Geometric Graphs Research Articles

A Survey on Variant Domination Problems in Geometric Intersection Graphs

Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.

Geometric molecular graph representation learning model for drug-drug interactions prediction.

Drug-drug interaction (DDI) can trigger many adverse effects in patients and has emerged as a threat to medicine and public health. Therefore, it is important to predict potential drug interactions since it can provide combination strategies of drugs for systematic and effective treatment. Existing deep learning-based methods often rely on DDI functional networks, or use them as an important part of the model information source. However, it is difficult to discover the interactions of a new drug. To address the above limitations, we propose a geometric molecular graph representation learning model (Mol-DDI) for DDI prediction based on the basic assumption that structure determines function. Mol-DDI only considers the covalent and non-covalent bond information of molecules, then it uses the pre-training idea of large-scale models to learn drug molecular representations and predict drug interactions during the fine-tuning process. Experimental results show that the Mol-DDI model outperforms others on the three datasets and performs better in predicting new drug interaction experiments.

Creating and analyzing datasets using eBPF technology in container infrastructures

Containerized environments such as Docker provide isolation and deployment of distributed services and applications. However, anomalies and attacks can still occur within containers and within containerized networks. This paper proposes the creation of a dataset using eBPF-based traceback to monitor communication between containers in a Docker environment. In the process, a Docker environment with containers providing different services is created and eBPF-based data collection programs are configured to capture packet-level data. The collected dataset contains characteristics such as process IDs, container IDs, network events, and function arguments. A detailed analysis of these characteristics can provide valuable information about container network behavior, performance issues and vulnerabilities. As a proof of concept, the paper evaluates the applicability of classifiers such as Naive Bayes, KNN, SVM and Random Forest to detect anomalies in the dataset. The results show accuracy in the range of 0.42-0.67 and F-measure performance in the range of 0.54-0.65. Overall, the study demonstrates the potential of creating eBPF-based datasets for container security monitoring and anomaly detection by further optimizing data analysis techniques. The proposed approach can help in securing, troubleshooting, and optimizing container deploymet., mobile network, visibility graph, geometric graph, random walks, simulation.

Expressiveness Analysis and Enhancing Framework for Geometric Knowledge Graph Embedding Models

Expressiveness Analysis and Enhancing Framework for Geometric Knowledge Graph Embedding Models

Lucky edge geometric mean labeling of graphs and applications in electrical networks

A Lucky Edge Geometric Mean Labeling (LEGML) is a function μ that assigns the integers to the vertices of graph G such that for every edge as ‘{E}’, uv∈E(G) there exists a function μ*:E(G) → N is defined by μ* (uv) = √μ(u)μ(v) (or) √μ(u)μ(v) with the state that μ* (uv) ≠ μ* (vw) whenever {u} and {vw} have a common vertex as {v}. A minor number ‘k→ E = {1, 2, 3 … k} is the LEGML of G and is signified by h¢GM(G). A result that accepts LEGML is the Lucky Edge Geometric Mean Graph (LEGMG). The idea of the Graph Theory (GT) concept is fused into the Electric Circuit (EC), and the related complicated problems are transformed into graph model representation problems; as a result, the valuation method is simplified and optimized. Network analysis means finding the current or voltage in each branch. One application of GT is the representation of EC. An ‘E’ in a diagram can signify anything in an EC. Each part of the diagram can represent a port or a terminal in the EC. In this article, we have to investigate the Middle Graph (MG) of the star, the Total Graph (TG) of the star, and the central graph (CG) of the star, which accepts LEGML and LEGMG. This method also computes the power consumed by all the resistors of the EC by LEGML.

The coreness and h-index of random geometric graphs

The coreness and h-index of random geometric graphs

Cucumber Seedling Segmentation Network Based on a Multiview Geometric Graph Encoder from 3D Point Clouds.

Plant phenotyping plays a pivotal role in observing and comprehending the growth and development of plants. In phenotyping, plant organ segmentation based on 3D point clouds has garnered increasing attention in recent years. However, using only the geometric relationship features of Euclidean space still cannot accurately segment and measure plants. To this end, we mine more geometric features and propose a segmentation network based on a multiview geometric graph encoder, called SN-MGGE. First, we construct a point cloud acquisition platform to obtain the cucumber seedling point cloud dataset, and employ CloudCompare software to annotate the point cloud data. The GGE module is then designed to generate the point features, including the geometric relationships and geometric shape structure, via a graph encoder over the Euclidean and hyperbolic spaces. Finally, the semantic segmentation results are obtained via a downsampling operation and multilayer perceptron. Extensive experiments on a cucumber seedling dataset clearly show that our proposed SN-MGGE network outperforms several mainstream segmentation networks (e.g., PointNet++, AGConv, and PointMLP), achieving mIoU and OA values of 94.90% and 97.43%, respectively. On the basis of the segmentation results, 4 phenotypic parameters (i.e., plant height, leaf length, leaf width, and leaf area) are extracted through the K-means clustering method; these parameters are very close to the ground truth, and the R 2 values reach 0.98, 0.96, 0.97, and 0.97, respectively. Furthermore, an ablation study and a generalization experiment also show that the SN-MGGE network is robust and extensive.

Stability and Ultimate Boundedness of Solutions of Certain Third Order Nonlinear Rectangular Matrix Differential Equations

We present in this paper the qualitative study of solutions of certain third order nonlinear matrix differential equations where the unknown function X is matrix-valued. The properties of solutions were investigated using Lyapunov’s direct method by employing the use of suitable Lyapunov functionals obtained from the differential equations describing the system satisfying certain requirements for establishing the stability and boundedness of solutions of the system considered. An example is given to demonstrate the significance of the results obtained as well as analysis through geometric graphs describing the dynamics of the system’s solutions. The results obtained are novel and will significantly enhance and extend the results of those mentioned in the literature.

Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-Offs

Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-Offs

MAEMOT: Pretrained MAE-Based Antiocclusion 3-D Multiobject Tracking for Autonomous Driving.

The existing 3-D multiobject tracking (MOT) methods suffer from object occlusion in real-world traffic scenes. However, previous works have faced challenges in providing a reasonable solution to the fundamental question: "How can the interference of the perception data loss caused by occlusion be overcome?" Therefore, this article attempts to provide a reasonable solution by developing a novel pretrained movement-constrained masked autoencoder (M-MAE) for an antiocclusion 3-D MOT called MAEMOT. Specifically, for the pretrained M-MAE, this article adopts an efficient multistage transformer (MST) encoder and a spatiotemporal-based motion decoder to predict and reconstruct occluded point cloud data, following the properties of object motion. Afterward, the well-trained M-MAE model extracts the global features of occluded objects, ensuring that the features of the intraobjects between interframes are as consistent as possible throughout the spatiotemporal sequence. Next, a proposal-based geometric graph aggregation (PG 2 A) module is utilized to extract and fuse the spatial features of each proposal, producing refined region-of-interest (RoI) components. Finally, this article designs an object association module that combines geometric and corner affinities, which helps to match the predicted occlusion objects more robustly. According to an extensive evaluation, the proposed MAEMOT method can effectively overcome the interference of occlusion and achieve improved 3-D MOT performance under challenging conditions.

Geometric Graph Representation with Learnable Graph Structure and Adaptive AU Constraint for Micro-Expression Recognition

Geometric Graph Representation with Learnable Graph Structure and Adaptive AU Constraint for Micro-Expression Recognition

Social adaptive behavior and oscillatory prevalence in an epidemic model on evolving random geometric graphs

Our recent experience with the COVID-19 pandemic amply shows that spatial effects like the mobility of agents and average interpersonal distance, together with the adaptation of agents, are very important in deciding the outcome of epidemic dynamics. Structural and dynamical aspects of random geometric graphs are widely employed in describing processes with a spatial dependence, such as the spread of an airborne disease. In this work, we investigate the interplay between spatial factors, such as agent mobility and average interpersonal distance, and the adaptive responses of individuals to an ongoing epidemic within the framework of random geometric graphs. We show that such spatial factors, together with the adaptive behavior of the agents in response to the prevailing level of global epidemic, can give rise to oscillatory prevalence even with the classical SIR framework. We characterize in detail the effects of social adaptation and mobility of agents on the disease dynamics and obtain the threshold values. We also study the effects of delayed adaptive response of agents on epidemic dynamics. We show that a delay in executing non-pharmaceutical spatial mitigation strategies can amplify oscillatory prevalence tendencies and can have non-linear effects on peak prevalence. This underscores the importance of early implementation of adaptive strategies coupled with the dissemination of real-time prevalence information to manage and control the epidemic effectively.

Elastic subspace diagnosis via graph-balanced discriminant projection

Subspace learning of high-dimensional vibration signals is crucial for mechanical fault diagnosis. However, traditional subspace learning methods are difficult to acquire and balance the adaptive global and local intrinsic graph structures of fault data. To address this problem, we propose a novel elastic subspace diagnosis method based on graph balanced discriminant projection (GbDP). By inter-sample collaborative representation and supervision information, the method constructs adaptive global and local geometric graphs and further gives a graph balancing strategy for the optimization model of elastic subspace learning. On the basis of capturing the inherent discriminative structure of fault data, GbDP improves the intra-class aggregation and inter-class separability of the elastic subspace and achieves a relative balance of global and local structures. The results of the experimental cases on bearing and gear faults show that GbDP can extract more discriminative low-dimensional sensitive features and can effectively improve fault classification accuracy.

Partial and Simultaneous Transitive Orientations via Modular Decompositions

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.

Properties of a Random Bipartite Geometric Associator Graph Inspired by Vehicular Networks.

We consider a point process (PP) generated by superimposing an independent Poisson point process (PPP) on each line of a 2D Poisson line process (PLP). Termed PLP-PPP, this PP is suitable for modeling networks formed on an irregular collection of lines, such as vehicles on a network of roads and sensors deployed along trails in a forest. Inspired by vehicular networks in which vehicles connect with their nearest wireless base stations (BSs), we consider a random bipartite associator graph in which each point of the PLP-PPP is associated with the nearest point of an independent PPP through an edge. This graph is equivalent to the partitioning of PLP-PPP by a Poisson Voronoi tessellation (PVT) formed by an independent PPP. We first characterize the exact distribution of the number of points of PLP-PPP falling inside the ball centered at an arbitrary location in R2 as well as the typical point of PLP-PPP. Using these distributions, we derive cumulative distribution functions (CDFs) and probability density functions (PDFs) of kth contact distance (CD) and the nearest neighbor distance (NND) of PLP-PPP. As intermediate results, we present the empirical distribution of the perimeter and approximate distribution of the length of the typical chord of the zero-cell of this PVT. Using these results, we present two close approximations of the distribution of node degree of the random bipartite associator graph. In a vehicular network setting, this result characterizes the number of vehicles connected to each BS, which models its load. Since each BS has to distribute its limited resources across all the vehicles connected to it, a good statistical understanding of load is important for an efficient system design. Several applications of these new results to different wireless network settings are also discussed.

Maximum Bipartite Subgraphs of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph ([Formula: see text]) problem, which is defined as follows. Given a set [Formula: see text] of [Formula: see text] geometric objects in the plane, we want to compute a maximum-size subset [Formula: see text] such that the intersection graph of the objects in [Formula: see text] is bipartite. We first give an [Formula: see text]-time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the [Formula: see text] problem is [Formula: see text]-hard on geometric graphs for which the maximum independent set is [Formula: see text]-hard (hence, it is [Formula: see text]-hard even on unit squares and unit disks). On the other hand, we give a [Formula: see text] for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph ([Formula: see text]) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the [Formula: see text] except the intersection graph induced by the set [Formula: see text] needs to be triangle-free only (instead of being bipartite).

Geometric Graph Learning to Predict Changes in Binding Free Energy and Protein Thermodynamic Stability upon Mutation.

Accurate prediction of binding free energy changes upon mutations is vital for optimizing drugs, designing proteins, understanding genetic diseases, and cost-effective virtual screening. While machine learning methods show promise in this domain, achieving accuracy and generalization across diverse data sets remains a challenge. This study introduces Geometric Graph Learning for Protein-Protein Interactions (GGL-PPI), a novel approach integrating geometric graph representation and machine learning to forecast mutation-induced binding free energy changes. GGL-PPI leverages atom-level graph coloring and multiscale weighted colored geometric subgraphs to capture structural features of biomolecules, demonstrating superior performance on three standard data sets, namely, AB-Bind, SKEMPI 1.0, and SKEMPI 2.0 data sets. The model's efficacy extends to predicting protein thermodynamic stability in a blind test set, providing unbiased predictions for both direct and reverse mutations and showcasing notable generalization. GGL-PPI's precision in predicting changes in binding free energy and stability due to mutations enhances our comprehension of protein complexes, offering valuable insights for drug design endeavors.

Some New Properties of Cyclotomic Cliques Arrangements

In this paper, we study the arrangement of lines in the euclidean plane constructed from the geometric clique graph generated by the regular -gon. The vertices of this clique arrangement are located on finite concentric circles that we call orbits. We focus especially on the number of orbits and the number of vertices inside and outside the regular -gon. Combinatorics in finite sets of quadruplets of integers provide information on the way the orbits are distributed. Next, using cyclotomic fields, we give galoisian properties of the radii of the orbits and their cardinalities. Keywords: arrangement of lines in the plane, cyclotomic fields, geometric graphs, Galois theory

Rates of convergence for regression with the graph poly-Laplacian

In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset {xi}i=1n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x_i\\}_{i=1}^n$$\\end{document} and a set of noisy labels {yi}i=1n⊂R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{y_i\\}_{i=1}^n\\subset \\mathbb {R}$$\\end{document} we let un:{xi}i=1n→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_n{:}\\{x_i\\}_{i=1}^n\\rightarrow \\mathbb {R}$$\\end{document} be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When yi=g(xi)+ξi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y_i = g(x_i)+\\xi _i$$\\end{document}, for iid noise ξi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi _i$$\\end{document}, and using the geometric random graph, we identify (with high probability) the rate of convergence of un\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_n$$\\end{document} to g in the large data limit n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

Rainbow Hamilton cycles in random geometric graphs

AbstractLet be chosen independently and uniformly at random from the unit ‐dimensional cube . Let be given and let . The random geometric graph has vertex set and an edge whenever . We show that if each edge of is colored independently from one of colors and has the smallest value such that has minimum degree at least two, then contains a rainbow Hamilton cycle asymptotically almost surely.