Random Geometric Graphs Research Articles

A Unified Framework for Correlation Mining in Ultra-High Dimension

Many applications benefit from theory relevant to the identification of variables having large correlations or partial correlations in high dimension. Recently there has been progress in the ultra-high dimensional setting when the sample size <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> is fixed and the dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> tends to infinity. Despite these advances, the correlation screening framework suffers from practical, methodological and theoretical deficiencies. For instance, previous correlation screening theory requires that the population covariance matrix be sparse and block diagonal. This block sparsity assumption is however restrictive in practical applications. As a second example, correlation and partial correlation screening requires the estimation of dependence measures, which can be computationally prohibitive. In this paper, we propose a unifying approach to correlation and partial correlation mining that is not restricted to block diagonal correlation structure, thus yielding a methodology that is suitable for modern applications. By making connections to random geometric graphs, the number of highly correlated or partial correlated variables are shown to have compound Poisson finite-sample characterizations, which hold for both the finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> case and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> tends to infinity. The unifying framework also demonstrates a duality between correlation and partial correlation screening with theoretical and practical consequences.

Normal approximation via non-linear exchangeable pairs

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.

Directed random geometric graphs: structural and spectral properties

In this work we analyze structural and spectral properties of a model of directed random geometric graphs: given n vertices uniformly and independently distributed on the unit square, a directed edge is set between two vertices if their distance is smaller than the connection radius , which is randomly drawn from a Pareto distribution. This Pareto distribution is characterized by the power-law decay α and the lower bound of its support ; thus the graphs depend on three parameters . By increasing , for fixed , the model transits from isolated vertices () to complete graphs (). We first propose a phenomenological expression for the average degree which works well for α > 3, when k is a self-averaging quantity. Then we numerically demonstrate that , for all α, where is the number of nonisolated vertices of G. Finally, we explore the spectral properties of by the use of adjacency matrices represented by diluted random matrix ensembles; a non-Hermitian and a Hermitian one. We find that is a good scaling parameter of spectral and eigenvector properties of G mainly for large α.

Efficiently generating geometric inhomogeneous and hyperbolic random graphs

AbstractHyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks. In particular, they have a power-law degree distribution with controllable exponent $\beta$ and high clustering that can be controlled via the temperature $T$ .We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to $T = 0$ . We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases.

Entropy of labeled versus unlabeled networks.

The structure of a network is an unlabeled graph, yet graphs in most models of complex networks are labeled by meaningless random integers. Is the associated labeling noise always negligible, or can it overpower the network-structural signal? To address this question, we introduce and consider the sparse unlabeled versions of popular network models and compare their entropy against the original labeled versions. We show that labeled and unlabeled Erdős-Rényi graphs are entropically equivalent, even though their degree distributions are very different. The labeled and unlabeled versions of the configuration model may have different prefactors in their leading entropy terms, although this remains conjectural. Our main results are upper and lower bounds for the entropy of labeled and unlabeled one-dimensional random geometric graphs. We show that their unlabeled entropy is negligible in comparison with the labeled entropy. This means that in sparse networks the entropy of meaningless labeling may dominate the entropy of the network structure. The main implication of this result is that the common practice of using exchangeable models to reason about real-world networks with distinguishable nodes may introduce uncontrolled aberrations into conclusions made about these networks, suggesting a need for a thorough reexamination of the statistical foundations and key results of network science.

The distribution of the number of isolated nodes in the 1-Dimensional soft random geometric graph

We study the number of isolated nodes in a soft random geometric graph whose vertices constitute a Poisson process on the torus of length L (the line segment [0,L] with periodic boundary conditions), and where an edge is present between two nodes with a probability which depends on the distance between them. Edges between distinct pairs of nodes are mutually independent. In a suitable scaling regime, we show that the number of isolated nodes converges in total variation to a Poisson random variable. The result implies an upper bound on the probability that the random graph is connected.

Localization game for random geometric graphs

The localization game is a two player combinatorial game played on a graph G=(V,E). The cops choose a set of vertices S1⊆V with |S1|=k. The robber then chooses a vertex v∈V whose location is hidden from the cops, but the cops learn the graph distance between the current position of the robber and the vertices in S1. If this information is sufficient to locate the robber, the cops win immediately; otherwise the cops choose another set of vertices S2⊆V with |S2|=k, and the robber may move to a neighboring vertex. The new distances to the robber are presented, and if the cops can deduce the new location of the robber based on all information they accumulated thus far, then they win; otherwise, a new round begins. If the robber has a strategy to avoid being captured, then she wins. The localization number is defined to be the smallest integer k so that the cops win the game. In this paper we determine the localization number (up to poly-logarithmic factors) of the random geometric graph G∈G(n,r) slightly above the connectivity threshold.

Hamiltonicity of graphs perturbed by a random geometric graph

AbstractWe study Hamiltonicity in graphs obtained as the union of a deterministic ‐vertex graph with linear degrees and a ‐dimensional random geometric graph , for any . We obtain an asymptotically optimal bound on the minimum for which a.a.s. is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.

An anomalous topological phase transition in spatial random graphs

Clustering–the tendency for neighbors of nodes to be connected–quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric to non-geometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finite-size scaling behavior of clustering. Moreover, a slow decay of clustering in the non-geometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.

Gromov centrality: A multiscale measure of network centrality using triangle inequality excess.

Centrality measures quantify the importance of a node in a network based on different geometric or diffusive properties, and focus on different scales. Here, we adopt a geometrical viewpoint to define a multiscale centrality in networks. Given a metric distance between the nodes, we measure the centrality of a node by its tendency to be close to geodesics between nodes in its neighborhood, via the concept of triangle inequality excess. Depending on the size of the neighborhood, the resulting Gromov centrality defines the importance of a node at different scales in the graph, and it recovers as limits well-known concepts such as the clustering coefficient and closeness centrality. We argue that Gromov centrality is affected by the geometric and boundary constraints of the network, and illustrate how it can help distinguish different types of nodes in random geometric graphs and empirical transportation networks.

Analysis and algorithms for ℓ-based semi-supervised learning on graphs

This paper addresses theory and applications of ℓ p -based Laplacian regularization in semi-supervised learning. The graph p -Laplacian for p > 2 has been proposed recently as a replacement for the standard ( p = 2 ) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence results for p -Laplace problems on k -nearest neighbor ( k -NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on k -NN graphs, the p -Laplacian retains information about the data distribution as p → ∞ and Lipschitz learning ( p = ∞ ) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the p -Laplacian forgets the data distribution as p → ∞ . We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity. In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic p -Laplace equations on weighted graphs for p > 2 . We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the p -Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ( p = ∞ ) performs well with very few labels on k -NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on k -NN graphs.

Cascading traffic jamming in a two-dimensional Motter and Lai model.

We study the cascading traffic jamming on a two-dimensional random geometric graph using the Motter and Lai model. The traffic jam is caused by a localized attack incapacitating a circular region or a line of a certain size, as well as a dispersed attack on an equal number of randomly selected nodes. We investigate if there is a critical size of the attack above which the network becomes completely jammed due to cascading jamming, and how this critical size depends on the average degree 〈k〉 of the graph, on the number of nodes N in the system, and the tolerance parameter α of the Motter and Lai model.

Fréchet distribution in geometric graphs for drone networks.

In this paper, we focus on the link density in random geometric graphs (RGGs) with a distance-based connection function. After deriving the link density in D dimensions, we focus on the two-dimensional (2D) and three-dimensional (3D) space and show that the link density is accurately approximated by the Fréchet distribution, for any rectangular space. We derive expressions, in terms of the link density, for the minimum number of nodes needed in the 2D and 3D spaces to ensure network connectivity. These results provide first-order estimates for, e.g., a swarm of drones to provide coverage in a disaster or crowded area.

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centered in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

Data fusion reconstruction of spatially embedded complex networks

Abstract We introduce a kernel Lasso (kLasso) approach which is a type of sparse optimization that simultaneously accounts for spatial regularity and structural sparsity to reconstruct spatially embedded complex networks from time-series data about nodal states. Through the design of a spatial kernel function motivated by real-world network features, the proposed kLasso approach exploits spatial embedding distances to penalize overabundance of spatially long-distance connections. Examples of both random geometric graphs and real-world transportation networks show that the proposed method improves significantly upon existing network reconstruction techniques that mainly concern sparsity but not spatial regularity. Our results highlight the promise of data and information fusion in the reconstruction of complex networks, by utilizing both microscopic node-level dynamics (e.g. time series data) and macroscopic network-level information (metadata or other prior information).

Sub-tree counts on hyperbolic random geometric graphs

AbstractThe hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.

Convergence of combinatorial gravity

We present a new regularization of Euclidean Einstein gravity in terms of (sequences of) graphs. In particular, we define a discrete Einstein-Hilbert action that converges to its manifold counterpart on sufficiently dense random geometric graphs (more generally on any sequence of graphs that converges suitably to the manifold in the sense of Gromov-Hausdorff). Our construction relies crucially on the Ollivier curvature of optimal transport theory. Our methods also allow us to define an analogous discrete action for Klein-Gordon fields. These results are part of the ongoing program combinatorial approach to quantum gravity where we seek to generate graphs that approximate manifolds as metric-measure structures.

The impact of geometry on monochrome regions in the flip Schelling process

Schelling's classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex.We investigate the probability that an edge {u,v} is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive.As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge {u,v} makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant c>0 such that the expected fraction of monochrome edges after the FSP is at least 1/2+c. (2) For Erdős–Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most 1/2+o(1). Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

On the spectrum of dense random geometric graphs

In this paper we study the spectrum of the random geometric graph G(n,r), in a regime where the graph is dense and highly connected. In the Erdős–Rényi G(n,p) random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around 1. We show that such concentration does not occur in the G(n,r) case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than 1. In the special case where the vertices are distributed uniformly in the unit cube and r=1, we show that for every 0≤k≤d there are at least ( d k) eigenvalues near 1−2−k, and the limiting spectral gap is exactly 1/2. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.

Community detection and percolation of information in a geometric setting

AbstractWe make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.