Geometric Graphs Research Articles

Finding geometric representations of apex graphs is NP-hard

Planar graphs can be represented as the intersection graphs of different types of geometric objects in the plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., SODA 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex graphs, i.e., graphs that can be made planar by removing one vertex from them. We show, somewhat surprisingly, that deciding whether geometric representations exist for apex graphs is NP-hard as well.More precisely, we show that for every fixed positive integer g and every graph class G such that ▪, it is NP-hard to decide whether an input graph belongs to the graph class G, even when the inputs are restricted to apex graphs of girth g. Here, ▪ is the class of intersection graphs of axis-parallel line segments (where horizontal segments intersect only vertical segments), and ▪ is the class of intersection graphs of simple curves (where two intersecting curves cross each other exactly once) in the plane. This partially answers an open question raised by Kratochvíl & Pergel (COCOON, 2007).Most known reductions for earlier proofs of NP-hardness for these problems are from variants of 3-SAT (mainly Planar 3-Connected 3-SAT). We reduce from the ▪▪▪ problem, which uses the more intuitive notion of planarity. As a result, our proof is much simpler and encapsulates several classes of geometric intersection graphs.

On asymptotic packing of convex geometric and ordered graphs

AbstractA convex geometric graph is said to be packable if there exist edge‐disjoint copies of in the complete convex geometric graph covering all but edges. We prove that every convex geometric graph with cyclic chromatic number at most 4 is packable. With a similar definition of packability for ordered graphs, we prove that every ordered graph with interval chromatic number at most 3 is packable. Arguments based on the average length of edges imply these results are the best possible. We also identify a class of convex geometric graphs that are packable due to having many “long” edges.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.

Human Mobility Modeling during the COVID-19 Pandemic via Deep Graph Diffusion Infomax

Non-Pharmaceutical Interventions (NPIs), such as social gathering restrictions, have shown effectiveness to slow the transmission of COVID-19 by reducing the contact of people. To support policy-makers, multiple studies have first modelled human mobility via macro indicators (e.g., average daily travel distance) and then study the effectiveness of NPIs. In this work, we focus on mobility modelling and, from a micro perspective, aim to predict locations that will be visited by COVID-19 cases. Since NPIs generally cause economic and societal loss, such a prediction benefits governments when they design and evaluate them. However, in real-world situations, strict privacy data protection regulations result in severe data sparsity problems (i.e., limited case and location information). To address these challenges and jointly model variables including a geometric graph, a set of diffusions and a set of locations, we propose a model named Deep Graph Diffusion Infomax (DGDI). We show the maximization of DGDI can be bounded by two tractable components: a univariate Mutual Information (MI) between geometric graph and diffusion representation, and a univariate MI between diffusion representation and location representation. To facilitate the research of COVID-19 prediction, we present two benchmarks that contain geometric graphs and location histories of COVID-19 cases. Extensive experiments on the two benchmarks show that DGDI significantly outperforms other competing methods.

A cosine rule-based discrete sectional curvature for graphs

Abstract How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analysing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this article is to introduce and validate a new estimator of discrete sectional curvature for random graphs with low metric-distortion. The latter are constructed via a specific graph sprinkling method on different manifolds with constant sectional curvature. We define a notion of metric distortion, which quantifies how well the graph metric approximates the metric of the underlying manifold. We show how graph sprinkling algorithms can be refined to produce hard annulus random geometric graphs with minimal metric distortion. We construct random geometric graphs for spheres, hyperbolic and Euclidean planes; upon which we validate our curvature estimator. Numerical analysis reveals that the error of the estimated curvature diminishes as the mean metric distortion goes to zero, thus demonstrating convergence of the estimate. We also perform comparisons to other existing discrete curvature measures. Finally, we demonstrate two practical applications: (i) estimation of the earth’s radius using geographical data; and (ii) sectional curvature distributions of self-similar fractals.

Heterogeneous Domain Adaptation for IoT Intrusion Detection: A Geometric Graph Alignment Approach

Data scarcity hinders the usability of data-dependent algorithms when tackling IoT intrusion detection (IID). To address this, we utilise the data rich network intrusion detection (NID) domain to facilitate more accurate intrusion detection for IID domains. In this paper, a Geometric Graph Alignment (GGA) approach is leveraged to mask the geometric heterogeneities between domains for better intrusion knowledge transfer. Specifically, each intrusion domain is formulated as a graph where vertices and edges represent intrusion categories and category-wise interrelationships, respectively. The overall shape is preserved via a confused discriminator incapable to identify adjacency matrices between different intrusion domain graphs. A rotation avoidance mechanism and a centre point matching mechanism is used to avoid graph misalignment due to rotation and symmetry, respectively. Besides, category-wise semantic knowledge is transferred to act as vertex-level alignment. To exploit the target data, a pseudo-label election mechanism that jointly considers network prediction, geometric property and neighbourhood information is used to produce fine-grained pseudo-label assignment. Upon aligning the intrusion graphs geometrically from different granularities, the transferred intrusion knowledge can boost IID performance. Comprehensive experiments on several intrusion datasets demonstrate state-of-the-art performance of the GGA approach and validate the usefulness of GGA constituting components.

Limit theory of sparse random geometric graphs in high dimensions

We study topological and geometric functionals of l∞-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

Non-Existence of Annular Separators in Geometric Graphs

Non-Existence of Annular Separators in Geometric Graphs

Geometric graph neural networks on multi-omics data to predict cancer survival outcomes

The advance of sequencing technologies has enabled a thorough molecular characterization of the genome in human cancers. To improve patient prognosis predictions and subsequent treatment strategies, it is imperative to develop advanced computational methods to analyze large-scale, high-dimensional genomic data. However, traditional machine learning methods face a challenge in handling the high-dimensional, low-sample size problem that is shown in most genomic data sets. To address this, our group has developed geometric network analysis techniques on multi-omics data in connection with prior biological knowledge derived from protein-protein interactions (PPIs) or pathways. Geometric features obtained from the genomic network, such as Ollivier-Ricci curvature and the invariant measure of the associated Markov chain, have been shown to be predictive of survival outcomes in various cancers. In this study, we propose a novel supervised deep learning method called geometric graph neural network (GGNN) that incorporates such geometric features into deep learning for enhanced predictive power and interpretability. More specifically, we utilize a state-of-the-art graph neural network with sparse connections between the hidden layers based on known biology of the PPI network and pathway information. Geometric features along with multi-omics data are then incorporated into the corresponding layers. The proposed approach utilizes a local–global principle in such a manner that highly predictive features are selected at the front layers and fed directly to the last layer for multivariable Cox proportional-hazards regression modeling. The method was applied to multi-omics data from the CoMMpass study of multiple myeloma and ten major cancers in The Cancer Genome Atlas (TCGA). In most experiments, our method showed superior predictive performance compared to other alternative methods.

Scaling of the Clustering Function in Spatial Inhomogeneous Random Graphs

We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection model, the infinite geometric inhomogeneous random graph, and the age-based random connection model. These infinite models arise as the local limit of the corresponding finite models. For these models we identify the asymptotics of the local clustering as a function of the degree of the root in different regimes in a unified way. We show that the scaling exhibits phase transitions as the interpolation parameter moves across different regimes. This allows us to draw conclusions on the geometry of a typical triangle contributing to the clustering in the different regimes.

Bootstrap percolation in random geometric graphs

AbstractFollowing Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is $a\log n$ for some fixed $a>1$ . Each vertex is added with probability p to a set $A_0$ of initially infected vertices. Vertices subsequently become infected if they have at least $ \theta a \log n $ infected neighbours. Here $p, \theta \in [0,1]$ are taken to be fixed constants.We show that if $\theta < (1+p)/2$ , then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for $ \theta > (1+p)/2$ , even if one adversarially infects every vertex inside a ball of radius $O(\sqrt{\log n} )$ , with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.In addition we give some bounds on the $(a, p, \theta)$ regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.

Partial Inverse Sturm-Liouville Problems

This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a finite interval or a geometric graph) from spectral characteristics, while the coefficients in the remaining part of the domain are known a priori. Usually, partial inverse problems require less spectral data than complete inverse problems. In this review, we pay considerable attention to partial inverse problems on graphs and to the unified approach based on the reduction of partial inverse problems to Sturm-Liouville problems with entire analytic functions in a boundary condition. We not only describe the results of selected studies but also compare them with each other and establish interconnections.

Universal geometric graphs

AbstractWe extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $\mathcal H$ . Our main result is that there exists a geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$ -vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that contains every $n$ -vertex forest as a subgraph. The upper bound of $O\!\left(n \log n\right)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$ -vertex convex geometric graph that is universal for $n$ -vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega _h(n^{2-1/h})$ , for every positive integer $h$ ; this almost matches the trivial $O(n^2)$ upper bound given by the $n$ -vertex complete convex geometric graph. Finally, we prove that there exists an $n$ -vertex convex geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$ -vertex caterpillars.

Conflict Optimization for Binary CSP Applied to Minimum Partition into Plane Subgraphs and Graph Coloring

CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments. Surprisingly, the top three teams used the same technique, called conflict optimization. This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem. In this article, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP). Then, the top three teams describe their different implementations of the same underlying strategy. We evaluate the performance of those implementations to vertex color not only geometric graphs, but also other types of graphs.

Space-efficient algorithms for reachability in directed geometric graphs

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem.For intersection graphs of Jordan regions, we show how to obtain a “good” vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m1/2log⁡n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m1/2log⁡n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ϵ>0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n1/4+ϵ) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.

Limiting shape for first-passage percolation models on random geometric graphs

AbstractLet a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

A geometric Chung–Lu model and the Drosophila medulla connectome

Abstract Many real-world graphs have edges correlated to the distance between them, but in an inhomogeneous manner. While the Chung–Lu model and the geometric random graph models both are elegant in their simplicity, they are insufficient to capture the complexity of these networks. In this article, we develop a generalized geometric random graph model that preserves many graph theoretic aspects of these real-world networks. We test the validity of this model on a graphical representation of the Drosophila medulla connectome.

Local limits of spatial inhomogeneous random graphs

AbstractConsider a set of n vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$ , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights.Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models.We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

A Multiverse Graph to Help Scientific Reasoning from Web Usage: Interpretable Patterns of Assessor Shifts in GRAPHYP

The digital support for scientific reasoning presents contrasting results. Bibliometric services are improving, but not academic assessment; no service for scholars relies on logs of web usage to base query strategies for relevance judgments (or assessor shifts). Our Scientific Knowledge Graph GRAPHYP innovates with interpretable patterns of web usage, providing scientific reasoning with conceptual fingerprints and helping identify eligible hypotheses. In a previous article, we showed how usage log data, in the form of ‘documentary tracks’, help determine distinct cognitive communities (called adversarial cliques) within sub-graphs. A typology of these documentary tracks through a triplet of measurements from logs (intensity, variety and attention) describes the potential approaches to a (research) question. GRAPHYP assists interpretation as a classifier, with possibilistic graphical modeling. This paper shows what this approach can bring to scientific reasoning; it involves visualizing complete interpretable pathways, in a multi-hop assessor shift, which users can then explore toward the ‘best possible solution’—the one that is most consistent with their hypotheses. Applying the Leibnizian paradigm of scientific reasoning, GRAPHYP highlights infinitesimal learning pathways, as a ‘multiverse’ geometric graph in modeling possible search strategies answering research questions.

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in \(\mathbb {R}^2 \) admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^2 \) has a separator of size \(O(\sqrt {k}) \) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in \(\mathbb {R}^d \) for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^d \) has a separator of size O ( k 1 − 1/ d ). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ -lanky : a geometric graph is τ -lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ -lanky geometric graph of n vertices in \(\mathbb {R}^d \) has a separator of size O ( τn 1 − 1/ d ). We then derive our main result by showing that the greedy spanner is O (1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in \(\mathbb {R}^d \) . Our technique naturally extends to doubling metrics. We use the τ -lanky criterion to show that there exists a (1 + ϵ)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size \(O(n^{1-\frac{1}{d}}) \) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n -point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta) \) where Δ is the spread of the metric; the factor log ( Δ ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree . Finally, we discuss algorithmic implications of our results.