Random Geometric Graphs Research Articles

The effects of random geometric graph structure and clustering on localizability of sensor networks

Graph rigidity provides the conditions of unique localizability for cooperative localization of wireless ad hoc and sensor networks. Specifically, redundant rigidity and 3-connectivity are necessary and sufficient conditions for unique localizability of generic configurations. In this article, we introduce a graph invariant for 3-connectivity, called 3-connectivity index. Using this index along with the rigidity and redundancy indices provided in previous work, we explore the rigidity and connectivity properties of two classes of graphs, namely, random geometric graphs and clustered graphs. We have found out that, in random geometric graphs and clustered graphs, it needs significantly less effort to achieve 3-connectivity once we obtain redundant rigidity. In reconsidering the general conditions for unique localizability, the most striking finding in random geometric graphs is that it is unlikely to observe a graph, in which 3-connectivity is satisfied before the graph becomes redundantly rigid. Therefore, in random geometric graphs, it is more likely sufficient to test only 3-connectivity for unique localizability. On the contrary to random geometric graphs, our findings indicate that 3-connectivity may be satisfied before the graph becomes redundantly rigid in clustered graphs, which means that, in clustered graphs, we have to test both redundant rigidity and 3-connectivity for unique localizability.

Estimating perimeter using graph cuts

AbstractWe investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆D= (0, 1)dwithd≥ 2, we are givennrandom independent and identically distributed points onDwhose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative toD) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices inD∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect tonand ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimensiond= 5: we show that in low dimensionsd= 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.

Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

Concentration bounds for the probabilities P(N≥M+r) and P(N≤M−r) are proved, where M is a median or the expectation of a subgraph count N associated with a random geometric graph built over a Poisson process. The lower tail bounds have a Gaussian decay and the upper tail inequalities satisfy an optimality condition. A remarkable feature is that the underlying Poisson process can have a.s. infinitely many points.The estimates for subgraph counts follow from tail inequalities for more general local Poisson U-statistics. These bounds are proved using recent general concentration results for Poisson U-statistics and techniques involving the convex distance for Poisson processes.

Fast Temporal Path Localization on Graphs via Multiscale Viterbi Decoding

We consider a problem of localizing a path-signal that evolves over time on a graph. A path-signal can be viewed as the trajectory of a moving agent on a graph in several consecutive time points. Combining dynamic programming and graph partitioning, we propose a path-localization algorithm with significantly reduced computational complexity. We analyze the localization error for the proposed approach both in the Hamming distance and the destination's distance between the path estimate and the true path using numerical bounds. Unlike usual theoretical bounds that only apply to restricted graph models, the obtained numerical bounds apply to all graphs and all non-overlapping graph-partitioning schemes. In random geometric graphs, we are able to derive a closed-form expression for the localization error bound, and a tradeoff between localization error and the computational complexity. Finally, we compare the proposed technique with the maximum likelihood estimate under the path constraint in terms of computational complexity and localization error, and show significant speedup (100 times) with comparable localization error (4 times) on a graph from real data. Variants of the proposed technique can be applied to tracking, road congestion monitoring, and brain signal processing.

The random connection model: Connectivity, edge lengths, and degree distributions

AbstractConsider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter Mn for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .

Infection dynamics on spatial small-world network models

The study of complex networks, and in particular of social networks, has mostly concentrated on relational networks, abstracting the distance between nodes. Spatial networks are, however, extremely relevant in our daily lives, and a large body of research exists to show that the distances between nodes greatly influence the cost and probability of establishing and maintaining a link. A random geometric graph (RGG) is the main type of synthetic network model used to mimic the statistical properties and behavior of many social networks. We propose a model, called REDS, that extends energy-constrained RGGs to account for the synergic effect of sharing the cost of a link with our neighbors, as is observed in real relational networks. We apply both the standard Watts-Strogatz rewiring procedure and another method that conserves the degree distribution of the network. The second technique was developed to eliminate unwanted forms of spatial correlation between the degree of nodes that are affected by rewiring, limiting the effect on other properties such as clustering and assortativity. We analyze both the statistical properties of these two network types and their epidemiological behavior when used as a substrate for a standard susceptible-infected-susceptible compartmental model. We consider and discuss the differences in properties and behavior between RGGs and REDS as rewiring increases and as infection parameters are changed. We report considerable differences both between the network types and, in the case of REDS, between the two rewiring schemes. We conclude that REDS represent, with the application of these rewiring mechanisms, extremely useful and interesting tools in the study of social and epidemiological phenomena in synthetic complex networks.

Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.

Scaling Limits and Generic Bounds for Exploration Processes

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.

Image-based model of the spectrin cytoskeleton for red blood cell simulation.

We simulate deformable red blood cells in the microcirculation using the immersed boundary method with a cytoskeletal model that incorporates structural details revealed by tomographic images. The elasticity of red blood cells is known to be supplied by both their lipid bilayer membranes, which resist bending and local changes in area, and their cytoskeletons, which resist in-plane shear. The cytoskeleton consists of spectrin tetramers that are tethered to the lipid bilayer by ankyrin and by actin-based junctional complexes. We model the cytoskeleton as a random geometric graph, with nodes corresponding to junctional complexes and with edges corresponding to spectrin tetramers such that the edge lengths are given by the end-to-end distances between nodes. The statistical properties of this graph are based on distributions gathered from three-dimensional tomographic images of the cytoskeleton by a segmentation algorithm. We show that the elastic response of our model cytoskeleton, in which the spectrin polymers are treated as entropic springs, is in good agreement with the experimentally measured shear modulus. By simulating red blood cells in flow with the immersed boundary method, we compare this discrete cytoskeletal model to an existing continuum model and predict the extent to which dynamic spectrin network connectivity can protect against failure in the case of a red cell subjected to an applied strain. The methods presented here could form the basis of disease- and patient-specific computational studies of hereditary diseases affecting the red cell cytoskeleton.

Locally Orthogonal Training Design for Cloud-RANs Based on Graph Coloring

We consider training-based channel estimation for a cloud radio access network (CRAN), in which a large amount of remote radio heads (RRHs) and users are randomly scattered over the service area. In this model, assigning orthogonal training sequences to all users will incur a substantial overhead to the overall network, and is even impossible when the number of users is large. Therefore, in this paper, we introduce the notion of local orthogonality, under which the training sequence of a user is orthogonal to those of the other users in its neighborhood. We model the design of locally orthogonal training sequences as a graph coloring problem. Then, based on the theory of random geometric graph, we show that the minimum training length scales in the order of $\ln K$, where $K$ is the number of users covered by a CRAN. This indicates that the proposed training design yields a scalable solution to sustain the need of large-scale cooperation in CRANs. Numerical results show that the proposed scheme outperforms other reference schemes.

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.

Quasirandom geometric networks from low-discrepancy sequences.

We define quasirandom geometric networks using low-discrepancy sequences, such as Halton, Sobol, and Niederreiter. The networks are built in d dimensions by considering the d-tuples of digits generated by these sequences as the coordinates of the vertices of the networks in a d-dimensional I^{d} unit hypercube. Then, two vertices are connected by an edge if they are at a distance smaller than a connection radius. We investigate computationally 11 network-theoretic properties of two-dimensional quasirandom networks and compare them with analogous random geometric networks. We also study their degree distribution and their spectral density distributions. We conclude from this intensive computational study that in terms of the uniformity of the distribution of the vertices in the unit square, the quasirandom networks look more random than the random geometric networks. We include an analysis of potential strategies for generating higher-dimensional quasirandom networks, where it is know that some of the low-discrepancy sequences are highly correlated. In this respect, we conclude that up to dimension 20, the use of scrambling, skipping and leaping strategies generate quasirandom networks with the desired properties of uniformity. Finally, we consider a diffusive process taking place on the nodes and edges of the quasirandom and random geometric graphs. We show that the diffusion time is shorter in the quasirandom graphs as a consequence of their larger structural homogeneity. In the random geometric graphs the diffusion produces clusters of concentration that make the process more slow. Such clusters are a direct consequence of the heterogeneous and irregular distribution of the nodes in the unit square in which the generation of random geometric graphs is based on.

Tree-ansatz percolation of hard spheres.

Suspensions of hard core spherical particles of diameter D with inter-core connectivity range δ can be described in terms of random geometric graphs, where nodes represent the sphere centers and edges are assigned to any two particles separated by a distance smaller than δ. By exploiting the property that closed loops of connected spheres become increasingly rare as the connectivity range diminishes, we study continuum percolation of hard spheres by treating the network of connected particles as having a tree-like structure for small δ/D. We derive an analytic expression of the percolation threshold which becomes increasingly accurate as δ/D diminishes and whose validity can be extended to a broader range of connectivity distances by a simple rescaling.

The Total Acquisition Number of Random Geometric Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs $\mathcal{G}(n,r)$ with $n$ vertices distributed uniformally at random in $[0,\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance is at most $r$. We show that asymptotically almost surely $a_t(\mathcal{G}(n,r)) = \Theta( n / (r \lg r)^2)$ for the whole range of $r=r_n \ge 1$ such that $r \lg r \le \sqrt{n}$. By monotonicity, asymptotically almost surely $a_t(\mathcal{G}(n,r)) = \Theta(n)$ if $r < 1$, and $a_t(\mathcal{G}(n,r)) = \Theta(1)$ if $r \lg r > \sqrt{n}$.

Super-Exponential Extinction Time of the Contact Process on Random Geometric Graphs

In this paper we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connection radius tending to infinity. We obtain that for any infection rate λ > 0, the contact process on these graphs survives a time super-exponential in the number of vertices.

Scalable Uplink Signal Detection in C-RANs via Randomized Gaussian Message Passing

Cloud Radio Access Network (C-RAN) is a promising architecture for unprecedented capacity enhancement in next-generation wireless networks thanks to the centralization and virtualization of base station processing. However, centralized signal processing in C-RANs involves high computational complexity that quickly becomes unaffordable when the network grows to a huge size. Among the first, this paper endeavours to design a scalable uplink signal detection algorithm, in the sense that both the complexity per unit network area and the total computation time remain constant when the network size grows. To this end, we formulate the signal detection in C-RAN as an inference problem over a bipartite random geometric graph. By passing messages among neighboring nodes, message passing (a.k.a. belief propagation) provides an efficient way to solve the inference problem over a sparse graph. However, the traditional message-passing algorithm is not guaranteed to converge, because the corresponding bipartite random geometric graph is locally dense and contains many short loops. As a major contribution of this paper, we propose a randomized Gaussian message passing (RGMP) algorithm to improve the convergence. Instead of exchanging messages simultaneously or in a fixed order, we propose to exchange messages asynchronously in a random order. The proposed RGMP algorithm demonstrates significantly better convergence performance than conventional message passing. The randomness of the message update schedule also simplifies the analysis, and allows the derivation of the convergence conditions for the RGMP algorithm. In addition, we propose a blockwise RGMP (B-RGMP) algorithm for practical implementation. The average computation time of B-RGMP remains constant when the network size increases.

Symmetric motifs in random geometric graphs

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen-Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit the probability that the closest nodes are symmetric approaches one, whilst in the intensive limit this probability depends upon the dimension.

Random Walker Coverage Analysis for Information Dissemination in Wireless Sensor Networks

The increasing technological progress in electronics provides network nodes with new and enhanced capabilities that allow the revisit of the traditional information dissemination (and collection) problem. The probabilistic nature of information dissemination using random walkers is exploited here to deal with challenges imposed by unconventional modern environments. In such systems, node operation is not deterministic (e.g., does not depend only on network nodes’ battery), but it rather depends on the particulars of the ambient environment (e.g., in the case of energy harvesting: sunshine, wind). The mechanism of information dissemination using one random walker is studied and analyzed in this paper under a different and novel perspective. In particular, it takes into account the stochastic nature of random walks, enabling further understanding of network coverage. A novel and original analysis is presented, which reveals the evolution network coverage by a random walker with respect to time. The derived analytical results reveal certain additional interesting aspects regarding network coverage, thus shedding more light on the random walker mechanism. Further analytical results, regarding the walker’s spatial movement and its associated neighborhood, are also confirmed through experimentation. Finally, simulation results considering random geometric graph topologies, which are suitable for modeling mobile environments, support and confirm the analytical findings.

Normal approximation for stabilizing functionals

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

AbstractGiven a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d‐dimensional cube for any fixed . Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 587–606, 2017