Edges Of Graph Research Articles

Abstract An ( r , z ; g )-mixed graph is a graph containing both edges and darts satisfying the regularity property that each vertex of the graph is incident to r edges, z ingoing and z outgoing darts (called total regularity), and being of oriented girth g , i.e., containing an oriented cycle of length g , and no shorter oriented cycles. The problem addressed in this paper is analogous to the Cage Problem and calls for determining the orders of the smallest totally regular ( r , z ; g )-mixed graphs. We derive several upper and lower bounds on the orders of such minimal graphs, study the relations between these extremal graphs and their non-oriented or digraphical counterparts, and focus on properties of totally regular mixed graphs obtained by replacing some of the edges of the incidence graphs of projective and biaffine planes by darts. We also introduce two constructions based on introducing additional edges or darts into induced subgraphs of these incidence graphs.

Partitioning vertices and edges of graphs into connected subgraphs

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such ’master variables’ are usually highly desirable in treatments of ‘large N’ statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with explicit control over the corrections. We use this advantage to compute the first subleading correction to the Park–Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale–Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c\sqrt{k}n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c\sqrt{k}n, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2-- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

Correspondence analysis (CA) and its covariate-based counterpart, canonical correspondence analysis (CCA), are classic yet popular scaling methods in the natural, social, and biomedical sciences to estimate latent gradients that drive the formation of edges in a bipartite graph. However, these methods struggle to identify latent gradients when they exist in sparse graphs where small subsets of nodes are hyperspecialized to each other. This article proposes a new computational method to prevent hyperspecialized nodes from obscuring latent gradient solutions based on a Markov chain interpretation of the CA eigenvalue problem. This approach identifies small subsets of hyperspecialized nodes with greater precision than traditional graph clustering techniques, and outperforms existing regularization techniques at identifying a latent gradient on a real-world political fundraising network of candidates for U.S. federal office, which spans three decades and includes nearly 20,000 candidates for federal office and 3 million of their donors. Supplementary materials for this article are available online.

The precise recognition of human lower limb movements based on wearable sensors is very important for human-computer interaction. However, the existing methods tend to ignore the dynamic spatial information in the process of executing human lower limb movements, leading to challenges such as reduced decoding accuracy and limited robustness. In this paper, we construct skeleton graph data based on inertial measurement unit (IMU) sensors. Also, a two-branch deep learning model, termed TCNN-MGCHN, is proposed to mine meaningful spatial and temporal feature representations from IMU-based skeleton graph data. Firstly, a temporal convolutional module (consisting of a multi-scale convolutional sub-module and an attention sub-module) is developed to extract temporal feature information with highly discriminative power. Secondly, a multi-scale graph convolutional module and a spatial graph edges’ importance weight assignment method based on body partitioning strategy are proposed to obtain intrinsic spatial feature information between different skeleton nodes. Finally, the fused spatio-temporal features are passed into the classification module to obtain the predicted gait movements and sub-phases. Extensive comparison and ablation studies are conducted on our self-constructed human lower limb movement dataset. The results demonstrate that TCNN-MGCHN delivers superior classification performance compared to the mainstream methods. This study can provide a benchmark for IMU-based human lower limb movement recognition and related deep-learning modeling works.

Generalizing and strengthening a classical result of Vizing, Rautenbach proved a linear relationship between the domination number, the maximum degree, the number of vertices, and the number of edges for graphs with no isolated vertices. The sharpest version of this result was established recently by Henning and the first author. In their paper, the question was raised of establishing the sharpest possible linear-Vizing inequality for other domination parameters; finding the minimum so that for every graph G and a domination parameter γ′. In particular, the question was raised of showing that a key portion of the inequality – the linear-Vizing constant – remained bounded when γ′ was distance-2 domination. In this note, we settle this question in the affirmative. The key ingredient arises from a simple new proof of a result of Henning and Lichiardopol bounding the distance-k domination number.

Abstract The effective analysis of high-dimensional Electronic Health Record (EHR) data, with substantial potential for healthcare research, presents notable methodological challenges. Employing predictive modeling guided by a knowledge graph (KG), which enables efficient feature selection, can enhance both statistical efficiency and interpretability. While various methods have emerged for constructing KGs, existing techniques often lack statistical certainty concerning the presence of links between entities, especially in scenarios where the utilization of patient-level EHR data is limited due to privacy concerns. In this paper, we propose the first inferential framework for deriving a sparse KG with statistical guarantee based on a dynamic log-linear topic model. Within this model, the KG embeddings are estimated by performing singular value decomposition on the empirical pointwise mutual information matrix, offering a scalable solution. We then establish entrywise asymptotic normality for the KG low-rank estimator, enabling the recovery of sparse graph edges with controlled type I error. Our work uniquely addresses the under-explored domain of statistical inference about non-linear statistics under the low-rank temporal dependent models, a critical gap in existing research. We validate our approach through extensive simulation studies and then apply the method to real-world EHR data in constructing clinical KGs and generating clinical feature embeddings.

Graph anomaly detection (GAD) refers to identifying abnormal graph nodes or edges that heavily deviate from normal observations. Existing approaches inevitably suffer from the influence of imbalanced data and privacy protection. This shortcoming poses challenges in optimizing node embeddings and detecting multitype anomalies simultaneously, resulting in decreased accuracy of existing GAD models. To address this shortcoming, we introduce a new federated learning model for graph anomaly detection (FedGAD). FedGAD enables collaborative unsupervised learning among decentralized data centers without requiring direct access to the distributed subgraphs. Specifically, FedGAD masks and reconstructs the neighborhood features to enhance the knowledge of node representations. Considering the data diversity across distributed clients, we also design a cross-clients' node representation module that enables nodes to reconstruct neighbors by leveraging information from other clients. Furthermore, we use a multiscale contrastive learning function, which includes both structure-level and contextual-level learning functions, to detect graph anomalies in the condition that subgraphs located at different clients show imbalanced data distributions. Experimental results on seven benchmark datasets demonstrate the superior performance of FedGAD compared with baseline methods, verifying its capability of improving GAD performance.

ProtGeoNet-Pocket is an innovative multimodal prediction framework designed for protein binding site recognition, effectively integrating sequence information, geometric features, and graph-based structural representations. To address the structural complexity of proteins and the diversity of binding pocket shapes, ProtGeoNet-Pocket leverages multiscale structural information for high-precision binding site prediction. It uses a PointNet module to extract geometric features from residue coordinates and enhances them using an attention mechanism. The geometric features are then fused with encoded sequence features and graph edge features. The combined features are fed into a Graph Isomorphism Network (GIN) to capture topological relationships via a message-passing mechanism. ProtGeoNet-Pocket achieved an F1 score of 72.87% on the scPDB training set and demonstrated strong predictive performance across five independent benchmark data sets: COACH420, HOLO4K, SC6K, PDBbind, and ApoHolo. Furthermore, the visualization results confirm a high spatial overlap between the predicted and actual binding sites, demonstrating the superior performance of this method compared to existing ones.

Developing new ethical drugs is exceedingly expensive in terms of both time and resources. A single drug can take up to a decade to bring to market, with costs soaring to over a billion dollars. Drug repositioning has thus become an attractive alternative to the development of new compounds, with growing interest in the use of in silico repositioning predictions. Bipartite graphs and efficient biclique enumeration algorithms can be used to study drug-protein or other pairwise crucial interactions. Extensions of this approach to datasets with three or more divergent data types have been hobbled, however, by a lack of effective analytics. To address this shortcoming, a highly innovative and efficient graph theoretical technique is introduced to impute potential edges (links) in an arbitrary multipartite graph. The utility of this method is demonstrated on five tripartite graphs, each comprised of three partite sets, one each for diseases, drugs, and gene products of interest, and with interpartite edges denoting known interactions or associations. Evidence for the reliability of imputed edges is also reported.

Graph labeling is the assigning of labels represented by integers or symbols to graph elements, edges and/or vertices (or both) of a graph. Consider a simple graph with a vertex-set and an edge-set . The order of graph , denoted by , is the number of vertices on . The prime labeling is a bijective function , such that the labels of any two adjacent vertices in G are relatively prime or , for every two adjacent vertices and in . If a graph can be labeled with prime labeling, then the graph can be said to be a prime graph. A flower graph is a graph formed by helm graph by connecting its pendant vertices (the vertices have degree one) to the central vertex of , such a flower graph is denoted as In this research, we employ constructive and analytical methods to investigate prime labelings on specific graph classes. Definitions, lemmas, and theorems are developed as the main results in this research. The amalgamation is a graph formed by taking all by taking all the and identifying their fixed vertices . If , then we write with . In previous research, it has been shown that the flower graphs , for are prime graphs. Continuing the research, we prove that two classes of amalgamation of flower graphs are prime graphs.

Local efficiency analysis of the emotion regulation network in younger and older adults experiencing sleep deprivation: A task-based fMRI study.

ABSTRACTA ‐star is a complete bipartite graph . A partial ‐star design of order is a pair where is a set of vertices and is a set of edge‐disjoint ‐stars whose vertex sets are subsets of . If each edge of the complete graph with vertex set is in some star in , then is a (complete) ‐star design. We say that is completable if there is a ‐star design such that . In this paper we determine, for all and , the minimum number of stars in an uncompletable partial ‐star design of order .

Abstract The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$ -free graph $G$ is close to the extremal graphs for $H$ , then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$ . As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.

Graph recommendation methods, representing a connected interaction perspective, reformulate user-item interactions as graphs to leverage graph structure and topology to recommend and have proved practical effectiveness at scale. Large language models, representing a textual generative perspective, excel at modeling user languages, understanding behavioral contexts, capturing user-item semantic relationships, analyzing textual sentiments, and generating coherent and contextually relevant texts as recommendations. However, there is a gap between the connected graph perspective and the text generation perspective as the task formulations are different. A research question arises: how can we effectively integrate the two perspectives for more personalized recsys? To fill this gap, we propose to incorporate graph-edge information into LLMs via prompt and attention innovations. We reformulate recommendations as a probabilistic generative problem using prompts. We develop a framework to incorporate graph edge information from the prompt and attention mechanisms for graph-structured LLM recommendations. We develop a new prompt design that brings in both first-order and second-order graph relationships; we devise an improved LLM attention mechanism to embed direct the spatial and connectivity information of edges. Our evaluation of real-world datasets demonstrates the framework's ability to understand connectivity information in graph data and to improve the relevance and quality of recommendation results. Our code is released at: https://github.com/anord-wang/LLM4REC.git .

Mechanical systems face a major drawback to fault diagnosis, as class imbalance greatly undermines it since minority class instances (critical faults) are underrepresented, resulting in biased predictions. This paper introduces a new Multiscale Receptive Fields and Dynamic Edge Weighting (MRS-GNN) framework, which fuses MRF and Dynamic Edge Weighting (DEW) on a GNN to improve classification performance in imbalanced datasets. Graph edge strengths are dynamically weighted with the learned node embeddings during training according to the DEW mechanism, and MRF allows the model to aggregate information from different neighborhood scopes for robust feature representation. In addition, a graph-specific oversampling algorithm, MR-SMOTE, was used to generate synthetic minority class nodes respecting and preserving the topology of the graph. The proposed model was evaluated through experiments on the 2009 PHM gearbox dataset and was found to have an accuracy of 92.1% and an AUC-ROC score of 0.95, better than traditional oversampling methods such as SMOTE, LR-SMOTE, and Graph-SMOTE. The results of an ablation study indicate that 3.7% and 2.4% accuracy drops occur in DEW and MRF removals, respectively, highlighting their importance. This study proposes a scalable and topology-preserving solution to the imbalanced fault diagnosis problem and makes substantial improvements compared to existing GNN-based methods.

Understanding the functions of proteins is of great importance for deciphering the mechanisms of life activities. To date, there have been over 200 million known proteins, but only 0.2% of them have well-annotated functional terms. By measuring the contacts among residues, proteins can be described as graphs so that the graph leaning approaches can be applied to learn protein representations. However, existing graph-based methods put efforts in enriching the residue node information and did not fully exploit the edge information, which leads to suboptimal representations considering the strong association of residue contacts to protein structures and to the functions. In this article, we propose SuperEdgeGO, which introduces the supervision of edges in protein graphs to learn a better graph representation for protein function prediction. Different from common graph convolution methods that uses edge information in a plain or unsupervised way, we introduce a supervised attention to encode the residue contacts explicitly into the protein representation. Comprehensive experiments demonstrate that SuperEdgeGO achieves state-of-the-art performance on all three categories of protein functions. Additional ablation analysis further proves the effectiveness of the devised edge supervision strategy. The implementation of edge supervision in SuperEdgeGO resulted in enhanced graph representations for protein function prediction, as demonstrated by its superior performance across all the evaluated categories. This superior performance was confirmed through ablation analysis, which validated the effectiveness of the edge supervision strategy. This strategy has a broad application prospect in the study of protein function and related fields.

Spatial morphoproteomic features predict disease states from tissue architectures.

Contamination of cultivated soils with potentially toxic elements (PTEs) poses a growing threat to global food security. Although existing risk assessments have examined the accumulation and toxicity of PTEs, their dynamic interplay with multidimensional drivers has remained inadequately characterized. Here, an innovative heuristic graph convolutional network (GCN) model is introduced by integrating adaptive graph topology with quantified directional feedback optimization to improve ecological risk prediction. Leveraging 466 spatially resolved soil samples and 28 environmental drivers of a typical rice production area Yangtze River Basin in China, the heuristic GCN model outperformed traditional approaches by 23.1% in predictive accuracy. A three-phase heuristic algorithm pruned 85.5% of spurious edges in the topological graph, and GCN adaptively quantified the directional feedback between environmental drivers and ecological risk. Topological networks and feature importance analysis jointly identified pH, base saturation, calcium carbonate, exchangeable bases, and soil organic carbon as pivotal regulators acting alongside geological factors. By linking mechanistic soil chemistry with machine-learning-based causal inference, our model supports streamlinedly simplified, directionally quantified, and dynamically adapted ecological risk prediction. This enables the screening of the most efficient pathway of risk management and provides more precise and integrated strategies for ecological risk control in agroecosystems.