Transfer learning with graph neural networks for pressure estimation in monitoring-limited water distribution networks.

This article addresses the fixed-time leaderless cluster synchronization of spatiotemporal community networks (SCNs) characterized by nonidentical node dynamics and reaction-diffusion feature. First, a signed SCN with reaction-diffusion effect is formulated, where the sign-based coupling is introduced to capture the dynamics of coopetition interactions among different communities. Second, to ensure the invariance of the synchronous manifold, an improved interdegree balance condition is proposed as a prerequisite for achieving cluster synchronization of the community network. Third, based on the local state information from adjacent nodes within each community, a time-limited controller is designed to enhance intracommunity coordination while avoiding the adverse effects of intercommunity competition on synchronization. Subsequently, with the help of the matrix decomposition technique and a Lyapunov-like method, several flexible leaderless cluster synchronization criteria are derived by establishing a nontrivial integral inequality and key properties of the intracommunity Laplacian matrix. Finally, the theoretical results are substantiated through a numerical example.

Abstract Knowledge graph-based recommendation systems have strong capabilities in deep association mining and structured reasoning, which effectively alleviate data sparsity and cold-start problems in recommendation. However, traditional knowledge graph-based recommendation considers each graph relation in isolation, failing to capture the potential semantic correlations between different relation types, which leads to incomplete semantic representations. In addition, existing methods generally ignore negative feedback signals in users’ historical interactions, resulting in biased preference modeling. To overcome the above challenges, we introduce S ign-aware R ecommendation based on V irtual S emantic K nowledge G raph(SRVSKG), which enhances recommendation performance by combining virtual semantic collaborative representations with sign-aware learning techniques. Firstly, we propose a virtual semantic subgraph collaborative representation module to learn user and item embeddings. In this module, we build virtual semantic subgraphs through relation clustering based on latent semantic similarity measurement. Then a hierarchical feature extraction mechanism based on graph attention network is applied to virtual semantic subgraphs, which captures semantic associations across relations and enriches item embedding. At the same time, we emphasize user preference for attributes to enrich user embedding. Secondly, we design a sign-aware learning module, which constructs a user-item signed graph, applies Laplacian matrix factorization to simultaneously model the topological features of positive and negative feedback, and introduces the transformer architecture to dynamically fuse signed information, effectively utilizing negative feedback information to eliminate bias in user preference modeling. Finally, we establish a multi-feature fusion mechanism that deeply combines features from the virtual semantic subgraph collaborative representation module and the sign-aware learning module to enable recommendation. Experiment results on three public datasets demonstrate that SRVSKG outperforms state-of-the-art recommendation baselines.

This research paper introduces a mathematical model and control framework for coordinating multiple drones (a swarm) using a Proportional–Derivative (PD) control approach grounded in graph theory. Each drone functions as an independent agent, communicating only with its nearby neighbors. This local interaction creates a decentralized and scalable system that eliminates the need for a central controller. The inter-drone connectivity and interactions are represented through the Laplacian matrix, while eigenvalue analysis helps in designing the control gains to ensure smooth and stable motion.The model is formulated using Newton’s equations of motion, expressed through differential equations that describe each drone’s dynamic behavior. The proposed approach is tested on a five-drone ring formation, demonstrating that the drones can maintain formation, avoid collisions, and reach the target position with stable convergence.

We propose a quantum algorithm for calculating the structural properties of complex networks and graphs. The corresponding protocol——is designed to perform large-scale community and botnet detection, where a specific subgraph of a larger graph is identified based on its properties. We construct a workflow relying on ground state preparation of the network modularity matrix or graph Laplacian. The corresponding maximum modularity vector is encoded into a log 2 ( N ) -qubit register that contains community information. We develop a strategy for “signing” this vector via quantum signal processing, such that it closely resembles a hypergraph state, and project it onto a suitable linear combination of such states to detect botnets. As part of the workflow, and of potential independent interest, we present a readout technique that allows filtering out the incorrect solutions deterministically. This can reduce the scaling for the number of samples from exponential to polynomial. The approach serves as a building block for graph analysis with quantum speedup and enables the cybersecurity of large-scale networks.

Let [Formula: see text] be a commutative ring with unity. The essential ideal graph [Formula: see text] of [Formula: see text] is a graph in which the vertex set comprises of set of all nonzero proper ideals of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we discuss the structure of an induced subgraph of the essential ideal graph of the ring [Formula: see text] as a [Formula: see text]-generalized join graph and thereby completely determine the structure of [Formula: see text]. Also, we prove a characterization of [Formula: see text] to be Laplacian integral in terms of the vertex-weighted Laplacian matrix of annihilating ideal graph of [Formula: see text] for [Formula: see text], [Formula: see text] are different prime numbers. Further, we estimate the eigenvalues of various matrices like adjacency matrix, Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the induced subgraph of the essential ideal graph of [Formula: see text]. Finally, we obtain the upper bounds of spectral radius and algebraic connectivity of [Formula: see text] and compute the values of [Formula: see text] for which these bounds are attained.

Graph theory plays a fundamental role in various fields of science and engineering, providing powerful tools for modeling and analyzing relationships among entities. One of the most effective ways to study graphs is through matrix representation. This paper explores the three primary matrix representations of graphs: the adjacency matrix, The adjacency matrix provides direct insight into vertex connectivity and the incidence matrix, the incidence matrix reflects the relationship between edges and vertices. and the Laplacian matrix defined as the difference between the degree matrix and the adjacency matrix, plays a central role in spectral graph theory. Matrix representations enable efficient storage, computation, and analysis of graphs using linear algebraic techniques. They form the basis for many modern algorithms in graph theory, This paper discusses the mathematical foundations, construction methods, and practical applications of these matrix forms, highlighting their essential role in both theoretical and applied graph analysis.

Spectral Analysis of Normalized Hermitian Laplacian Matrices in Random Mixed Graphs

Multi-scale signed graph convolutional network based on framelet.

The Gaussian network model (GNM) has been successful in explaining protein dynamics by modeling proteins as elastic networks of alpha carbons connected by harmonic springs. However, its uniform interaction assumption and neglect of higher-order correlations limit its accuracy in predicting experimental B-factors and residue cross-correlations critical for understanding allostery and information transfer. This study introduces an information-theoretic enhancement to the GNM, incorporating mutual information-based corrections to the Kirchhoff matrix to account for multi-body interactions and contextual residue dynamics. By iteratively optimizing B-factor predictions and applying a Monte Carlo-driven maximum entropy approach to refine covariances, our method achieves significant improvements, reducing RMSDs between predicted and experimental B-factors by 26%-46% across nine representative proteins. The model contextualizes residue assignments based on local density, solvent exposure, and allosteric roles, showing complex dynamic patterns beyond simple neighbor counts. Enhanced predictions of mutual information and entropy perturbations in proteins like KRAS improve the identification of spanning trees containing key residues, which may correspond to allosteric communication pathways. This evolvable framework, capable of incorporating additional effects and utilizing contextual residue assignments, enables precise studies of mutation effects on protein dynamics, with improved cross-correlation predictions potentially increasing accuracy in drug design and function prediction.

The Behavioral Inhibition and Activation Systems (BIS/BAS) are central to emotional regulation, yet their genetic and neural mechanisms remain unclear. In this study, we examined how polygenic risk for BIS/BAS relates to striatal structural connectivity and emotional vulnerability during adolescence—a critical period for affective problems onset. Using data from 1,929 adolescents (average age 9.95 years, 50.39% female) in the ABCD Study, we found that BIS and BAS polygenic risks exerted dissociable influences on striatal structural gradient (SSG), delineating distinct neurobiological pathways to anxiety and depression. Elevated BIS risk was linked to greater anxiety symptoms, whereas increased BAS risk related more strongly to depressive and withdrawn features. Altogether, our findings elucidate a neuroanatomical pleiotropy mechanism, whereby shared striatal connectivity architectures differentially channel genetic susceptibilities into distinct symptom dimensions, thereby advancing our understanding of the neural pathways underlying emotional vulnerability across diverse psychiatric phenotypes.(a) Construction of PRS. Variants associated with BIS and BAS were extracted from GWAS data. Effect weights were calculated, and a weighted sum was performed to derive the PRS scores, with detailed procedures outlined in the Methods section. (b) Framework for striatal structural connectopic mapping. Probabilistic tractography was conducted from the striatal volumetric seed region. The resulting connectivity matrix, denoted as A, was subjected to dimensionality reduction using Singular Value Decomposition (SVD), resulting in matrix B. Subsequently, the similarity matrix S was computed utilizing the η² Coefficient. The graph's Laplacian was then decomposed into its eigenvectors, which correspond to the connectopic maps of the seed region. (c) Simplified Schematic of a Multivariate Model Investigating Gene–Brain–Behavior Correlations. Green arrows indicate mediation pathways involving BIS PRS. Orange dashed arrows indicate mediation pathways involving BAS PRS. Grey arrows denote significant associations between variables without mediation. GWAS, Genome-Wide Association Study. PRS, Polygenic Risk Scores. BIS, Behavioral Inhibition System. BAS, Behavioral Activation System. BAS_rr, BAS Reward Responsiveness.

This research compares Graph Neural Networks reduction techniques for computational modeling in structural engineering. The proposed Finite Element Shape Logic Graph technology serves as a benchmark, capturing intricate details through fully connected logic graphs derived from Finite Element Models. Multiple graph reduction methods are introduced, each addressing computational efficiency: shortest path connectivity, Laplacian Matrix for Dimensionality Reduction, and an approach exploiting graph sparsity. This research contributes to structural engineering and offers insight for diverse applications in the evolving landscape of digital twins, particularly within the specific context of graph optimization and the efficiency of physical neural networks. Our proposed techniques are tested on datasets consisting of four different finite element models. Our study suggests that, while the shortest path method is a promising starting point for graph reduction in mechanical system modeling, leveraging Laplacian spectral reduction techniques is essential for achieving optimal solutions, particularly for larger finite element models. This approach served as the basis for the development of digital twins of mechanical structures, ensuring accurate representation and efficient analysis of complex engineering systems.

The primary objective of this study was to examine changes in brain network architecture across multiple frequency bands using spectral analysis of both weighted and binarized functional connectivity networks. This cross-sectional observational study, conducted as a secondary analysis of a publicly available EEG dataset, analyzed spectral coherence measurements from 25 patients with Alzheimer's disease (AD) and 25 age- and sex-matched healthy controls (HC). Nevertheless, the modest sample size and cultural homogeneity of the dataset may limit the statistical power and generalizability of the results. A data-driven thresholding approach was employed to generate binary networks, allowing a robust comparison of connectivity disruptions associated with AD. Brain network features derived from the graph Laplacian, including weighted Fiedler value, spectral range, and Middle Eigenvalue, were analyzed across seven frequency layers: delta, theta, alpha1, alpha2, beta1, beta2, and gamma. For binary networks, the Fiedler value was calculated after thresholding. Statistical group comparisons between AD and HC were performed using t-tests (p < 0.05), and each feature was assessed based on the number of frequency bands showing significant differences. Among all features, the weighted Fiedler value was the most discriminative, showing significant reductions in AD patients within the alpha2 and beta1 bands. In binary networks, the Fiedler value remained significantly lower in AD within the alpha2 band, confirming topological degradation even without edge weight information. Other spectral features showed similar trends, but did not reach statistical significance in the binary networks. The consistent decline in Fiedler value across both weighted and binary networks indicates a global reduction in connectivity characteristic of AD. These spectral markers offer a quantitative and interpretable framework for understanding the progressive disconnection syndrome in AD. This study demonstrates significant alterations in Laplacian spectral features of brain networks between the AD and HC groups across specific frequency bands. These exploratory findings indicate that the spectral features, particularly the Fiedler value, consistently differentiate AD patients from healthy controls across frequency bands, suggesting its potential as a biomarker. However, larger and longitudinal studies are needed to confirm its diagnostic and prognostic utility. The combined use of weighted and binarized connectivity matrices enhances analytical sensitivity and facilitates the application of spectral graph theory for the early detection and monitoring of AD.

Spatial transcriptomics allows researchers to visualize and analyze gene expression within the precise location of tissues or cells. It provides spatially resolved gene expression data but often lacks cellular resolution, necessitating cell type deconvolution to infer cellular composition at each spatial location. In this paper we propose BASIN for cell type deconvolution, which models deconvolution as a nonnegative matrix factorization (NMF) problem incorporating graph Laplacian prior. Rather than find a deterministic optima like other recent methods, we propose a matrix variate Bayesian NMF method with nonnegativity and sparsity priors, in which the variables are maintained in their matrix form to derive a more efficient matrix normal posterior. BASIN employs a Gibbs sampler to approximate the posterior distribution of cell type proportions and other parameters, offering a distribution of possible solutions, enhancing robustness and providing inherent uncertainty quantification. The performance of BASIN is evaluated on different spatial transcriptomics datasets and outperforms other deconvolution methods in terms of accuracy and efficiency. The results also show the effect of the incorporated priors and reflect a truncated matrix normal distribution as we expect.

Accurate differentiation between Left Bundle Branch Block (LBBB) and its strict subtype (sLBBB) is essential for optimizing patient selection for Cardiac Resynchronization Therapy (CRT), yet remains clinically challenging. This study proposes and compares two graph-theory-based pipelines for automated classification of 12-lead electrocardiograms (ECGs) into Healthy, LBBB, and sLBBB categories. Functional connectivity graphs were constructed from inter-lead measures, including Pearson correlation, cross-correlation, and phase difference. The first approach combines Graph Signal Processing (GSP) with machine learning. Graph filtering was performed via spectral decomposition of the Laplacian matrix, selecting dominant eigenmodes and reconstructing signals through the inverse Graph Fourier Transform—integrating spatial and temporal features. The second approach converted connectivity matrices into grayscale images, classified using a Convolutional Neural Network (CNN), and incorporated Explainable AI (XAI) via Grad-CAM to visualize inter-lead interactions and enhance model transparency. The GSP-based method using phase difference and a Support Vector Machine achieved the highest performance (mean balanced accuracy = 0.8317 ), while the CNN-based approach with cross-correlation images reached 0.7646 , offering improved interpretability. Both methods distinguished pathological from healthy cases, but precise classification between LBBB and sLBBB remains challenging. These results highlight the complementary value of graph-based ECG analysis and support future hybrid models for CRT stratification.

Let $G$ be a simple undirected graph, $\theta(G)$ be the circuit rank of $G$, $\eta_M(G)$ be the nullity of a graph matrix $M(G)$, and $m_M(G,\lambda)$ be the multiplicity of eigenvalue $\lambda$ of $M(G)$. In the case $M(G)$ is the adjacency matrix $A(G)$, (the Laplacian matrix $L(G)$, or the signless Laplacian matrix $Q(G)$) we find bounds to $m_M(G,\lambda)$ in terms of $\theta(G)$ when $\lambda$ is an integer (even integer, respectively). We also demonstrate that when $\alpha$ and $\lambda$ are rational numbers, similar bounds can be obtained for $m_{A_{\alpha}}(G,\lambda)$, where $A_{\alpha}(G)$ is the generalized adjacency matrix of $G$. Distinctively, our bounds involve only $\theta(G)$, not a multiple of it. Previous bounds for $m_A(G,\lambda)$ (and later $m_{A_\alpha}(G,\lambda)$) in terms of the circuit rank have all included $2\theta(G)$ with the sole exception of the case $\lambda=0$. Wong et al. (2022) showed that $\eta_A(G_c)\leq \theta(G_c)+1$, where $G_c$ is a connected cactus whose blocks are even cycles. Our result, in particular, generalizes and extends this result to the multiplicity of any even eigenvalue of A(G) of any even connected graph $G$, as well as to any even eigenvalue of $L(G)$ and $Q(G)$ for any connected graph $G$. They also showed that $\eta_A(G_c)\leq 1$ when every block of the cactus is an odd cycle. This also aligns with a special case of our bound.

Let G be a connected threshold graph. We derive a concise formula to find the eigenvalues of the distance Laplacian matrix of G and show that this formula characterizes all connected threshold graphs. Additionally, by this formula, we give a simple procedure to determine the number of dominant and pendant vertices of G. We then classify all the dominant vertices of G from the eigenvectors of the distance Laplacian matrix. Finally, we prove that, among all connected threshold graphs, the coefficients of the characteristic polynomial of the distance Laplacian matrix attain the minimum and maximum values at the complete graph and the star tree, respectively.

Threshold graphs are generated from one node by repeatedly adding a node that links to all existing nodes or adding a node without links. In the weighted threshold graph, we add a new node in step $i$, which is linked to all existing nodes by a link of weight $w_i$. In this work, we consider the set ${\cal A}_N$ that contains all Laplacian matrices of weighted threshold graphs of order $N$. We show that ${\cal A}_N$ forms a commutative algebra. Using this, we find a common basis of eigenvectors for the matrices in ${\cal A}_N$. It follows that the eigenvalues of each matrix in ${\cal A}_N$ can be represented as a linear transformation of the link weights. In addition, we prove that, if there are just three or fewer different weights, two weighted threshold graphs with the same Laplacian spectrum must be isomorphic.

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are \textsl{degree similar} if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-\mu D_1$ and $A_2-\mu D_2$ are similar over the field of rational functions $\mathbb{Q}(\mu)$ if and only if the Smith normal forms of the matrices $tI-(A_1-\mu D_1)$ and $tI-(A_2-\mu D_2)$ are equal.

A ring polymer is a random walk whose steps obey a single linear condition; their sum vanishes. Factoring the graph Laplacian into the product of the incidence matrix and its transpose allows us to show that for a more complicated network, the steps must lie in a linear subspace determined by the graph topology. This provides a useful new perspective on the James-Guth theory of phantom elastic networks. In particular, we formulate phantom networks which are free from the constraints of fixed crosslinks. For a given network the solution of the loop constraints makes the partition function finite-valued in the path integral formulation without applying any external forces or fixing any monomer positions. The resulting probability distribution on edge displacements is rotationally invariant, which is practically quite useful for generating unbiased random samples of edge displacements and monomer positions. Furthermore, one can exactly calculate many physical quantities such as correlation functions with respect to this distribution. Finally, this reformulation lends itself well to the case of non-Gaussian distributions. We illustrate this by computing the expected radius of gyration of a ring polymer in a wide variety of models.