Random Geometric Graphs Research Articles

Reconstruction of random geometric graphs: Breaking the [formula omitted] distortion barrier

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r=nα for any 0<α<1/2, with high probability reconstructs the vertex positions with a maximum error of O(nβ) where β=1/2−(4/3)α, until α≥3/8 where β=0 and the error becomes O(logn). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R3 and to hypercubes in any constant fixed dimension.

The longest edge of the one-dimensional soft random geometric graph with boundaries

The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger[ 2 ].

Dispersion entropy for graph signals

We present a novel method, called Dispersion Entropy for Graph Signals, DEG, as a powerful tool for analysing the irregularity of signals defined on graphs. DEG generalizes the classical dispersion entropy concept for univariate time series, enabling its application in diverse domains such as image processing, time series analysis, and network analysis. Furthermore, DEG establishes a theoretical framework that provides insights into the irregularities observed in graph centrality measures and in the spectra of operators acting on graphs. We demonstrate the effectiveness of DEG in detecting changes in the dynamics of signals defined on both synthetic and real-world graphs, by defining a mix process on random geometric graphs or those exhibiting small-world properties. Our results indicate that DEG effectively captures the irregularity of graph signals across various network configurations, successfully differentiating between distinct levels of randomness and connectivity. Consequently, DEG provides a comprehensive framework for entropy analysis of various data types, enabling new applications of dispersion entropy not previously feasible, and uncovering nonlinear relationships between graph signals and their graph topology.

Threshold for detecting high dimensional geometry in anisotropic random geometric graphs

AbstractIn the anisotropic random geometric graph model, vertices correspond to points drawn from a high‐dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős‐Rényi graph with the same edge probability. If is the number of vertices and is the vector of eigenvalues, Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017 shows that detection is possible when and impossible when . We show detection is impossible when , closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer, Geo. Aspects Func. Analysis: Israel seminar, 2017.

On the first and second largest components in the percolated random geometric graph

The percolated random geometric graph Gn(λ,p) has vertex set given by a Poisson Point Process in the square [0,n]2, and every pair of vertices at distance at most 1 independently forms an edge with probability p. For a fixed p, Penrose proved that there is a critical intensity λc=λc(p) for the existence of a giant component in Gn(λ,p). Our main result shows that for λ>λc, the size of the second-largest component is a.a.s. of order (logn)2. Moreover, we prove that the size of the largest component rescaled by n converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of G(λ,p) (which is the infinite volume version of Gn(λ,p)). Moreover, we prove that for a large class of graphs converging in a suitable sense to G(λ,1), the corresponding critical percolation thresholds converge as well to the ones of G(λ,1).

CoRMAC: A Connected Random Topology Formation With Maximal Area Coverage in Wireless Ad-Hoc Networks

R1C2Random geometric graphs can be used in constructing topology formations for wireless ad-hoc networks (WANETs), including wireless sensor networks, flying ad-hoc networks, and others. They are useful for energy-saving schemes where a randomly alternating subset of deployed nodes is turned off temporarily (without compromising the network connectivity). This also improves network security by periodically re-routing the data flow, which makes it harder for third parties to run a traffic analysis. In the area and target coverage scenarios, a WANET deployment is desired to have maximal area coverage efficiency, which implies covering the largest possible area using the fewest number of nodes by maintaining the connectivity as well. Although deterministic topology formations offering optimal area coverage are known, a random node deployment method that grants connectivity and area coverage maximality has not been presented, to the best of our knowledge. This study introduces a novel topology formation method, called CoRMAC, that consistently yields tree-formed random geometric graphs that are guaranteed to be connected and offer maximal area coverage for any given area size and node cardinality. The area coverage maximality of CoRMAC is formally proven. Extensive theoretical and computational analyses have been provided to demonstrate its graph-theoretic, Euclidean, and networking features. Moreover, comparisons were made with other approaches to show the effectiveness of the proposed method.

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.

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.

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.

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 $a&gt;1$ . 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 $\theta &lt; (1+p)/2$ , 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 &gt; (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.

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.

An analysis of probabilistic forwarding of coded packets on random geometric graphs

We consider the problem of energy-efficient broadcasting on large ad-hoc networks. Ad-hoc networks are generally modelled using random geometric graphs (RGGs). Here, nodes are deployed uniformly in a square area around the origin, and any two nodes which are within Euclidean distance of 1 are assumed to be able to receive each other’s broadcast. A source node at the origin encodes k data packets of information into n(>k) coded packets and transmits them to all its one-hop neighbours. The encoding is such that, any node that receives at least k out of the n coded packets can retrieve the original k data packets. Every other node in the network follows a probabilistic forwarding protocol; upon reception of a previously unreceived packet, the node forwards it with probability p and does nothing with probability 1−p. We are interested in the minimum forwarding probability which ensures that a large fraction of nodes can decode the information from the source. We deem this a near-broadcast. The performance metric of interest is the expected total number of transmissions at this minimum forwarding probability, where the expectation is over both the forwarding protocol as well as the realization of the RGG. In comparison to probabilistic forwarding with no coding, our treatment of the problem indicates that, with a judicious choice of n, it is possible to reduce the expected total number of transmissions while ensuring a near-broadcast.

Rates of convergence for Laplacian semi-supervised learning with low labeling rates

We investigate graph-based Laplacian semi-supervised learning at low labeling rates (ratios of labeled to total number of data points) and establish a threshold for the learning to be well posed. Laplacian learning uses harmonic extension on a graph to propagate labels. It is known that when the number of labeled data points is finite while the number of unlabeled data points tends to infinity, the Laplacian learning becomes degenerate and the solutions become roughly constant with a spike at each labeled data point. In this work, we allow the number of labeled data points to grow to infinity as the total number of data points grows. We show that for a random geometric graph with length scale $$\varepsilon >0$$ , if the labeling rate $$\beta \ll \varepsilon ^2$$ , then the solution becomes degenerate and spikes form. On the other hand, if $$\beta \gg \varepsilon ^2$$ , then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of $$O(\varepsilon \beta ^{-1/2})$$ for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study p-Laplacian regularization and show the same degeneracy result when $$\beta \ll \varepsilon ^p$$ . The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use $$\Gamma $$ -convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.

On the clique number of noisy random geometric graphs

AbstractLet be a random geometric graph, and then for we construct a ‐perturbed noisy random geometric graph where each existing edge in is removed with probability , while and each non‐existent edge in is inserted with probability . We give asymptotically tight bounds on the clique number for several regimes of parameter.

Structural Complexity of One-Dimensional Random Geometric Graphs

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> nodes randomly scattered in [0, 1] that connect if they are within the connection range <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r\in [{0,1}]$ </tex-math></inline-formula> . We provide bounds on the number of possible structures which give universal upper bounds on the structural entropy that hold for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> and distribution of the node locations. For fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> , the number of structures is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Theta (a^{2n})$ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a=a(r)=2 \cos {\left ({\frac {\pi }{\lceil 1/r \rceil +2}}\right)}$ </tex-math></inline-formula> , and therefore the structural entropy is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2n\log _{2} a(r) + O(1)$ </tex-math></inline-formula> . For large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , we derive bounds on the structural entropy normalized by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , and evaluate them for independent and uniformly distributed node locations. When the connection range <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r_{n}$ </tex-math></inline-formula> is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(1/n)$ </tex-math></inline-formula> , the obtained upper bound is given in terms of a function that increases with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n r_{n}$ </tex-math></inline-formula> and asymptotically attains 2 bits per node. If the connection range is bounded away from zero and one, the upper and lower bounds decrease linearly with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> , as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2(1-r)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(1-r)\log _{2} e$ </tex-math></inline-formula> , respectively. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r_{n}$ </tex-math></inline-formula> is vanishing but dominates <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/n$ </tex-math></inline-formula> (e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r_{n} \propto \ln n / n$ </tex-math></inline-formula> ), the normalized entropy is between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2} e \approx 1.44$ </tex-math></inline-formula> and 2 bits per node. We also give a simple encoding scheme for random structures that requires 2 bits per node. The upper bounds in this paper easily extend to the entropy of the labeled random graph model, since this is given by the structural entropy plus a term that accounts for all the permutations of node labels that are possible for a given structure, which is no larger than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2}(n!) = n \log _{2} n {-} n + O(\log _{2} n)$ </tex-math></inline-formula> .

Large deviations for the volume of k-nearest neighbor balls

This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.

Phase transition in noisy high-dimensional random geometric graphs

Phase transition in noisy high-dimensional random geometric graphs