the feeling that it seems reasonable is not enough. One needs clear-cut definitions and formal conclusions. This is exactly the purpose of this paper. Previously, we have represented Classical Software by the Laplacian Matrix. More recently, we have shown that Quantum Software is faithfully represented by Density Matrices. It turns out that a Laplacian Matrix normalized by the Laplacian Trace easily obtains a Density Matrix. This opens the horizons for Quantum Software operations — such as unitary and reversible evolution — not naturally available with the classical Laplacian. This paper provides the necessary definitions and conclusions, illustrating the more general Quantum operations with a relevant case study, playing the double role of both classical and quantum software. [ABSTRACT FROM AUTHOR]

In recent years, the heat kernel has proven to be a valuable tool for graph characterization and graph-based object clustering. It serves as the fundamental solution to the heat diffusion equation associated with the discrete graph Laplacian, describing how information flows across graph edges over time by exponentiating the Laplacian eigensystem. This paper focuses on the novel concept of the generalized heat kernel of a network, recently introduced by Kalala Mutombo et al. These authors build upon the k-path Laplacian operator for graphs, pioneered by Estrada et al., to incorporate long-range interactions (LRI) into information transmission across nodes and edges of a network/graph. LRI are handled through the use of Mellin and Laplace transforms. This paper makes a notable contribution by highlighting the practical application of the generalized heat kernel. We achieve this by demonstrating the effectiveness of the generalised heat kernel invariants in object clustering with real data using principal component analysis (PCA). Moreover, we investigate how incorporating long-range interactions (LRI) impacts object characterization and clustering, revealing superior results compared to conventional diffusion methods. Through experimentation, we show that object clustering remains achievable even with small values of the Mellin and Laplace parameters, contrasting with the requirement of an infinite value in the absence of LRI. • Application of graph generalised heat kernel invariants in object clustering via PCA. • Long-range interactions enhance graph-based object clustering. • Object clustering achieved with small Mellin/Laplace parameters. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

This paper concerns the consensus problem of linear time-invariant multi-agent systems (MASs) with multiple state delays and communicate delays. Consensus control is widely applied in spacecraft formation, sensor networks, robotic manipulators, autonomous vehicles, and others. By introducing a linear transformation, the consensus problem of the delayed MAS under an undirected network was converted into a robust asymptotic stability problem associated with the eigenvalues of the normalized Laplacian matrix of the network. By means of the argument principle and optimization technologies, a numerical controller design method was presented for the delayed MAS to reach consensus. The effectiveness of the proposed approach was illustrated by some numerical examples. The proposed approach may be applied to multi-agent systems with distributed delays. [ABSTRACT FROM AUTHOR]

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group : Here, , , , , and . By means of variational techniques, we extended the arguments developed in [1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms. [ABSTRACT FROM AUTHOR]

This paper investigates the multiple-bipartite consensus problem without considering neighbours' velocity information in networked Lagrangian systems (NLSs). A distributed adaptive control algorithm without using neighbours' velocity information is proposed, which facilitates the practical configuration deployments in the coopetition networks. By borrowing a subtle vector composed of the eigenvector components associated with zero eigenvalue of Laplacian matrix, a novel reference estimated vector is introduced to conduct the stability analysis step-by-step in the coopetition networks. Finally, simulations are provided to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem. [ABSTRACT FROM AUTHOR]

To address the problem that traditional spectral clustering algorithms cannot obtain the complete structural information of networks, this paper proposes a spectral clustering community detection algorithm, PMIK-SC, based on the point-wise mutual information (PMI) graph kernel. The kernel is constructed according to the point-wise mutual information between nodes, which is then used as a proximity matrix to reconstruct the network and obtain the symmetric normalized Laplacian matrix. Finally, the network is partitioned by the eigendecomposition and eigenvector clustering of the Laplacian matrix. In addition, to determine the number of clusters during spectral clustering, this paper proposes a fast algorithm, BI-CNE, for estimating the number of communities. For a specific network, the algorithm first reconstructs the original network and then runs Monte Carlo sampling to estimate the number of communities by Bayesian inference. Experimental results show that the detection speed and accuracy of the algorithm are superior to other existing algorithms for estimating the number of communities. On this basis, the spectral clustering community detection algorithm PMIK-SC also has high accuracy and stability compared with other community detection algorithms and spectral clustering algorithms. [ABSTRACT FROM AUTHOR]

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [2]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs R(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and R(G2) and S(H1) and R(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs. [ABSTRACT FROM AUTHOR]

The modern power grid features the high penetration of power converters, which widely employ a phase-locked loop (PLL) for grid synchronization. However, it has been pointed out that PLL can give rise to small-signal instabilities under weak grid conditions. This problem can be potentially resolved by operating the converters in grid-forming mode, namely, without using a PLL. Nonetheless, it has not been theoretically revealed how the placement of grid-forming converters enhances the small-signal stability of power systems integrated with large-scale PLL-based converters. This paper aims at filling this gap. Based on matrix perturbation theory, we explicitly demonstrate that the placement of grid-forming converters is equivalent to increasing the power grid strength and thus improving the small-signal stability of PLL-based converters. Furthermore, we investigate the optimal locations to place grid-forming converters by increasing the smallest eigenvalue of the weighted and Kron-reduced Laplacian matrix of the power network. The analysis in this paper is validated through high-fidelity simulation studies on a modified two-area test system and a modified 39-bus test system. This paper potentially lays the foundation for understanding the interaction between PLL-based (i.e., grid-following) converters and grid-forming converters, and coordinating their placements in future converter-dominated power systems. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

Hyper-networks tend to perform better in representing multivariate relationships among nodes. Yet, due to the complexity of the hyper-network structure, research in synchronization dynamics is rarely involved. In this paper, a Kuramoto model more suitable for k-uniform hyper-networks is proposed. And the generalized Laplacian matrix expression of the k-uniform hyper-network is present. We use the eigenvalue ratio of the generalized Laplacian matrix to quantify synchronization. And we studied the effects of some important structure parameters on the synchronization of three types of k-uniform hyper-networks. And obtained different relationships between synchronization and these parameters. The results show the synchronization of the k-uniform hyper-networks is related to both structure and parameters. And as the size of the nodes increases, the synchronization ability gradually increases for ER random hyper-network, while that gradually decreases for NW small-world hyper-network and BA scale-free hyper-network. As the uniformity increases, the synchronization ability of all three types of uniform hyper-networks increases. In addition, when the structure and node size are fixed, the synchronization ability increases with the increase of the hyper-clustering coefficient in BA scale-free hyper-network and ER random hyper-network, while it decreases with the increase of the hyper-clustering coefficient in NW small-world hyper-network. [ABSTRACT FROM AUTHOR]

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, ..., n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs. [ABSTRACT FROM AUTHOR]

Laplacian matrix and its spectrum are commonly used for giving a measure in networks in order to analyse its topological properties. In this paper, Laplacian matrix of graphs and their spectrum are studied. Laplacian energy of a graph G of order n is defined as LE (G) = ∑ i = 1 n | λ i (L) - d ¯ | , where λ i (L) is the i-th eigenvalue of Laplacian matrix of G, and d ¯ is their average. If LE (G) = LE (K n) for the complete graph K n of order n, then G is known as L-borderenergetic graph. In the first part of this paper, we construct three infinite families of non-complete disconnected L-borderenergetic graphs: Λ 1 = { G b , j , k = [ (((j - 2) k - 2 j + 2) b + 1) K (j - 1) k - (j - 2) ] ∪ b (K j × K k) | b , j , k ∈ Z + } , Λ 2 = { G 2 , b = [ K 6 ∇ b (K 2 × K 3) ] ∪ (4 b - 2) K 9 | b ∈ Z + } , Λ 3 = { G 3 , b = [ b K 8 ∇ b (K 2 × K 4) ] ∪ (14 b - 4) K 8 b + 6 | b ∈ Z + } , where ∇ is join operator and × is direct product operator on graphs. Then, in the second part of this work, we construct new infinite families of non-complete connected L-borderenergetic graphs Ω 1 = { K 2 ∇ a K 2 r ¯ | a ∈ Z + } , Ω 2 = { a K 3 ∪ 2 (K 2 × K 3) ¯ | a ∈ Z + } and Ω 3 = { a K 5 ∪ (K 3 × K 3) ¯ | a ∈ Z + } , where G ¯ is the complement operator on G. [ABSTRACT FROM AUTHOR]

This paper considers the secure distributed control consensus problem of linear multi‐agent systems (MASs) under switching topologies, subject to intermittent sequential scaling attacks, which was compelled to scaling factor, attack frequency, duration and cooperative‐competitive networks. First, the scenario of a fixed topology is considered, and a novel control protocol combined with a logarithmic quantizer and relative state measurements of neighbouring agents is discussed. The sighed graph is utilized to characterize the communication topology determined by the information flow directions and captured by the graph Laplacian matrix. After that, sufficient conditions for effectiveness of the developed control methods in guiding the MASs to secure bipartite leader‐following consensus are constructed. Second, the scenario of switching topologies is considered, and it is derived that the secure bipartite consensus will be achieved if the designed state feedback control protocol with the scaling factor, the attack duration, attack frequency and switching signal are selected properly. At last, to prove the effectiveness of the designed controllers, a simulation example depended on the real‐word actual military is introduced. [ABSTRACT FROM AUTHOR]

Recently, graph theory has become a promising tool for biomedical signal analysis, wherein the signals are transformed into a graph network and represented as either adjacency or Laplacian matrices. However, as the size of the time series increases, the dimensions of transformed matrices also expand, leading to a significant rise in computational demand for analysis. Therefore, there is a critical need for efficient feature extraction methods demanding low computational time. This paper introduces a new feature extraction technique based on the Gershgorin Circle theorem applied to biomedical signals, termed Gershgorin Circle Feature Extraction (GCFE). The study makes use of two publicly available datasets: one including synthetic neural recordings, and the other consisting of EEG seizure data. In addition, the efficacy of GCFE is compared with two distinct visibility graphs and tested against seven other feature extraction methods. In the GCFE method, the features are extracted from a special modified weighted Laplacian matrix from the visibility graphs. This method was applied to classify three different types of neural spikes from one dataset, and to distinguish between seizure and non-seizure events in another. The application of GCFE resulted in superior performance when compared to seven other algorithms, achieving a positive average accuracy difference of 2.67% across all experimental datasets. This indicates that GCFE consistently outperformed the other methods in terms of accuracy. Furthermore, the GCFE method was more computationally-efficient than the other feature extraction techniques. The GCFE method can also be employed in real-time biomedical signal classification where the visibility graphs are utilized such as EKG signal classification. [ABSTRACT FROM AUTHOR]

In this paper, the secure containment control issue for heterogeneous multi-agent systems subject to hybrid attacks is studied. In the network channels among agents, adversaries launch replay attacks such that the followers can only retrieve the previous information of their neighbors. To characterize the effects of replay attacks, a distributed auxiliary system with heterogeneous time-varying delay is constructed to estimate the convex hull of the leaders' state. The control input of each follower may be contaminated by the malicious false data injection (FDI) attacks with unknown bounded time-varying signals, which are intended to compromise the containment performance of the whole system. A local adaptive compensator is designed for each follower to eliminate the adverse effects of the FDI attacks. Based on the distributed auxiliary system and the local adaptive compensator, a secure distributed control protocol is proposed to ensure that the output trajectories of each follower can converge to the dynamic convex hull spanned by the leader outputs. Finally, two illustrative examples are provided to show the feasibility of the proposed secure control scheme. • This paper introduces a secure control protocol to dispose a more complicated attack situation. • The original Laplacian matrix is divided into several sub-Laplacian matrices associated with the heterogeneous time-varying delays. • The designed adaptive compensator can fully eliminate the adverse effects of the FDI attacks without using any information of the FDI attacks. [ABSTRACT FROM AUTHOR]

For sparse subspace clustering methods, it is crucial to develop a good representation matrix to capture the data structure. In this paper, we incorporated the label information into sparse representation and proposed a new semi-supervised sparse subspace clustering method, named semi-supervised sparse subspace clustering with manifold regularization (S 4 CMR). When developing the sparse self-expressive matrix, the S 4 CMR method utilized the label information to constrain the development of expressiveness coefficients. The local manifold regularization was also integrated to enhance clustering stability and local consistency. By utilizing the Alternating Direction Method of Multipliers (ADMM), the convex optimization problem associated with linear constraints can be easily resolved. The developed similarity matrix can provide strong discriminant information, making it more effective for semi-supervised tasks. The effectiveness of the proposed algorithm is demonstrated through experiments on benchmark data sets, such as motion segmentation and image clustering. [ABSTRACT FROM AUTHOR]

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks. [ABSTRACT FROM AUTHOR]

Over the past few decades, there have been rapid advances in solid-state technology as well as a reduction in cost. This, coupled with the functionality and efficiency improvements they afford, has resulted in a massive increase in the use of electronic devices. Where traditionally, there were a few well-known nonlinear loads that needed to be considered, now there are numerous low-power devices. Although individually insignificant, collectively, they are very significant. This paper presents a tensor-based harmonic analysis approach that is capable of capturing important interactions while being computationally efficient enough to model a large distribution system. Numerical experiments are used to highlight the advantages of the tensor framework. Numerous papers have investigated the tensor parametrisation or its mathematical equivalent—harmonically coupled admittance matrices (also known as frequency coupling matrices). However, this paper, for the first time, demonstrates how these models can be applied to perform harmonic modelling of a complete low voltage (LV) distribution system. [ABSTRACT FROM AUTHOR]

In this paper, a fully distributed strategy for the economic dispatch problem (EDP) in the smart grid is proposed. The economic dispatch model considers both traditional thermal generators and wind turbines (WTs), integrating generation costs, carbon trading expenses, and the expected costs associated with the unpredictability of wind power. The EDP is transformed into an equivalent optimization problem with only an equality constraint and thus can be solved by an alternating-direction method of multipliers (ADMM). Then, to tackle this problem in a distributed manner, the outer-layer framework of the proposed strategy adopts a parallel ADMM, where different variables can be calculated simultaneously. And the inner-layer framework adopts a finite-step consensus algorithm. Convergence to the optimal solution is achieved within a finite number of communication iterations, which depends on the scale of the communication network. In addition, leveraging local and neighbor information, a distributed algorithm is designed to compute the eigenvalues of the Laplacian matrix essential for the finite-step algorithm. Finally, several numerical examples are presented to verify the correctness and effectiveness of the proposed strategy. [ABSTRACT FROM AUTHOR]

The optimal selection of pinning nodes for multi-agent flocking is a challenging NP-hard problem. Current pinning node selection strategies mainly rely on centrality measures of complex networks, which lack rigorous mathematical proof for effective flocking control. This paper proposes a pinning node selection strategy based on matrix eigenvalue theory. First, the effect of the pinning node on the eigenvalue of the Laplacian matrix is analyzed. Then, a synchronization index representing the topology uniformity of the multi-agent system is proposed to exert maximum influence on the system synchronizability. A practical optimal pinning node selection method based on the synchronization index is proposed and analyzed using the eigenvalue perturbation method. Finally, simulations demonstrate that the convergence rate of the system obtained using the optimal synchronizability pinning node selection method is better than that achieved with the maximum degree centrality node selection strategy. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlinear higher-order Kirchhoff-type equation with dissipation in a bounded domain. By establishing a first order differential inequality technique, a lower bound for the blow-up time is obtained when the blow-up of solution occurs. [ABSTRACT FROM AUTHOR]

Let R be a ring and U (R) be the set of unit elements of R. The unit graph G(R) of R is the graph whose vertices are all the elements of R, defining distinct vertices x and y to be adjacent if and only if x + y ∈ U (R). The Laplacian spectrum of G(ℤn) was studied when n = pm, where p is a prime and m is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of G(ℤn), for n = p1p2...pk, where pi are distinct primes and i = 1,2, ..., k. [ABSTRACT FROM AUTHOR]

Discrete-time architectures offer a distinct advantage over their continuous counterparts, as they can be seamlessly implemented on embedded hardware without the necessity for discretization processes. Yet, because of the difficulty of ensuring Lyapunov difference expressions, their designs, which are based on quadratic Lyapunov-based frameworks, are highly complex. As a result, various existing continuous-time results using adaptive control methods to deal with system uncertainties and coupled dynamics in agents of a multiagent system cannot be directly applied to the discrete-time context. Furthermore, compared to their continuous-time equivalent, discrete-time information exchange based on periodic time intervals is more practical in the control of multiagent systems. Motivated by these standpoints, in this paper, we first introduce a discrete-time adaptive control architecture designed for uncertain scalar multiagent systems without coupled dynamics as a preliminary result. We then introduce another discrete-time adaptive control approach for uncertain multiagent systems in the presence of coupled dynamics. Our approach incorporates observer dynamics to manage unmeasurable coupled dynamics, along with a user-assigned Laplacian matrix to induce cooperative behaviors among multiple agents. Our solution includes Lyapunov analysis with logarithmic and quadratic Lyapunov functions for guaranteeing asymptotic stability with both controllers. To demonstrate the effectiveness of the proposed control architectures, we provide an illustrative example. The illustrative numerical example shows that the standard discrete-time adaptive control in the absence of observer dynamics cannot guarantee the reference state vector tracking, while the proposed discrete-time adaptive control can ensure the tracking objective. [ABSTRACT FROM AUTHOR]

In this paper, a fully distributed control protocol is presented based on adaptive technology to solve the time‐varying formation problem of multi‐unmanned aerial vehicle (UAV) systems, where the coupling weights between neighbouring UAVs can be adjusted through adaptive method. Firstly, the proposed control protocol eliminates the need for global information on the interaction topology of multi‐UAV systems, the calculation of Laplacian matrix eigenvalues is avoided and reduces the computational complexity of the system. Secondly, an algorithm with two steps is proposed to determine the distributed adaptive control gain matrices, where the feasibility condition of time‐varying formation of multi‐UAV systems is given. Then, the closed‐loop stability of multi‐UAV systems is proved through Lyapunov theory. Finally, numerical simulation is conducted to verify the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Unsupervised spectral feature selection (USFS) methods could output interpretable and discriminative results by embedding a Laplacian regularizer in the framework of sparse feature selection to keep the local similarity of the training samples. To do this, USFS methods usually construct the Laplacian matrix using either a general-graph or a hyper-graph on the original data. Usually, a general-graph could measure the relationship between two samples while a hyper-graph could measure the relationship among no less than two samples. Obviously, the general-graph is a special case of the hyper-graph and the hyper-graph may capture more complex structure of samples than the general graph. However, in previous USFS methods, the construction of the Laplacian matrix is separated from the process of feature selection. Moreover, the original data usually contain noise. Each of them makes difficult to output reliable feature selection models. In this paper, we propose a novel feature selection method by dynamically constructing a hyper-graph based Laplacian matrix in the framework of sparse feature selection. Experimental results on real datasets showed that our proposed method outperformed the state-of-the-art methods in terms of both clustering and segmentation tasks. [ABSTRACT FROM AUTHOR]

This paper treats a parallel 12-pulse rectifier as the study subject. The influence of unbalanced terminal voltage, the triggering angle, as well as the commutation angle of a thyristor-controlled rectifier circuit are considered in switching functions. Then, a frequency-domain harmonic coupled admittance matrix under the effect of positive sequence and negative sequence voltages are proposed, respectively. With the integration of the above matrices, a harmonic coupled admittance matrix model of a parallel 12-pulse rectifier under unbalanced voltage is proposed. On this basis, a simplified harmonic model of a 12-pulse rectifier is established, which only considers the effect of major elements through analyzing the elements' distribution and magnitude of the harmonic coupled admittance matrix, as well as the AC current harmonic producing mechanism. All the models proposed in this paper are verified with high accuracy through simulation in MATLAB/Simulink. The models proposed in this paper can be used to guide the selection of harmonic suppression measures and equipment. [ABSTRACT FROM AUTHOR]

It is known that many statistical and machine learning approaches heavily rely on pairwise distance between data points. The choice of distance function on the underlying manifold has a fundamental impact on performance of these processes. This is closely related to questions of how to appropriately calculate distances, and hence, fundamental solutions (heat kernels) for heat operators can be obtained. In general, it is not so easy to obtain a closed form for heat kernels. We first survey results of heat kernels on radially symmetric Riemannian manifolds, e.g., Euclidean spaces and unit spheres in Rn. For the cases n = 1;2; 3, we may construct the heat kernel explicitly. But, the computation is much more complicated when n > 3. However, by results of Nagase, we may construct parametrices for the heat kernel by using elementary functions so that the error terms can be under controlled. In the second part of the paper, we discuss some results on subRiemannian manifolds, especially 3-dimensional sphere in C² as a CR-manifold. We study geodesics connecting two given points on S³ respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of S³ and some partial results are obtained in the general case. The Carnot-Carath'eodory distance is calculated. We also present some motivations related to quantum mechanics. Then we give a brief discussion of Greiner's methods on the heat kernel for the Cauchy-Riemann subLaplacian on S2n+1. We provide a brief discussion on applications of these heat kernels to graph kernels in the last part of the paper. [ABSTRACT FROM AUTHOR]

The goal of this paper is to discuss some of the results in the author’s previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in [47] is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see [40], is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as t → +∞, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity’s sake, we restrict our attention to the case of spherical target manifolds Sn−1, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds N (cf. [48]). [ABSTRACT FROM AUTHOR]

The present paper deals with the external identification of a reciprocal, special passive, 2 n -port network under measurement uncertainties. In the present context, the multiport model is represented by an admittance matrix and the condition that the network is 'reciprocal special passive' refers to the assumption that the real part of the admittance matrix is symmetric and positive-definite. The key point is to reformulate the identification problem as a matrix optimization program over the matrix manifold S + (2 n) × S (2 n) . The optimization problem requires a least-squares criterion function designed to cope with over-determinacy due to the incoherent data pairs whose cardinality exceeds the problem's number of degrees of freedom. The present paper also proposes a numerical solution to such an optimization problem based on the Riemannian-gradient steepest descent method. The numerical results show that the proposed method is effective as long as reasonable measurement error levels and problem sizes are being dealt with. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well... ...where a and b are positive constants, λ > 0 is a parameter, 1 < q < 2, the potential V ∈ C(R³;R) and V-1(0) has a nonempty interior. The functions f and g are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as λ→∞ is also explored. [ABSTRACT FROM AUTHOR]

This paper presents an FPGA-based Digital Real-Time Simulation (DRTS) environment for the analysis of Electromagnetic Transients (EMTs) of power systems that include multiple Power Electronic Converters (PECs). The main features of the proposed DRTS platform are that it i) employs two-value resistor switch model for PECs and thus avoids inaccuracies of Associate Discrete Circuit (ADC) model and ii) does not require separation of PECs by transmission lines (stub lines) and thus accurately represents PECs in close electrical proximity of each other. These features respectively result in a time-variant and large system admittance matrix which significantly increases the computation burden in each simulation time-step and can render it infeasible for real-time applications. To overcome this problem this paper introduces i) Switching-Network Partitioning (SNP) method which is tailored for PECs representation in FPGA and ii) Reformulated Modified Nodal Analysis (RMNA) method to enable computational parallelism between the solution of network and component models in FPGA. SNP enables solution-level partitioning of the PECs and the rest of the system. The developed DRTS platform enables sub-microsecond simulation time-step and provides a fixed hardware irrespective of the system configuration. Performance of the platform is evaluated for a study system and verified versus off-line simulation studies. [ABSTRACT FROM AUTHOR]

We define a new family of random geometric graphs which we call random covering graphs and study its statistical properties. To the best of our knowledge, this family of graphs has not been explored in the past. Our experimental results suggest that there are striking deviations in the expected number of edges, degree distribution, spectrum of adjacency/normalized Laplacian matrix associated with the new family of graphs as compared to both the well-known Erdos–Renyi random graphs and the general random geometric graphs as originally defined by Gilbert. Particularly, degree distribution of the graphs shows some interesting features in low dimensions. To the more applied end, we believe that our random graph family might be effective in modelling some practically useful networks (world wide web, social networks, railway or road networks, etc.). It is observed that the degree distribution of some complex networks arising in practice follow power law distribution or log power distribution; they tend to be right skewed, having a heavy tail unlike the degree distribution of Erdos–Renyi graphs or general geometric random graphs (which follow exponential distribution with a sharp tail). The degree distribution of our random graph family significantly deviates from that of Erdos–Renyi graphs or general geometric random graphs and is closer to a right-skewed power law distribution with a heavy tail. Thus, we believe that this new family of graphs might be more effective in modelling the typical real-world networks mentioned above. The key contribution of the paper is introducing this new random graph family and studying some of its properties experimentally, further investigation into which would be interesting from a purely mathematical perspective. Also, it might be of practical interest in terms of modelling real-world networks. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph of order n. In this paper, we introduce a new distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as UD (G) = αTr (G) + βD(G) + γJ + δI, where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix and α, β, γ, δ ∈ R and β ̸= 0. This unified definition enables us to derive the spectra of different matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal distance matrix namely, {ρ1, ρ2,.. ., ρn} is known as the universal distance spectrum of G. As a consequence, by taking appropriate values for α, β, γ, δ ∈ R and β ̸= 0, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of graph complements. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H using the Schur complement of a block matrix. [ABSTRACT FROM AUTHOR]

Controllability is a fundamental issue in the field of fractional complex network control, yet it has not received adequate attention in the past. This paper is dedicated to exploring the controllability of complex networks involving the Caputo fractional derivative. By utilizing the Cayley–Hamilton theorem and Laplace transformation, a concise proof is given to determine the controllability of linear fractional complex networks. Subsequently, leveraging the Schauder Fixed-Point theorem, controllability Gramian matrix, and fractional calculus theory, we derive controllability conditions for nonlinear fractional complex networks with a weighted adjacency matrix and Laplacian matrix, respectively. Finally, a numerical method for the controllability of fractional complex networks is obtained using Matlab (2021a)/Simulink (2021a). Three examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: ... where (-Δ)ps is the fractional p-Laplacian on the Heisenberg group ℍN, M is the Kirchhoff function, V (ξ) is the potential function, 0 < s < 1,1 < p < N/s, μ > 0, f (ξ, u) is the nonlinear function, 0 < λ < Q, Q = 2N + 2, and Qλ* = 2Q-λ/Q-2 is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if μ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has m pairs of solutions if μ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions to the following Kirchhoff type equation with zero mass and Hardy-Littlewood-Sobolev critical nonlinearity ... where N ≥ 3, λ > 0, μ ∈ (0, min{N, 4}), 2μ* = 2n-μ/n-2 is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, and m satisfies some local monotonicity conditions near zero. The non- linearity f is odd in u and satisfies some classical superlinear and quasi-critical growth conditions. For any given k ∈ ℕ, k pairs of nontrivial solutions are obtained for λ large enough by a version of the symmetric mountain pass theorem and a version of the second concentration compactness principle. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of Harary Laplacian-energy-like for a simple undirected and connected graph G with order n. We also establish novel matrix results in this regard. Furthermore, by employing matrix order reduction techniques, we derive upper and lower bounds utilizing existing graph invariants and vertex connectivity. Finally, we characterize the graphs that achieve the aforementioned bounds by considering the generalized join operation of graphs. [ABSTRACT FROM AUTHOR]

Broad Learning System (BLS) is proposed as an alternative to deep learning. It has a fast adaptive model selection and online incremental learning capability, which has been successfully applied in many fields. In this paper, the BLS model is introduced into intrusion detection, and considering the weakness of the BLS model in mining the internal structural information of samples, this paper proposes a Manifold Regularized Broad Learning System based on LU decomposition (LU-MRBLS) intrusion detection. Based on the manifold hypothesis, the LU-MRBLS model firstly constructs the graph Laplacian operator in the data input space to mine the potential information of the data. Then, under the manifold regularized framework, the feature nodes, enhancement nodes, and Laplacian matrix are combined to construct the objective function to regularize and optimize the BLS model to avoid the model falling into local optimization. Finally, the LU decomposition method is used to solve the output weight matrix of the MRBLS model, shorten the training time of the MRBLS model, avoid singular value problems of the solution process, and improve the intrusion detection performance of the model. In this paper, we use the KDD Cup99 dataset for parameter selection and apply it to other network models. Through rigorous experiments, the LU-MRBLS model is applied to KDD Cup99, NSL-KDD, UNSW-NB15, and CIDDS-001 datasets with better detection results than the classical machine learning models and the latest intrusion detection models. [ABSTRACT FROM AUTHOR]

Algorithms that use tensor decompositions are widely used due to how well they perfor with large amounts of data. Among them, we find the algorithms that search for the solution of a linear system in separated form, where the greedy rank-one update method stands out, to be the starting point of the famous proper generalized decomposition family. When the matrices of these systems have a particular structure, called a Laplacian-like matrix which is related to the aspect of the Laplacian operator, the convergence of the previous method is faster and more accurate. The main goal of this paper is to provide a procedure that explicitly gives, for a given square matrix, its best approximation to the set of Laplacian-like matrices. Clearly, if the residue of this approximation is zero, we will be able to solve, by using the greedy rank-one update algorithm, the associated linear system at a lower computational cost. As a particular example, we prove that the discretization of a general partial differential equation of the second order without mixed derivatives can be written as a linear system with a Laplacian-type matrix. Finally, some numerical examples based on partial differential equations are given. [ABSTRACT FROM AUTHOR]