Random Geometric Graphs Research Articles

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.

Asymptotic distribution of the friendship paradox of a random geometric graph

Asymptotic distribution of the friendship paradox of a random geometric graph

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”

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.

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Geometric random intersection graphs with general connection probabilities

Abstract Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$ . A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$ . This gives rise to a random intersection graph on $\mathbb{R}^d$ . Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

RGG-PSO+: Random Geometric Graphs Based Particle Swarm Optimization Method for UAV Path Planning

Evolutionary algorithms, such as particle swarm optimization (PSO), are widely applied to UAV path planning problems. However, the fixed particle length of PSO, which may not be suitable for the scenario, will compromise the search efficiency. This paper proposes the RGG-PSO+ method, which adapts to scenarios by dynamically adjusting the number of waypoints. Random geometric graphs (RGG) and the divide-and-conquer paradigm are involved in improving the proposed method. Comparative analyses with established heuristic methods demonstrate RGG-PSO+’s superior performance in complex environments, particularly in terms of convergence speed and path length. The implementation of RGG significantly improves the F-Measure, indicating a shift from exploration to exploitation of PSO’s iterations, and the implementation of the divide-and-conquer paradigm is evident in the improved mean and variance of normalized path lengths.

Sharp threshold for embedding balanced spanning trees in random geometric graphs

AbstractA rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph . In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on Rd. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

Covariance matrices of length power functionals of random geometric graphs – an asymptotic analysis

Asymptotic properties of a vector of length power functionals of random geometric graphs are investigated. Algebraic properties of the asymptotic covariance matrix are studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a systematic discussion of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the formulation of the results a case distinction is necessary. In the three possible regimes the respective covariance matrix is of quite different nature which leads to different statements. Stochastic consequences for random geometric graphs are derived.

On the Distances Within Cliques in a Soft Random Geometric Graph

We study the distances of vertices within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We obtain the scaling of the maximal distance between any two points within a clique of size k. Moreover, we show that asymptotically in all cliques with large distances, there is only one remote point and all other points are nearby. Furthermore, we prove that a re-scaled version of the maximal k-clique distance converges in distribution to a Fréchet distribution. Thereby, we describe the order of magnitude according to which the largest distance between two points in a clique decreases with the clique size.

Exploiting sparsity for localisation of large‐scale wireless sensor networks

AbstractWireless Sensor Network (WSN) localisation refers to the problem of determining the position of each of the agents in a WSN using noisy measurement information. In many cases, such as in distance and bearing‐based localisation, the measurement model is a non‐linear function of the agents' positions, leading to pairwise interconnections between the agents. As the optimal solution for the WSN localisation problem is known to be computationally expensive in these cases, an efficient approximation is desired. The authors show that the inherent sparsity in this problem can be exploited to greatly reduce the computational effort of using an Extended Kalman Filter (EKF) for large‐scale WSN localisation. In the proposed method, which the authors call the L‐Banded Extended Kalman Filter (LB‐EKF), the measurement information matrix is converted into a banded matrix by relabelling (permuting the order of) the vertices of the graph. Using a combination of theoretical analysis and numerical simulations, it is shown that typical WSN configurations (which can be modelled as random geometric graphs) can be localised in a scalable manner using the proposed LB‐EKF approach.

The coreness and h-index of random geometric graphs

The coreness and h-index of random geometric graphs

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.

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.

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.

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.

Optimal transport methods for combinatorial optimization over two random point sets

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} where the edge cost between two points is given by a pth power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d≥3,1≤p<d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3, 1\\le p<d$$\\end{document}. Previous results were limited to the range p<d/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<d/2$$\\end{document}. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.

Constructing training set using distance between learnt graphical models of time series data on patient physiology, to predict disease scores.

Interventional endeavours in medicine include prediction of a score that parametrises a new subject's susceptibility to a given disease, at the pre-onset stage. Here, for the first time, we provide reliable learning of such a score in the context of the potentially-terminal disease VOD, that often arises after bone marrow transplants. Indeed, the probability of surviving VOD, is correlated with early intervention. In our work, the VOD-score of each patient in a retrospective cohort, is defined as the distance between the (posterior) probability of a random graph variable-given the inter-variable partial correlation matrix of the time series data on variables that represent different aspects of patient physiology-and that given such time series data of an arbitrarily-selected reference patient. Such time series data is recorded from a pre-transplant to a post-transplant time, for each patient in this cohort, though the data available for distinct patients bear differential temporal coverage, owing to differential patient longevities. Each graph is a Soft Random Geometric Graph drawn in a probabilistic metric space, and the computed inter-graph distance is oblivious to the length of the time series data. The VOD-score learnt in this way, and the corresponding pre-transplant parameter vector of each patient in this retrospective cohort, then results in the training data, using which we learn the function that takes VOD-score as its input, and outputs the vector of pre-transplant parameters. We model this function with a vector-variate Gaussian Process, the covariance structure of which is kernel parametrised. Such modelling is easier than if the score variable were the output. Then for any prospective patient, whose pre-transplant variables are known, we learn the VOD-score (and the hyperparameters of the covariance kernel), using Markov Chain Monte Carlo based inference.

Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

Let G be a graph obtained as the union of some n-vertex graph Hn with minimum degree δ(Hn)≥αn and a d-dimensional random geometric graph Gd(n,r). We investigate under which conditions for r the graph G will a.a.s. contain the kth power of a Hamilton cycle, for any choice of Hn. We provide asymptotically optimal conditions for r for all values of α, d and k. This has applications in the containment of other spanning structures, such as F-factors.