Weighted Adjacency Matrices Research Articles

Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs

Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs

Unified approach for spectral properties of weighted adjacency matrices for graphs with degree-based edge-weights

Let G be a graph and di be the degree of a vertex vi in G. For a symmetric real function f(x,y), one can get an edge-weighted graph in such a way that for each edge vivj of G, the weight of vivj is assigned by f(di,dj). Hence, we have a weighted adjacency matrix Af(G) of G, in which the ij-entry is equal to f(di,dj) if vivj∈E(G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of Af(G) for f(x,y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights.

Transfer-Aware Graph U-Net with Cross-Level Interactions for PolSAR Image Semantic Segmentation

Although graph convolutional networks have found application in polarimetric synthetic aperture radar (PolSAR) image classification tasks, the available approaches cannot operate on multiple graphs, which hinders their potential to generalize effective feature representations across different datasets. To overcome this limitation and achieve robust PolSAR image classification, this paper proposes a novel end-to-end cross-level interaction graph U-Net (CLIGUNet), where weighted max-relative spatial convolution is proposed to enable simultaneous learning of latent features from batch input. Moreover, it integrates weighted adjacency matrices, derived from the symmetric revised Wishart distance, to encode polarimetric similarity into weighted max-relative spatial graph convolution. Employing end-to-end trainable residual transformers with multi-head attention, our proposed cross-level interactions enable the decoder to fuse multi-scale graph feature representations, enhancing effective features from various scales through a deep supervision strategy. Additionally, multi-scale dynamic graphs are introduced to expand the receptive field, enabling trainable adjacency matrices with refined connectivity relationships and edge weights within each resolution. Experiments undertaken on real PolSAR datasets show the superiority of our CLIGUNet with respect to state-of-the-art networks in classification accuracy and robustness in handling unknown imagery with similar land covers.

The Index Weighted Hermitian Adjacency Matrices for Mixed Graphs

Based on the Hermitian adjacency matrices of second kind introduced by Mohar [1] and weighted adjacency matrices introduced in [2], we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.

Unveiling Gene Regulatory Networks That Characterize Difference of Molecular Interplays Between Gastric Cancer Drug Sensitive and Resistance Cell Lines.

Gastric cancer is a leading cause of cancer-related deaths globally and chemotherapy is widely accepted as the standard treatment for gastric cancer. However, drug resistance in cancer cells poses a significant obstacle to the success of chemotherapy, limiting its effectiveness in treating gastric cancer. Although many studies have been conducted to unravel the mechanisms of acquired drug resistance, the existing studies were based on abnormalities of a single gene, that is, differential gene expression (DGE) analysis. Single gene-based analysis alone is insufficient to comprehensively understand the mechanisms of drug resistance in cancer cells, because the underlying processes of the mechanism involve perturbations of the molecular interactions. To uncover the mechanism of acquired gastric cancer drug resistance, we perform for identification of differentially regulated gene networks between drug-sensitive and drug-resistant cell lines. We develop a computational strategy for identifying phenotype-specific gene networks by extending the existing method, CIdrgn, that quantifies the dissimilarity of gene networks based on comprehensive information of network structure, that is, regulatory effect between genes, structure of edge, and expression levels of genes. To enhance the efficiency of identifying differentially regulated gene networks and improve the biological relevance of our findings, we integrate additional information and incorporate knowledge of network biology, such as hubness of genes and weighted adjacency matrices. The outstanding capabilities of the developed strategy are validated through Monte Carlo simulations. By using our strategy, we uncover gene regulatory networks that specifically capture the molecular interplays distinguishing drug-sensitive and drug-resistant profiles in gastric cancer. The reliability and significance of the identified drug-sensitive and resistance-specific gene networks, as well as their related markers, are verified through literature. Our analysis for differentially regulated gene network identification has the capacity to characterize the drug-sensitive and resistance-specific molecular interplays related to mechanisms of acquired drug resistance that cannot be revealed by analysis based solely on abnormalities of a single gene, for example, DGE analysis. Through our analysis and comprehensive examination of relevant literature, we suggest that targeting the suppressors of the identified drug-resistant markers, such as the Melanoma Antigen (MAGE) family, Trefoil Factor (TFF) family, and Ras-Associated Binding 25 (RAB25), while enhancing the expression of inducers of the drug sensitivity markers [e.g., Serum Amyloid A (SAA) family], could potentially reduce drug resistance and enhance the effectiveness of chemotherapy for gastric cancer. We expect that the developed strategy will serve as a useful tool for uncovering cancer-related phenotype-specific gene regulatory networks that provide essential clues for uncovering not only drug resistance mechanisms but also complex biological systems of cancer.

A graph theory based similarity metric enables comparison of subpopulation psychometric networks.

Network psychometrics leverages pairwise Markov random fields to depict conditional dependencies among a set of psychological variables as undirected edge-weighted graphs. Researchers often intend to compare such psychometric networks across subpopulations, and recent methodological advances provide invariance tests of differences in subpopulation networks. What remains missing, though, is an analogue to an effect size measure that quantifies differences in psychometric networks. We address this gap by complementing recent advances for investigating whether psychometric networks differ with an intuitive similarity measure quantifying the extent to which networks differ. To this end, we build on graph-theoretic approaches and propose a similarity measure based on the Frobenius norm of differences in psychometric networks' weighted adjacency matrices. To assess this measure's utility for quantifying differences between psychometric networks, we study how it captures differences in subpopulation network models implied by both latent variable models and Gaussian graphical models. We show that a wide array of network differences translates intuitively into the proposed measure, while the same does not hold true for customary correlation-based comparisons. In a simulation study on finite-sample behavior, we show that the proposed measure yields trustworthy results when population networks differ and sample sizes are sufficiently large, but fails to identify exact similarity when population networks are the same. From these results, we derive a strong recommendation to only use the measure as a complement to a significant test for network similarity. We illustrate potential insights from quantifying psychometric network similarities through cross-country comparisons of human values networks. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Graph theory applications in congenital heart disease

Graph theory can be used to address problems with complex network structures. Congenital heart diseases (CHDs) involve complex abnormal connections between chambers, vessels, and organs. We proposed a new method to represent CHDs based on graph theory, wherein vertices were defined as the spaces through which blood flows and edges were defined by the blood flow between the spaces and direction of the blood flow. The CHDs of tetralogy of Fallot (TOF) and transposition of the great arteries (TGA) were selected as examples for constructing directed graphs and binary adjacency matrices. Patients with totally repaired TOF, surgically corrected d-TGA, and Fontan circulation undergoing four-dimensional (4D) flow magnetic resonance imaging (MRI) were included as examples for constructing the weighted adjacency matrices. The directed graphs and binary adjacency matrices of the normal heart, extreme TOF undergoing a right modified Blalock–Taussig shunt, and d-TGA with a ventricular septal defect were constructed. The weighted adjacency matrix of totally repaired TOF was constructed using the peak velocities obtained from 4D flow MRI. The developed method is promising for representing CHDs and may be helpful in developing artificial intelligence and conducting future research on CHD.

Some interlacing results on weighted adjacency matrices of graphs with degree-based edge-weights

Some interlacing results on weighted adjacency matrices of graphs with degree-based edge-weights

Altered longitudinal structural connectome in paediatric mild traumatic brain injury: an Advancing Concussion Assessment in Paediatrics study.

Advanced diffusion-weighted imaging techniques have increased understanding of the neuropathology of paediatric mild traumatic brain injury (i.e. concussion). Most studies have examined discrete white-matter pathways, which may not capture the characteristically subtle, diffuse and heterogenous effects of paediatric concussion on brain microstructure. This study compared the structural connectome of children with concussion to those with mild orthopaedic injury to determine whether network metrics and their trajectories across time post-injury differentiate paediatric concussion from mild traumatic injury more generally. Data were drawn from of a large study of outcomes in paediatric concussion. Children aged 8-16.99 years were recruited from five paediatric emergency departments within 48 h of sustaining a concussion (n = 360; 56% male) or mild orthopaedic injury (n = 196; 62% male). A reliable change score was used to classify children with concussion into two groups: concussion with or without persistent symptoms. Children completed 3 T MRI at post-acute (2-33 days) and/or chronic (3 or 6 months, via random assignment) post-injury follow-ups. Diffusion-weighted images were used to calculate the diffusion tensor, conduct deterministic whole-brain fibre tractography and compute connectivity matrices in native (diffusion) space for 90 supratentorial regions. Weighted adjacency matrices were constructed using average fractional anisotropy and used to calculate global and local (regional) graph theory metrics. Linear mixed effects modelling was performed to compare groups, correcting for multiple comparisons. Groups did not differ in global network metrics. However, the clustering coefficient, betweenness centrality and efficiency of the insula, cingulate, parietal, occipital and subcortical regions differed among groups, with differences moderated by time (days) post-injury, biological sex and age at time of injury. Post-acute differences were minimal, whereas more robust alterations emerged at 3 and especially 6 months in children with concussion with persistent symptoms, albeit differently by sex and age. In the largest neuroimaging study to date, post-acute regional network metrics distinguished concussion from mild orthopaedic injury and predicted symptom recovery 1-month post-injury. Regional network parameters alterations were more robust and widespread at chronic timepoints than post-acutely after concussion. Results suggest that increased regional and local subnetwork segregation (modularity) and inefficiency occurs across time after concussion, emerging after post-concussive symptom resolve in most children. These differences persist up to 6 months after concussion, especially in children who showed persistent symptoms. While prognostic, the small to modest effect size of group differences and the moderating effects of sex likely would preclude effective clinical application in individual patients.

Extremal trees and unicyclic graphs with respect to spectral radius of weighted adjacency matrices with property $$P^{*}$$

Extremal trees and unicyclic graphs with respect to spectral radius of weighted adjacency matrices with property $$P^{*}$$

Improving fraud detection via hierarchical attention-based Graph Neural Network

Fraud has seriously influenced the social media ecosystems, and malicious users pursue high profit by disseminating fake information. Graph neural networks (GNN) have shown a promising potential for fraud detection tasks, where fraudulent nodes are identified by aggregating the neighbors that share similar feedbacks and relations. However, crafty fraudsters can trivially get around such detection via seemingly legitimate feedbacks once connected to legitimate users. In this paper, we leverage Relational Density Theory and propose a Hierarchical Attention-based Graph Neural Network (HA-GNN) for fraud detection, which incorporates weighted adjacency matrices across different relations against camouflage. This is motivated by the fact that there are dense connections between fraudsters who collectively participate in fraud activities. Specifically, we design a relation attention module to reflect the tie strength between two nodes, while a neighborhood attention module to capture the long-range structural affinity associated with the graph. We generate node embeddings by aggregating information from local/long-range structures and original node features. Experiments on three real-world datasets demonstrate that our approach achieves 3.21 – 9.97% RUC improvement compared with the state-of-the-arts.

UAV Formation Control under Fixed and Variable Adjacency Based Directed Network Topologies

The UAV formation control is one of the key aspects in several applications like surveillance, moving target tracking, load-transportation, and delivery systems etc. These situations demand the multiple UAVs to manoeuvre in a desired formation. To address this problem, a distributed formation control scheme is proposed incorporating the details about the state of the neighbouring UAVs. The communication network topology among the UAVs is considered to be directed with the constant and the weighted adjacency matrices. The nonholonomic constraints are considered while deriving the desired Euler angles. Satisfying the conditions of Lyapunov provides necessary proof of stability along the positional and the attitude subsystems. Simulation results demonstrate that the desired tetrahedron, octahedron, and cube shapes are attained and maintained by the UAVs successfully. Also, the designed formation paradigm works proficiently for both the constant and the weighted adjacency matrices based directed network topologies. The performance validation is done through extensive comparative analysis for varying network connections.

Spectral Clustering via Adaptive Layer Aggregation for Multi-Layer Networks

One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches based on effective convex layer aggregations. Our aggregation methods are strongly motivated by a delicate asymptotic analysis of the spectral embedding of weighted adjacency matrices and the downstream k-means clustering, in a challenging regime where community detection consistency is impossible. In fact, the methods are shown to estimate the optimal convex aggregation, which minimizes the misclustering error under some specialized multi-layer network models. Our analysis further suggests that clustering using Gaussian mixture models is generally superior to the commonly used k-means in spectral clustering. Extensive numerical studies demonstrate that our adaptive aggregation techniques, together with Gaussian mixture model clustering, make the new spectral clustering remarkably competitive compared to several popularly used methods. Supplementary materials for this article are available online.

A graph neural network framework based on preference-aware graph diffusion for recommendation.

Transforming user check-in data into graph structure data is a popular and powerful way to analyze users' behaviors in the field of recommendation. Graph-based deep learning methods such as graph embeddings and graph neural networks have shown promising performance on the task of point-of-interest recommendation in recent years. Despite effectiveness, existing methods fail to capture deep graph structural information, leading the suboptimal representations. In addition, they lack the ability of learning the influences of both global preference and user preference on the check-in behavior. To address the aforementioned issues, we propose a general framework based on preference-aware graph diffusion, named PGD. We first construct two types of graphs to represent the global preference and user preference. Then, we apply a graph diffusion process to capture the structural information of the generated graphs, resulting in weighted adjacency matrices. Finally, graph neural network-based backbones are introduced to learn the representations of users and POIs on weighted adjacency matrices. A learnable aggregation module is developed to learn the final representations from global preference and user preference adaptively. Extensive experiments on four real-world datasets demonstrate the superiority of PGD on POI recommendation, compared with the mainstream graph-based deep learning methods.

Non-Bayesian Social Learning on Random Digraphs With Aperiodically Varying Network Connectivity

In this article, we study non-Bayesian social learning on random directed graphs and show that under mild connectivity assumptions, all the agents almost surely learn the true state of the world asymptotically in time if the sequence of the associated weighted adjacency matrices belongs to Class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {P}^*$</tex-math></inline-formula> (a broad class of stochastic chains that subsumes uniformly strongly connected chains). We show that uniform strong connectivity, while being unnecessary for asymptotic learning, ensures that all the agents’ beliefs converge to a consensus almost surely, even when the true state is not identifiable. We then provide a few corollaries of our main results, some of which apply to the variants of the original update rule, such as inertial non-Bayesian learning and learning via diffusion and adaptation. Others include the extensions of known results on social learning. We also show that if the network of influences is balanced in a certain sense, then asymptotic learning occurs almost surely even in the absence of uniform strong connectivity.

Deep deterministic policy gradient and graph convolutional network for bracing direction optimization of grid shells

In this paper, we propose a method for bracing direction optimization of grid shells using a Deep Deterministic Policy Gradient (DDPG) and Graph Convolutional Network (GCN). DDPG allows simultaneous adjustment of variables during the optimization process, and GCN allows the DDPG agent to receive data representing the whole structure to determine its actions. The structure is interpreted as a graph where nodes, element properties, and internal forces are represented by the node feature matrix, adjacency matrices, and weighted adjacency matrices. DDPG agent is trained to optimize the bracing directions. The trained agent can find sub-optimal solutions with moderately small computational cost compared to the genetic algorithm. The trained agent can also be applied to structures with different sizes and boundary conditions without retraining. Therefore, when various types of braced grid shells have to be considered in the design process, the proposed method can significantly reduce computational cost for structural analysis.

An efficient supply management in water flow network using graph spectral techniques.

An efficient way of distributing water in urban cities is one of the major global challenges of our time. A Water Distribution Network (WDN) is a vital infrastructure, intended to provide fresh water to households within a city or designated boundary. Given the WDN's complexity, effective numerical techniques are required to aid in the development of an ideal monitoring system. Flow meters and gate valves should be placed in low-connected areas of the WDN with water flow reaching several regions of the network. This study proposes a general strategy for assisting water utilities in making decisions for effective water supply. The aim of the research is to use weighted spectral clustering algorithms to outline water districts while addressing hydraulic restrictions via weighted adjacency and Laplacian matrices of the weighted network. This work aims to identify influenced nodes in the network based on Eigen centrality and effectively distribute water across those nodes. This project also looks at how to measure the network's connection strength to avoid water leaks. The best clusters are found using Eigenvalues and Eigenvectors of weighted adjacency matrices and Laplacian matrices of the water network in the proposed graph spectral framework. In order to establish the optimum water network division approach, topological and graph metrics were used to compare multiple spectral clustering techniques. The proposed graph spectral approach is tested using a genuine water distribution network serving an urban area of Coimbatore city in India and offering a method for partitioning complex networks that employ the spectral graph partitioning algorithm.

Multisite Harmonization of Structural DTI Networks in Children: An A-CAP Study.

The analysis of large, multisite neuroimaging datasets provides a promising means for robust characterization of brain networks that can reduce false positives and improve reproducibility. However, the use of different MRI scanners introduces variability to the data. Managing those sources of variability is increasingly important for the generation of accurate group-level inferences. ComBat is one of the most promising tools for multisite (multiscanner) harmonization of structural neuroimaging data, but no study has examined its application to graph theory metrics derived from the structural brain connectome. The present work evaluates the use of ComBat for multisite harmonization in the context of structural network analysis of diffusion-weighted scans from the Advancing Concussion Assessment in Pediatrics (A-CAP) study. Scans were acquired on six different scanners from 484 children aged 8.00–16.99 years [Mean = 12.37 ± 2.34 years; 289 (59.7%) Male] ~10 days following mild traumatic brain injury (n = 313) or orthopedic injury (n = 171). Whole brain deterministic diffusion tensor tractography was conducted and used to construct a 90 x 90 weighted (average fractional anisotropy) adjacency matrix for each scan. ComBat harmonization was applied separately at one of two different stages during data processing, either on the (i) weighted adjacency matrices (matrix harmonization) or (ii) global network metrics derived using unharmonized weighted adjacency matrices (parameter harmonization). Global network metrics based on unharmonized adjacency matrices and each harmonization approach were derived. Robust scanner effects were found for unharmonized metrics. Some scanner effects remained significant for matrix harmonized metrics, but effect sizes were less robust. Parameter harmonized metrics did not differ by scanner. Intraclass correlations (ICC) indicated good to excellent within-scanner consistency between metrics calculated before and after both harmonization approaches. Age correlated with unharmonized network metrics, but was more strongly correlated with network metrics based on both harmonization approaches. Parameter harmonization successfully controlled for scanner variability while preserving network topology and connectivity weights, indicating that harmonization of global network parameters based on unharmonized adjacency matrices may provide optimal results. The current work supports the use of ComBat for removing multiscanner effects on global network topology.

A Graph Convolutional Incorporating GRU Network for Landslide Displacement Forecasting Based on Spatiotemporal Analysis of GNSS Observations

Landslide displacement prediction is crucial for the early warning of slope failure but remains a challenging task due to its spatiotemporal complexity. Although temporal dependency has been well studied and discussed, spatial dependence is relatively less explored due to its significant variations of the spatial structure of landslides. In this study, a novel graph convolutional incorporating GRU network (GC-GRU-N) is proposed and applied to landslide displacement forecasts. The model conducts attribute-augmented graph convolution (GC) operations on GNSS displacement data with weighted adjacency matrices and an attribute-augmented unit to combine features, including the displacements, the distance, and other external influence factors to capture spatial dependence. The output of multi-weight graph convolution is then applied to the gated recurrent unit (GRU) network to learn temporal dependencies. The related optimal hyper-parameters are determined by comparison experiments. When applied to two typical landslide sites in the Three Gorge Reservoir (TGR), China, GC-GRU-N outperformed the comparative models in both cases. The ablation experiment results also show that the attribute augmentation, which considers external factors of landslide displacement, can further improve the model’s prediction performance. We conclude that the GC-GRU-N model can provide robust landslide displacement forecasting with high efficiency.

Spectral analysis of weighted Laplacians arising in data clustering

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. The link between a three-parameter family of graph Laplacians and a three-parameter family of differential operators is explained. The spectral properties of these differential operators are analyzed in the situation where the data comprises of two nearly separated clusters, in a sense which is made precise. In particular, we investigate how the spectral gap depends on the three parameters entering the graph Laplacian, and on a parameter measuring the size of the perturbation from the perfectly clustered case. Numerical results are presented which exemplify the analysis and which extend it in the following ways: the computations study situations in which there are two nearly separated clusters, but which violate the assumptions used in our theory; situations in which more than two clusters are present, also going beyond our theory; and situations which demonstrate the relevance of our studies of differential operators for the understanding of finite data problems via the graph Laplacian. The findings provide insight into parameter choices made in learning algorithms which are based on weighted adjacency matrices; they also provide the basis for analysis of the consistency of various unsupervised and semi-supervised learning algorithms, in the large data limit.