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We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn–Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using ab initio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals. [ABSTRACT FROM AUTHOR]

The Q -index of a graph G is the largest eigenvalue of the signless Laplacian matrix of G. A graph is leaf-free if it has no pendent vertices. In this paper, we give sharp upper bounds on the Q -index of leaf-free graphs with given size, and characterize the corresponding extremal graphs completely. As a consequence, the 2-edge-connected graph (resp. Euler graph) with given size having the largest Q -index is also characterized. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the Laplacian eigenvalues of the extension of the zero divisor graph, Γ e (Z n) , for some particular values of n. We characterize the values of n that give the equality of the spectral radius and the second-smallest eigenvalue of Γ e (Z n). Finding Wiener index of Γ e (Z n) in terms of its Laplacian eigenvalues is another objective of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index n ≥ 49. To investigate how the nodes are connected, we analytically describe that the link density saturates to 6 / π 2 , whereas the average degree increases linearly with slope 6 / π 2 with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for 49 ≤ n ≤ 288 and it is at most 2 for n ≥ 289. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of r -length labeled cycles, which indicates the existence of a cycle of length at most O (log n). Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erdős–Rényi random network and Barabási–Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator–prey networks which can easily avoid the global synchronization. [ABSTRACT FROM AUTHOR]

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]

In this work, we deal with the stabilisation issue for a class of highly nonlinear stochastic coupled systems on networks (SCSNs, in short). Specifically, we obtain the criteria for the finite-time stabilisation of the highly nonlinear SCSNs based on two methods such as the Lyapunov-type condition related with Kirchhoff's Matrix Tree Theorem and coefficients-type criterion. Further, highly nonlinear SCSNs is finite-time stabilised through designing a state feedback controller. To demonstrate the effectiveness of the developed result, as an application, we show the finite-time stabilisation for a class of stochastic coupled oscillators whose coefficients satisfy highly nonlinear coefficients. [ABSTRACT FROM AUTHOR]

For a simple graph on n vertices, any of its signless Laplacian eigenvalues is in the interval [ 0 , 2 n − 2 ]. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in [ 0 , 2 n − 2 ] and graph invariants including matching number and diameter. [ABSTRACT FROM AUTHOR]

This article investigates the leader‐follower consensus control problem of a feedforward nonlinear multiagent system under denial‐of‐service (DoS) attacks. Compared with the existing results on the consensus control of multiagent systems under DoS attacks, this article considers a class of feedforward multiagent systems with nonlinear functions that are less conservative, and takes into account asynchronous DoS attacks with attack detection delay. To address these problems, firstly, a new class of topological switching sequences and Laplacian matrices are designed based on the properties of the considered DoS attacks to convert the DoS attacks with attack detection delay problems into a class of topological switching problems. Then, a distributed dynamic gain compensator and an output feedback controller are constructed for each follower by using the output information of neighbor agents that are not intercepted by the attacks. Afterwards, by using piecewise Lyapunov functions, it is proven that if the DoS signal satisfies the condition of time‐varying average dwell time (ADT), then the followers can asymptotically track the state of the leader. Finally, a numerical example is provided to verify the effectiveness of the control algorithm. [ABSTRACT FROM AUTHOR]

Normal and aberrant cognitive functions are the result of the dynamic interplay between large‐scale neural circuits. Describing the nature of these interactions has been a challenging task yet important for neurodegenerative disease evolution. Fusing modern dynamic graph network theory techniques and control theory applied on complex brain networks creates a new framework for neurodegenerative disease research by determining disease evolution at the subject level, facilitating a predictive treatment response and revealing key mechanisms responsible for disease alterations. It has been shown that two types of controllability—the average and the modal controllability—are relevant for the mechanistic explanation of how the brain navigates between cognitive states. The average controllability favors highly connected areas which move the brain to easily reachable states, while the modal controllability favors weakly connected areas representative for difficult‐to‐reach states. We propose two different techniques to achieve these two types of controllability: a centrality measure based on a sensitivity analysis of the Laplacian matrix is employed to determine the average controllability, while graph distances form the basis of the modal controllability. The concepts of "choosing the best driver set" and "graph distances" are applied to measure the average controllability and the modal controllability, respectively. Based on these new techniques, we obtain important disease descriptors that visualize alterations in the disease trajectory by revealing densely connected hubs or sparser areas. Our results suggest that these two techniques can accurately describe the different node roles in controlling trajectories of brain networks. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

Turing instability in complex networks is known to be dependent on the degree distribution, and the necessary conditions for Turing instability have been shown in the literature to have an explicit dependence on the eigenvalues of the Laplacian matrix, which, in turn, depends on the network topology. This study reveals that these conditions are not sufficient, and another global network measure—the nodal clustering—also plays a crucial role. Analytical and numerical results are presented to explain the effects of clustering for several network topologies, ranging from the S 1 / H 2 hyperbolic geometric networks that enable modeling the naturally occurring clustering in real-world networks, as well as the random and scale-free networks, which are obtained as limiting cases of the S 1 / H 2 model. Analysis of the Laplacian eigenvector localization properties in these networks is shown to reveal distinct signatures that enable identifying the so called Turing patterns even in complex networks. [ABSTRACT FROM AUTHOR]

This study proposes a frequency coupling suppression strategy for voltage source converters (VSCs) with DC‐link dynamics to address the challenges posed by the asymmetric system structures for system modelling and control design. The strategy introduces a reshaping outer loop into the q‐axis, which eliminates the conjugate voltages and currents induced by the DC‐link dynamics, to achieve decoupling of the DC‐link voltage control (DVC). Therefore, the VSC is modelled as a single‐input single‐output (SISO) admittance model, which provides clear physical insights for the analysis of admittance characteristics. Compared with the admittance matrix, the SISO model reveals the dominance of each control dynamics on VSC admittance in different frequency bands. This insight facilitates the exploration of optimized control strategies to improve the system stability margin. Based on the SISO model, a virtual admittance for compensating the negative resistive effect of the PLL is designed to illustrate the advantages of the proposed method in providing design‐oriented analysis. Experimental results verify the effectiveness of the proposed control strategy. [ABSTRACT FROM AUTHOR]

The optimal leaderless and leader-following time-varying formation (TVF) control problems for second-order multiagent systems (MASs) are investigated, where two optimal TVF control protocols are proposed to achieve the desired formations as well as minimize the comprehensive optimization function that contain the cooperative performance index and the control energy index. For leaderless case, the optimal formation control problem is reformulated as an infinite-time state regulator problem by employing the state space decomposition method, which is subject to specified constraints on energy and performance indices, and the analytic criterion for optimal TVF achievability is subsequently proposed. Then, the results of optimal leaderless TVF control are extended to the leader-following case with switching topologies, where the main challenge is changed to find the optimal controller rather than the optimal gain matrix, and the optimal value of the comprehensive index is accurately determined. Finally, two simulation cases are proposed to validate the effectiveness of the theoretical results, and comparisons with previous works are presented to expound the optimality of the proposed formation control method. • The specific gain matrix of optimal leaderless/leader-following formation is deduced, which can not only make the multiagent system achieve time-varying formation control, but also minimize the value of the objective optimization function. • The relationships among the gain matrix, the Laplacian matrix, the regulation parameters and the minimum value of the objective optimization function are revealed. • Sufficient conditions for optimal time-varying formation achievability are proposed, which are analytic. [ABSTRACT FROM AUTHOR]

We deduce a new property exhibited by the resistance matrices of connected graphs. Specifically, we show that if R=(rij) is the resistance matrix of a connected graph on n vertices, then every off-diagonal entry in the Moore-Penrose inverse of ... is negative. Thus, we establish that the Moore-Penrose inverse of the resistance Laplacian matrices are M-matrices. [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]

This paper investigates a multi‐pursuer single‐evader nonzero‐sum differential game. Unlike traditional treatments, cooperative tracking is considered to design the optimal pursuit strategy by introducing a novel weighted topology and a scalar coupling gain. First, the distributed control strategy is designed based on the Hamilton–Jacobi–Isaacs (HJI) under the nonzero‐sum differential game. Then the modified LQR‐based cooperative tracking strategy is applied to achieve cooperativity among all the pursuers. The interception condition is also analyzed using a coupling gain based on the Laplacian matrix. Several illustrative examples under different scenarios are presented, and the results show that the pursuers with the proposed strategy can successfully intercept the evader. [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]

Several factors significantly contribute to the onset of infectious diseases, including direct and indirect transmissions and their respective impacts on incubation periods. The intricate interplay of these factors within social networks remains a puzzle yet to be unraveled. In this study, we conduct a stability analysis within a network-organized SIR model incorporating dual delays to explore the influence of direct and indirect incubation periods on disease spread. Additionally, we investigate how compound networks affect the critical incubation period. Our findings reveal several vital insights. First, by examining crossing curves and the dispersion equation, we establish the conditions for Turing instability and delineate the stable regions associated with dual delays. Second, we ascertain that the critical incubation value exhibits an inverse relationship with a network's eigenvalues, indicating that the Laplacian matrix does not solely dictate periodic behavior in the context of delays. Furthermore, our study elucidates the impact of delays and networks on pattern formation, revealing distinct pattern types across different regions. Specifically, our observations suggest that effectively curtailing the spread of infectious diseases during an outbreak is more achievable when the incubation period for indirect contact is shorter and for direct contact is longer. Namely, our network framework enables regulation of the optimal combination of (τ 1 , τ 2) to mitigate the risk of infectious diseases. In summary, our results offer valuable theoretical insights that can inform strategies for preventing and managing infectious diseases. [ABSTRACT FROM AUTHOR]

Constructing well‐behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the de facto standard linear finite elements exhibit bias on tetrahedralized regular grids, motivating the development of finite‐volume methods. In this paper, we place existing methods into a common construction, showing how their differences amount to the choice of simplex centers. These choices lead to satisfaction or breakdown of important properties: continuity with respect to vertex positions, positive semi‐definiteness of the implied Dirichlet energy, positivity of the mass matrix, and unbiased‐ness on regular grids. Based on this analysis, we propose a new method for constructing dual‐volumes which explicitly satisfy all of these properties via convex optimization. [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]

We consider the Grover walk on the infinite graph in which an internal finite subgraph receives the inflow from the outside with some frequency and also radiates the outflow to the outside. To characterize the stationary state of this system, which is represented by a function on the arcs of the graph, we introduce a kind of discrete gradient operator twisted by the frequency. Then, we obtain a circuit equation which shows that (i) the stationary state is described by the twisted gradient of a potential function which is a function on the vertices; (ii) the potential function satisfies the Poisson equation with respect to a generalized Laplacian matrix. Consequently, we characterize the scattering on the surface of the internal graph and the energy penetrating inside it. Moreover, for the complete graph as the internal graph, we illustrate the relationship of the scattering and the internal energy to the frequency and the number of tails. [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]

The Laplacian matrix L of a signed graph G may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of L. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of G. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas. [ABSTRACT FROM AUTHOR]

Graph-based semi-supervised learning plays an important role in large scale image classification tasks. However, the problem becomes very challenging in the presence of noisy labels and outliers. Moreover, traditional robust semi-supervised learning solutions suffers from prohibitive computational burdens thus cannot be computed for streaming data. Motivated by that, we present a novel unified framework robust structure-aware semi-supervised learning called Unified RSSL (URSSL) for batch processing and recursive processing robust to both outliers and noisy labels. Particularly, URSSL applies joint semi-supervised dimensionality reduction with robust estimators and network sparse regularization simultaneously on the graph Laplacian matrix iteratively to preserve the intrinsic graph structure and ensure robustness to the compound noise. First, in order to relieve the influence from outliers, a novel semi-supervised robust dimensionality reduction is applied relying on robust estimators to suppress outliers. Meanwhile, to tackle noisy labels, the denoised graph similarity information is encoded into the network regularization. Moreover, by identifying strong relevance of dimensionality reduction and network regularization in the context of robust semi-supervised learning (RSSL), a two-step alternative optimization is derived to compute optimal solutions with guaranteed convergence. We further derive our framework to adapt to large scale semi-supervised learning particularly suitable for large scale image classification and demonstrate the model robustness under different adversarial attacks. For recursive processing, we rely on reparameterization to transform the formulation to unlock the challenging problem of robust streaming-based semi-supervised learning. Last but not least, we extend our solution into distributed solutions to resolve the challenging issue of distributed robust semi-supervised learning when images are captured by multiple cameras at different locations. Extensive experimental results demonstrate the promising performance of this framework when applied to multiple benchmark datasets with respect to state-of-the-art approaches for important applications in the areas of image classification and spam data analysis. [ABSTRACT FROM AUTHOR]

The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph. Entropy functions have been used successfully to capture different aspects of graph complexity. The generalized graph entropies result from applying information measures to a graph using various schemes for defining probability distributions over the elements of the graph. In this paper, we investigate the complexity of a class of composite graphs based on subdivision graphs and corona product evaluating the generalized graph entropies, and we present explicit formulae for the complexity of subdivision-corona type product graphs. [ABSTRACT FROM AUTHOR]

Let X be a tree with n vertices and L (X) be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of L (X) × K m is equal to m − 1 , where × denotes the Kronecker product. We provide a few necessary and sufficient conditions for L (X) × K m to be Laplacian integral. The algebraic connectivity of L (X) × K m , where X is a tree of diameter 4 and k -book graph is discussed. [ABSTRACT FROM AUTHOR]

Voltage source converter based high voltage direct current (VSC-HVDC) can participate in voltage regulation by flexible control to help maintain the voltage stability of the power grid. In order to quantitatively evaluate its influence on the voltage interaction between VSC-HVDC and line commutated converter based high voltage direct current (LCC-HVDC), this paper proposes a hybrid multi-infeed interaction factor (HMIIF) calculation method considering the voltage regulation control characteristics of VSC-HVDC. Firstly, for a hybrid multi-infeed high voltage direct current system, an additional equivalent operating admittance matrix is constructed to characterize HVDC equipment characteristics under small disturbance. Secondly, based on the characteristic curve between the reactive power and the voltage of a certain VSC-HVDC project, the additional equivalent operating admittance of VSC-HVDC is derived. The additional equivalent operating admittance matrix calculation method is proposed. Thirdly, the equivalent bus impedance matrix is obtained by modifying the alternating current (AC) system admittance matrix with the additional equivalent operating admittance matrix. On this basis, the HMIIF calculation method based on the equivalent bus impedance ratio is proposed. Finally, the effectiveness of the proposed method is verified in a hybrid dual-infeed high voltage direct current system constructed in Power Systems Computer Aided Design (PSCAD), and the influence of voltage regulation control on HMIIF is analyzed. [ABSTRACT FROM AUTHOR]

Network representation learning aims to map the relationship between network nodes and context nodes to a low-dimensional representation vector space. Directed network representation learning considers mapping directional of node vector. Currently, only sporadic work on direct network representation has been reported. In this work, we propose a novel algorithm that takes into account the direction of the edge with text's attribute of the node in directed network representation learning. We then define the matrix based on in-degree of Laplacian and signless Laplacian for digraph, and it utilizes web page datasets from universities in the USA to evaluate the performance of vertex classification. We compare our algorithm with other directed representation learning algorithms. The experimental results show that our algorithm outperforms the baseline by over 20% when the training ratio ranges from 10 to 90%. We apply the in-degree-Laplacian and In-degree-signless-Laplacian to directed representation learning, which is one of the main contributions of this algorithm. Additionally, we incorporate text information through matrix completion in directed network representation learning and the experimental results show an increase in performance of up to 20% compared to the baseline, especially when the training ratio is 10%. [ABSTRACT FROM AUTHOR]

Marine transportation is pivotal in the rapid development of global trade, significantly enhancing international economic and trade connectivity and impacting the sustainable development of the global economy. In this study, we developed a novel technical framework based on the Laplacian matrix to evaluate the node significance and edge connectivity of the global shipping network using 2019 vessel schedule data from the top 30 liner shipping companies, as ranked by Alphaliner. Our analyses were conducted in both L-space, characterizing the connectivity function, and P-space, characterizing the transfer function. The findings indicate the following. (1) There is no consistent relationship between node significance and centrality for most ports. Ports with high node significance are mainly located in the Asia–Pacific region, with Singapore Port being the port with the highest node significance in L-space and Shanghai Port being the port with the highest node significance in P-space. (2) In L-space, the structures with significant improvements in edge connectivity in the shipping network have at least one port node that exhibits both low degree centrality and low betweenness centrality; these are primarily found on East African routes. (3) In P-space, the structures with significant improvements in edge connectivity in the shipping network are more complex but are notably linked to the ports of Assaluyeh and Bandar Abbas in Iran. The proposed node evaluation and edge addition strategy effectively analyze port significance and edge connectivity, providing decision-making support for optimizing port layouts, supporting container route planning, and enhancing the overall performance of the shipping network. [ABSTRACT FROM AUTHOR]

Multi-Task Learning tries to improve the learning process of different tasks by solving them simultaneously. A popular Multi-Task Learning formulation for SVM is to combine common and task-specific parts. Other approaches rely on using a Graph Laplacian regularizer. Here we propose a combination of these two approaches that can be applied to L1, L2 and LS-SVMs. We also propose an algorithm to iteratively learn the graph adjacency matrix used in the Laplacian regularization. We test our proposal with synthetic and real problems, both in regression and classification settings. When the task structure is present, we show that our model is able to detect it, which leads to better results, and we also show it to be competitive even when this structure is not present. [ABSTRACT FROM AUTHOR]

This work develops and evaluates a self‐navigated variable density spiral (VDS)‐based manifold regularization scheme to prospectively improve dynamic speech magnetic resonance imaging (MRI) at 3 T. Short readout duration spirals (1.3‐ms long) were used to minimize sensitivity to off‐resonance. A custom 16‐channel speech coil was used for improved parallel imaging of vocal tract structures. The manifold model leveraged similarities between frames sharing similar vocal tract postures without explicit motion binning. The self‐navigating capability of VDS was leveraged to learn the Laplacian structure of the manifold. Reconstruction was posed as a sensitivity‐encoding–based nonlocal soft‐weighted temporal regularization scheme. Our approach was compared with view‐sharing, low‐rank, temporal finite difference, extra dimension‐based sparsity reconstruction constraints. Undersampling experiments were conducted on five volunteers performing repetitive and arbitrary speaking tasks at different speaking rates. Quantitative evaluation in terms of mean square error over moving edges was performed in a retrospective undersampling experiment on one volunteer. For prospective undersampling, blinded image quality evaluation in the categories of alias artifacts, spatial blurring, and temporal blurring was performed by three experts in voice research. Region of interest analysis at articulator boundaries was performed in both experiments to assess articulatory motion. Improved performance with manifold reconstruction constraints was observed over existing constraints. With prospective undersampling, a spatial resolution of 2.4 × 2.4 mm2/pixel and a temporal resolution of 17.4 ms/frame for single‐slice imaging, and 52.2 ms/frame for concurrent three‐slice imaging, were achieved. We demonstrated implicit motion binning by analyzing the mechanics of the Laplacian matrix. Manifold regularization demonstrated superior image quality scores in reducing spatial and temporal blurring compared with all other reconstruction constraints. While it exhibited faint (nonsignificant) alias artifacts that were similar to temporal finite difference, it provided statistically significant improvements compared with the other constraints. In conclusion, the self‐navigated manifold regularized scheme enabled robust high spatiotemporal resolution dynamic speech MRI at 3 T. [ABSTRACT FROM AUTHOR]

We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The power graph P (Z n) of Z n for a finite cyclic group Z n is a simple undirected connected graph such that two distinct nodes x and y in Z n are adjacent in P (Z n) if and only if x ≠ y and x i = y or y i = x for some non-negative integer i . In this article, we find the Laplacian eigenvalues of P (Z n) and show that P (Z n) is Laplacian integral (integer algebraic connectivity) if and only if n is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019). [ABSTRACT FROM AUTHOR]

Data are an essential component for building machine learning models. Linearly separable high-quality data are conducive to building efficient classification models. However, the collected dataset is not of high quality, and the number of instances for difference class is not absolutely consistent. Therefore, models built on these datasets are vulnerable to problems such as class-imbalance, class-overlap, and other problems. Traditional instance selection algorithms mainly determine whether there is redundancy or overlap in instances based on the degree of similarity between instances. Therefore, these methods only focus on the local information of the dataset and ignore the global approximate relationship of the instances in the dataset. In this paper, an instance selection method based on the global relationship of instances in the dataset is proposed, called instance selection based on Laplacian matrix decomposition with weighted k-nearest-neighbor graph (LMD-WNG). First, this method tries to construct a new distance-weighted Laplacian matrix using the weighted k-nearest-neighbor graph. Then, the distance-weighted Laplacian matrix is decomposed using a Schur decomposition method. Finally, according to the eigenvalues of the decomposed real matrix, a training dataset suitable for model learning is selected, and a classifier is constructed on the new training data. The experimental results show that as the imbalance ratio increases, LMD-WNG becomes more sensitive to parameter k. When the significance level is p = 0.05 , the analysis results using Friedman ranking and the Holm's post hoc test show that LMD-WNG is significantly better than or similar to other state-of-the-art algorithms on 30 datasets. [ABSTRACT FROM AUTHOR]

In this paper, we proposed several disturbance observer‐based matrix‐weighted consensus algorithms. A new disturbance observer is firstly designed for linear systems with unknown matched or mismatched disturbances representable as the multiplication of a known time‐varying matrix with a unknown constant vector. Under some assumptions on the boundedness and persistent excitation of the regression matrix, the disturbances can be estimated at an exponential rate. Then, a suitable compensation input is provided to compensate the unknown disturbances. Second, disturbance‐observer based consensus algorithms are proposed for matrix‐weighted networks of single‐ and double‐integrators with matched or mismatched disturbances. We show that both matched and mismatched disturbances can be estimated and actively compensated, and the consensus system uniformly globally asymptotically converges to a fixed point in the kernel of the matrix‐weighted Laplacian. Depending on the network connectivity, the system can asymptotically achieve a consensus or a cluster configuration. The disturbance‐observer based consensus design is further extended for a network of higher‐order integrators subjected to disturbances. Finally, simulation results are provided to support the mathematical analysis. [ABSTRACT FROM AUTHOR]

In recent years, with a large number of distributed energy sources connected to the grid, the distribution network has shown a clear trend towards the use of power electronics. As a power conversion interface, the stability of the converter's interaction with the grid has received considerable attention. Adequate admittance (impedance) modeling of the three‐phase grid‐connected converter is a prerequisite for analyzing the stability and resonance characteristics of the grid‐connected system. Understanding the internal mechanisms and proposing effective solutions are crucial to enable power systems to effectively cope with the impact of power electronics. Therefore, by studying the admittance equivalence relationship and equivalent circuit of three‐phase AC systems in various coordinate systems, this paper focuses on solving the non‐uniqueness problem of impedance admittance modeling of three‐phase converter system. It is proposed that the input admittance of the three‐phase asymmetrical system in the dq and fb rotating coordinate systems is a 2‐order matrix and equivalent. The input admittance of the three‐phase unbalanced system in the αβ and “12” static coordinate systems is a 4‐order matrix and equivalent. From both theoretical analysis and simulation verification, it is shown that the number of eigenvalues of the three‐phase asymmetrical system in the stationary coordinate system is twice that in the rotating coordinate system, and the overall damping of the system is improved. The control strategy based on the stationary coordinate system will make the system more stable. The second contribution of this paper is to establish the equivalent circuit of the three‐phase symmetric converter system by the node admittance matrix and extend it to the multi‐machine system to explain the system instability mechanism by the circuit resonance mechanism. Finally, the frequency coupling phenomenon is studied for the three‐phase asymmetrical converter system, and it is explained that it cannot establish an equivalent circuit similar to the three‐phase symmetrical converter system. [ABSTRACT FROM AUTHOR]

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical 'Dirac walk' diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such 'Dirac walks' on the Bethe lattice, a d -regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( d s = 1 for d = 2 and d s = 3 for d ⩾ 3 ). [ABSTRACT FROM AUTHOR]

Uncovering acquired drug resistance mechanisms has garnered considerable attention as drug resistance leads to treatment failure and death in patients with cancer. Although several bioinformatics studies developed various computational methodologies to uncover the drug resistance mechanisms in cancer chemotherapy, most studies were based on individual or differential gene expression analysis. However the single gene-based analysis is not enough, because perturbations in complex molecular networks are involved in anti-cancer drug resistance mechanisms. The main goal of this study is to reveal crucial molecular interplay that plays key roles in mechanism underlying acquired gastric cancer drug resistance. To uncover the mechanism and molecular characteristics of drug resistance, we propose a novel computational strategy that identified the differentially regulated gene networks. Our method measures dissimilarity of networks based on the eigenvalues of the Laplacian matrix. Especially, our strategy determined the networks' eigenstructure based on sparse eigen loadings, thus, the only crucial features to describe the graph structure are involved in the eigenanalysis without noise disturbance. We incorporated the network biology knowledge into eigenanalysis based on the network-constrained regularization. Therefore, we can achieve a biologically reliable interpretation of the differentially regulated gene network identification. Monte Carlo simulations show the outstanding performances of the proposed methodology for differentially regulated gene network identification. We applied our strategy to gastric cancer drug-resistant-specific molecular interplays and related markers. The identified drug resistance markers are verified through the literature. Our results suggest that the suppression and/or induction of COL4A1, PXDN and TGFBI and their molecular interplays enriched in the Extracellular-related pathways may provide crucial clues to enhance the chemosensitivity of gastric cancer. The developed strategy will be a useful tool to identify phenotype-specific molecular characteristics that can provide essential clues to uncover the complex cancer mechanism. [ABSTRACT FROM AUTHOR]

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks, since the possible removal of M edges requires that one compute all the eigenvalues of M distinct matrices. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and has a computational complexity that scales as O(MN) for networks with N nodes. Focusing on a form of VNE that is associated with a transport operator e-βL, where L is a graph Laplacian matrix and β> 0 is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate, and long timescales β for diffusion. [ABSTRACT FROM AUTHOR]

This paper investigates the adaptive consensus with limited information for a class of uncertain nonlinear multi-agent systems (MASs). The limited information, meaning the incomplete information transmitted between agents, stems from nonzero-kernel weight matrices of communication networks, which deserves concerns owing to its practical interest such as in privacy preserving and information compression. Apart from the limited/incomplete information, the paper considers nonlinear MASs with large uncertainties (without known upper bounds). This is essentially different from the existing results of linear systems and poses challenges for the design and analysis of adaptive consensus. As for fully distributed protocol design, dynamic high gains are introduced to compensate for the system uncertainties and for the unavailable global graph information. In closed-loop consensus analysis, a new Lyapunov function is constructed by utilising weight and incidence matrices instead of Laplacian matrix. It turns out that the designed protocol can guarantee the boundedness of closed-loop MASs signals and the convergent consensus. Two numerical examples are given to demonstrate the effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

We extend the notion of balance from the realm of signed and gain graphs to conjugate skew gain graphs which are skew gain graphs where the labels on the oriented edges get conjugated when we reverse the orientation. We characterize the balance in a conjugate skew gain graph in several ways especially by dealing with its adjacency matrix and the g -Laplacian matrix. We also deal with the concept of anti-balance in conjugate skew gain graphs. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a group G is a simple undirected graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we obtain the Laplacian spectrum of the enhanced power graph of certain non-abelian groups, viz. semidihedral, dihedral and generalized quaternion. Also, we obtained the metric dimension and the resolving polynomial of the enhanced power graphs of these groups. At the final part of this paper, we study the distant properties and the detour distant properties, namely: closure, interior, distance degree sequence, eccentric subgraph of the enhanced power graph of semidihedral group, dihedral group and generalized quaternion group, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern the fixed-time consensus control problem for nonlinear leader-following multi-agent systems(MASs). An edge-based adaptive control protocol is presented with part of parameters updated by relative information of system state. Compared with the common node-based adaptive protocols, the proposed edge-based adaptive control protocol assigning different time-varying weights to different communication edges is more accurate. By selecting special Lyapunov functions and theoretical analysis, the obtained conclusions indicate that both sufficient conditions and the settling time of the fixed-time consensus are independent of the system's global information such as the spectrum of Laplacian matrix, which implies that the given protocol is fully distributed. Finally, numerical simulation is illustrated to support the correctness of the main theoretical results. [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]

Let A = (aij) and H = (hij) be positive semidefinite matrices of the same order. If aij ≥ |hij| for all i, j; A is diagonally dominant and all row sums of H are equal to zero, then we show that the sum of all k x k principal minors of A is greater than or equal to the sum of all k x k principal minors of H. [ABSTRACT FROM AUTHOR]

In this paper, the unique graph with the minimal second smallest normalized Laplacian eigenvalue of unicyclic graphs with fixed girth is determined. [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]

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees. [ABSTRACT FROM AUTHOR]