Geometric Graphs Research Articles

The algorithm and complexity of co-secure domination in geometric intersection graphs

The algorithm and complexity of co-secure domination in geometric intersection graphs

Maximal cliques in scale-free random graphs

Abstract We investigate the number of maximal cliques, that is, cliques that are not contained in any larger clique, in three network models: Erdős–Rényi random graphs, inhomogeneous random graphs (IRGs) (also called Chung–Lu graphs), and geometric inhomogeneous random graphs (GIRGs). For sparse and not-too-dense Erdős–Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (G)IRGs are sparse, we give super-polynomial lower bounds for these models. This comes from the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (G)IRGs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.

Online landmark replacement for out-of-sample dimensionality reduction methods

A strategy to assist visualization and analysis of large and complex datasets is dimensionality reduction, with which one maps each data point into a low-dimensional manifold. However, various dimensionality reduction techniques are computationally infeasible for large data. Out-of-sample techniques aim to resolve this difficulty; they only apply the dimensionality reduction technique on a small portion of data, referred to as landmarks, and determine the embedding coordinates of the other points using landmarks as references. Out-of-sample techniques have been applied to online settings, or when data arrive as time series. However, existing online out-of-sample techniques use either all the previous data points as landmarks or the fixed set of landmarks and therefore are potentially not good at capturing the geometry of the entire dataset when the time series is non-stationary. To address this problem, we propose an online landmark replacement algorithm for out-of-sample techniques using geometric graphs and the minimal dominating set on them. We mathematically analyse some properties of the proposed algorithm, particularly focusing on the case of landmark multi-dimensional scaling as the out-of-sample technique, and test its performance on synthetic and empirical time-series data.

VIEPred: Predicting Viral Immune Evasion with Multimodal Representation Learning

Abstract Predicting coronaviral immune evasion is crucial for identifying and responding to new COVID-19 variants in advance, thereby optimizing vaccine development and public health strategies to prevent further outbreaks. Hence, we present VIEPred, a model designed to predict mutations in every amino acid of the RBD. However, protein representation learning is a challenge task due to the hierarchical structure of proteins. Our method uses a sequence encoder based on dilated convolutions and attention mechanisms to generate sequence embeddings and a structure encoder employing geometric vector perceptrons (GVP) and graph neural networks (GNN) to produce protein structure representations. These features are combined using a cross-attention fusion module and subsequently processed through a fully connected layer to output the probability of viral immune evasion. Extensive ablation studies led to an optimal model with ACC of 88.4% and AUC of 86.8%. Compared to existing prediction methods, VIEPred demonstrates superior predictive performance across most evaluation metrics for both SARS-CoV-2 and influenza viruses.

Strategic allocation of landmarks to reduce uncertainty in indoor navigation

Indoor navigation systems often rely on verbal, turn-based route instructions. These can, at times, be ambiguous at complex decision points with multiple paths intersecting under angles that are not well distinguished by the turn grammar used. Landmarks can be included into turn instructions to reduce this ambiguity. Here, we propose an approach to optimize landmark allocation to improve the clarity of route instructions. This study assumes that landmark locations are constrained to a pre-determined set of slots. We select a minimum-size subset of the set of all slots and allocate it with landmarks, such that the navigation ambiguity is resolved. Our methodology leverages computational geometric analysis, graph algorithms, and optimization formulations to strategically incorporate landmarks into indoor route instructions. We propose a method to optimize landmark allocation in indoor navigation guidance systems, improving the clarity of route instructions at complex decision points that are inadequately served by turn-based instructions alone.

GGN-GO: geometric graph networks for predicting protein function by multi-scale structure features.

Recent advances in high-throughput sequencing have led to an explosion of genomic and transcriptomic data, offering a wealth of protein sequence information. However, the functions of most proteins remain unannotated. Traditional experimental methods for annotation of protein functions are costly and time-consuming. Current deep learning methods typically rely on Graph Convolutional Networks to propagate features between protein residues. However, these methods fail to capture fine atomic-level geometric structural features and cannot directly compute or propagate structural features (such as distances, directions, and angles) when transmitting features, often simplifying them to scalars. Additionally, difficulties in capturing long-range dependencies limit the model's ability to identify key nodes (residues). To address these challenges, we propose a geometric graph network (GGN-GO) for predicting protein function that enriches feature extraction by capturing multi-scale geometric structural features at the atomic and residue levels. We use a geometric vector perceptron to convert these features into vector representations and aggregate them with node features for better understanding and propagation in the network. Moreover, we introduce a graph attention pooling layer captures key node information by adaptively aggregating local functional motifs, while contrastive learning enhances graph representation discriminability through random noise and different views. The experimental results show that GGN-GO outperforms six comparative methods in tasks with the most labels for both experimentally validated and predicted protein structures. Furthermore, GGN-GO identifies functional residues corresponding to those experimentally confirmed, showcasing its interpretability and the ability to pinpoint key protein regions. The code and data are available at: https://github.com/MiJia-ID/GGN-GO.

Accurately predicting enzyme functions through geometric graph learning on ESMFold-predicted structures

Enzymes are crucial in numerous biological processes, with the Enzyme Commission (EC) number being a commonly used method for defining enzyme function. However, current EC number prediction technologies have not fully recognized the importance of enzyme active sites and structural characteristics. Here, we propose GraphEC, a geometric graph learning-based EC number predictor using the ESMFold-predicted structures and a pre-trained protein language model. Specifically, we first construct a model to predict the enzyme active sites, which is utilized to predict the EC number. The prediction is further improved through a label diffusion algorithm by incorporating homology information. In parallel, the optimum pH of enzymes is predicted to reflect the enzyme-catalyzed reactions. Experiments demonstrate the superior performance of our model in predicting active sites, EC numbers, and optimum pH compared to other state-of-the-art methods. Additional analysis reveals that GraphEC is capable of extracting functional information from protein structures, emphasizing the effectiveness of geometric graph learning. This technology can be used to identify unannotated enzyme functions, as well as to predict their active sites and optimum pH, with the potential to advance research in synthetic biology, genomics, and other fields.

Pretrainable geometric graph neural network for antibody affinity maturation

Increasing the binding affinity of an antibody to its target antigen is a crucial task in antibody therapeutics development. This paper presents a pretrainable geometric graph neural network, GearBind, and explores its potential in in silico affinity maturation. Leveraging multi-relational graph construction, multi-level geometric message passing and contrastive pretraining on mass-scale, unlabeled protein structural data, GearBind outperforms previous state-of-the-art approaches on SKEMPI and an independent test set. A powerful ensemble model based on GearBind is then derived and used to successfully enhance the binding of two antibodies with distinct formats and target antigens. ELISA EC50 values of the designed antibody mutants are decreased by up to 17 fold, and KD values by up to 6.1 fold. These promising results underscore the utility of geometric deep learning and effective pretraining in macromolecule interaction modeling tasks.

Learning Joint 2-D and 3-D Graph Diffusion Models for Complete Molecule Generation.

Designing new molecules is essential for drug discovery and material science. Recently, deep generative models that aim to model molecule distribution have made promising progress in narrowing down the chemical research space and generating high-fidelity molecules. However, current generative models only focus on modeling 2-D bonding graphs or 3-D geometries, which are two complementary descriptors for molecules. The lack of ability to jointly model them limits the improvement of generation quality and further downstream applications. In this article, we propose a joint 2-D and 3-D graph diffusion model (JODO) that generates geometric graphs representing complete molecules with atom types, formal charges, bond information, and 3-D coordinates. To capture the correlation between 2-D molecular graphs and 3-D geometries in the diffusion process, we develop a diffusion graph transformer (DGT) to parameterize the data prediction model that recovers the original data from noisy data. The DGT uses a relational attention mechanism that enhances the interaction between node and edge representations. This mechanism operates concurrently with the propagation and update of scalar attributes and geometric vectors. Our model can also be extended for inverse molecular design targeting single or multiple quantum properties. In our comprehensive evaluation pipeline for unconditional joint generation, the experimental results show that JODO remarkably outperforms the baselines on the QM9 and GEOM-Drugs datasets. Furthermore, our model excels in few-step fast sampling, as well as in inverse molecule design and molecular graph generation. Our code is provided in https://github.com/GRAPH-0/JODO.

Asymptotic distribution of the friendship paradox of a random geometric graph

Asymptotic distribution of the friendship paradox of a random geometric graph

On the Complexity of Target Set Selection in Simple Geometric Networks

We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”

Distributed Independent Sets in Interval and Segment Intersection Graphs

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Spatial invasion of cooperative parasites

In this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN with rN of order N(β−1)/n for some 0<β<1. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time.An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation. Furthermore, we build on proof techniques developed in Brouard and Pokalyuk (2022), where an analogous invasion process has been studied for host populations structured according to a configuration model.We substantiate our results with simulations.

PANDA-3D: protein function prediction based on AlphaFold models.

Previous protein function predictors primarily make predictions from amino acid sequences instead of tertiary structures because of the limited number of experimentally determined structures and the unsatisfying qualities of predicted structures. AlphaFold recently achieved promising performances when predicting protein tertiary structures, and the AlphaFold protein structure database (AlphaFold DB) is fast-expanding. Therefore, we aimed to develop a deep-learning tool that is specifically trained with AlphaFold models and predict GO terms from AlphaFold models. We developed an advanced learning architecture by combining geometric vector perceptron graph neural networks and variant transformer decoder layers for multi-label classification. PANDA-3D predicts gene ontology (GO) terms from the predicted structures of AlphaFold and the embeddings of amino acid sequences based on a large language model. Our method significantly outperformed a state-of-the-art deep-learningmethod that was trained with experimentally determined tertiary structures, and either outperformed or was comparable with several other language-model-based state-of-the-art methods with amino acid sequences as input.PANDA-3D is tailored to AlphaFold models, and the AlphaFold DB currently contains over 200 million predicted protein structures (as of May 1st, 2023), making PANDA-3D a useful tool that can accurately annotate the functions of a large number of proteins. PANDA-3D can be freely accessed as a web server from http://dna.cs.miami.edu/PANDA-3D/ and as a repository from https://github.com/zwang-bioinformatics/PANDA-3D.

A note on the k-colored crossing ratio of dense geometric graphs

A geometric graph is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer k≥2, there exists a constant c>0 such that the following holds. The edges of every dense geometric graph, with sufficiently many vertices, can be colored with k colors, such that the number of pairs of edges of the same color that cross is at most (1/k−c) times the total number of pairs of edges that cross. The case when k=2 and G is a complete geometric graph, was proved by Aichholzer et al. (2019) [2].

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Asymptotic dynamic graph order evolution analysis

In this work, we investigate the analysis of generators for dynamic graphs, which are defined as graphs whose topology changes over time. We focus on generated graphs whose order (number of nodes) varies over time. We use a concept called “sustainability” to qualify the long-term evolution of dynamic graphs. A dynamic graph is considered sustainable if its evolution does not result in a static, empty, or periodic graph. To illustrate how the analysis can be conducted, a parameterized and probability-based generator, named D3G3 (Degree-Driven Dynamic Geometric Graphs Generator), has been introduced in a recent work. From this model, we derive multiple scenarios that correspond to three trends in graph order evolution. Our central contribution lies in a mathematical framework that provides an expectation of the order of the graph at time step t+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t+1$$\\end{document}, given its order at time step t. Nevertheless, our analysis underscores the challenge of characterizing the sustainability of dynamic graphs, even when a formal mathematical model for graph order evolution is known.

Geometric localized graph convolutional network for multi-view semi-supervised classification

Multi-view learning has received increasing attention in recent years due to its ability to leverage valuable patterns hidden in heterogeneous data sources. While existing studies have achieved encouraging results, especially those based on graph convolutional networks, they are still limited in their ability to fully exploit the connectivity relationships between samples and are susceptible to noise. To address the aforementioned limitations, we propose a framework called geometric localized graph convolutional network for multi-view semi-supervised classification. This framework utilizes a diffusion map to obtain the geometric structure of the feature space of multiple views and constructs a stable distance matrix that considers the local connectivity of nodes on the geometric structure. Additionally, we propose a truncated diffusion correlation function that maps the distance matrix of each view into correlations between samples to obtain a reliable sparse graph. To fuse the features, we use learnable weights to concatenate the coordinates of the geometric structure. Finally, we obtain a graph embedding of the fused feature and topology by using graph convolutional networks. Comprehensive experiments demonstrate the superiority of the proposed method over other state-of-the-art methods.

A note on the 2-colored rectilinear crossing number of random point sets in the unit square

Let S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} defines a pair of crossing edges of the same color is equal to 1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/4$$\\end{document}. This is connected to a recent result of Aichholzer et al. [1] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation 12-750\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2}-\\frac{7}{50}$$\\end{document} of the total number of crossings.