Random Geometric Graphs Research Articles

An architecture for resilient intrusion detection in ad-hoc networks

We study efficient and lightweight Intrusion Detection Systems (IDS) for ad-hoc networks via the prism of IPv6-enabled Wireless Sensor Actuator Networks. These networks consist of highly constrained devices able to communicate wirelessly in an ad-hoc fashion, thus following the architecture of ad-hoc networks. Current state-of-the-art (IDS) has been developed taking into consideration the architecture of conventional computer networks, and as such they do not efficiently address the paradigm of ad-hoc networks, that is highly relevant in emergent networks, such as the Internet of Things (IoT). In this context, the network properties of resilience and redundancy have not been studied yet. In this work, we firstly identify a trade-off between the communication overhead and energy consumption of an IDS (as captured by the number of active IDS agents in the network) and the performance of the system in terms of successfully identifying attacks. In order to fine tune this trade-off, we model such networks as Random Geometric Graphs; a rigorous approach that allows us to capture underlying structural properties of the network. We then introduce a novel IDS architectural approach that consists of a central IDS agent a set of distributed IDS agents deployed uniformly at random over the network area. These nodes are able to efficiently detect attacks at the networking layer in a collaborative manner by monitoring locally available network information provided by IoT routing protocols such as RPL. Our detailed experimental evaluation demonstrates significant performance gains in terms of communication overhead and energy consumption while maintaining high detection rates. We also show that the performance of our IDS in ad-hoc networks does not rely on the size of the network but on fundamental underling network properties, such as the network topology and the average degree of the nodes. Conducted experiments show that our proposed IDS architecture is resilient against frequent topology changs due to nodes failures.

Connectivity of dynamic graphs in multiply connected spaces

Dynamic geometric graphs are natural mathematical models of many real-world systems placed and moving in space: computer ad hoc networks, transport systems, territorial distributed systems for various purposes. An important property of such graphs is connectivity, which is difficult to maintain during movement due to the presence of obstacles on the ground. In this paper, a model of a multiply connected region with obstacles of the “city blocks” type is constructed and the behavior of the characteristics of dynamic graphs located in such domains is studied. A probabilistic approach to the study of graphs is proposed, in which their characteristics are considered as random processes. For graphs of different scales, dependences of the connectivity probability, the number of components on the parameters of a multiply connected region, and the radius of stable signal reception / transmission were found. The mathematical expectation of the number of components in the starting random geometric graph is found. The significant influence not only of geometrical parameters, but also of the topological characteristics of a multiply-connected domain has been revealed. Graphs of changes in the probability of connectedness of a dynamic graph over time are constructed on the basis of calculating the average value over the set of realizations of the random process of moving network nodes. They are characterized by a periodic component that correlates with the structure of a multiply connected region, and a component that exponentially decreases with time. The dependence of the probability of connectedness of the graph on the direction of the network displacement vector was studied, which turned out to be very significant. The results obtained give an idea of the influence of a multiply-connected domain on the dynamics of graphs, and can be used in control algorithms for mobile distributed systems to ensure their spatial connectivity.

Learning random points from geometric graphs or orderings

Let Xv for v∈V be a family of n iid uniform points in the square . Suppose first that we are given the random geometric graph , where vertices u and v are adjacent when the Euclidean distance dE(Xu,Xv) is at most r. Let n3/14≪r≪n1/2. Given G (without geometric information), in polynomial time we can with high probability approximately reconstruct the hidden embedding, in the sense that “up to symmetries,” for each vertex v we find a point within distance about r of Xv; that is, we find an embedding with “displacement” at most about r. Now suppose that, instead of G we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v. Then, with high probability, in polynomial time we can find an embedding with displacement .

Scheduling Wireless Links in the Physical Interference Model by Fractional Edge Coloring

We consider the problem of scheduling the links of wireless mesh networks for capacity maximization in the physical interference model. We represent such a network by an undirected graph $G$, with vertices standing for network nodes and edges for links. We define network capacity to be $1/\chi'^*_\mathrm{phys}(G)$, where $\chi'^*_\mathrm{phys}(G)$ is a novel edge-chromatic indicator of $G$, one that modifies the notion of $G$'s fractional chromatic index. This index asks that the edges of $G$ be covered by matchings in a certain optimal way. The new indicator does the same, but requires additionally that the matchings used be all feasible in the sense of the physical interference model. Sometimes the resulting optimal covering of $G$'s edge set by feasible matchings is simply a partition of the edge set. In such cases, the index $\chi'^*_\mathrm{phys}(G)$ becomes the particular case that we denote by $\chi'_\mathrm{phys}(G)$, a similar modification of $G$'s well-known chromatic index. We formulate the exact computation of $\chi'^*_\mathrm{phys}(G)$ as a linear programming problem, which we solve for an extensive collection of random geometric graphs used to instantiate networks in the physical interference model. We have found that, depending on node density (number of nodes per unit deployment area), often $G$ is such that $\chi'^*_\mathrm{phys}(G)<\chi'_\mathrm{phys}(G)$. This bespeaks the possibility of increased network capacity by virtue of simply defining it so that edges are colored in the fractional, rather than the integer, sense.

Localization in random geometric graphs with too many edges

We consider a random geometric graph $G(\chi_{n},r_{n})$, given by connecting two vertices of a Poisson point process $\chi_{n}$ of intensity $n$ on the $d$-dimensional unit torus whenever their distance is smaller than the parameter $r_{n}$. The model is conditioned on the rare event that the number of edges observed, $|E|$, is greater than $(1+\delta )\mathbb{E}(|E|)$, for some fixed $\delta >0$. This article proves that upon conditioning, with high probability there exists a ball of diameter $r_{n}$ which contains a clique of at least $\sqrt{2\delta \mathbb{E}(|E|)}(1-\varepsilon )$ vertices, for any given $\varepsilon >0$. Intuitively, this region contains all the “excess” edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be nonconvex.

Optimal Cheeger cuts and bisections of random geometric graphs

Let $d\geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on $n$ random points in a $d$-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of $n$) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large $n$ to an analogous Cheeger-type constant of the domain. Previously, García Trillos et al. had shown this for $d\geq 3$ but had required an extra condition on the distance parameter when $d=2$.

Probabilistic Forwarding of Coded Packets on Networks

We consider a scenario of broadcasting information over a network of nodes connected by noiseless communication links. A source node in the network has some data packets to broadcast. It encodes these data packets into $n$ coded packets in such a way that any node in the network that receives any $k$ out of the $n$ coded packets will be able to retrieve all the original data packets. The source transmits the $n$ coded packets to its one-hop neighbours. Every other node in the network follows a probabilistic forwarding protocol, in which it forwards a previously unreceived packet to all its neighbours with a certain probability $p$ . We say that the information from the source undergoes a “near-broadcast” if the expected fraction of nodes that receive at least $k$ of the $n$ coded packets is close to 1. The forwarding probability $p$ is chosen so as to minimize the expected total number of transmissions needed for a near-broadcast. We study how, for a given $k$ , this minimum forwarding probability and the associated expected total number of packet transmissions varies with $n$ . We specifically analyze the probabilistic forwarding of coded packets on two network topologies: binary trees and square grids. For trees, our analysis shows that for fixed $k$ , the expected total number of transmissions increases with $n$ . On the other hand, on grids, a judicious choice of $n$ significantly reduces the expected total number of transmissions needed for a near-broadcast. Behaviour similar to that of the grid is also observed in other well-connected network topologies such as random geometric graphs and random regular graphs.

Testing multivariate uniformity based on random geometric graphs

We present new families of goodness-of-fit tests of uniformity on a full-dimensional set $W\subset \mathbb{R}^{d}$ based on statistics related to edge lengths of random geometric graphs. Asymptotic normality of these statistics is proven under the null hypothesis as well as under fixed alternatives. The derived tests are consistent and their behaviour for some contiguous alternatives can be controlled. A simulation study suggests that the procedures can compete with or are better than established goodness-of-fit tests. We show with a real data example that the new tests can detect non-uniformity of a small sample data set, where most of the competitors fail.

Lower large deviations for geometric functionals

This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson–Voronoi cells, as well as power-weighted edge lengths in the random geometric, $k$-nearest neighbor and relative neighborhood graph.

Homophily as a Process Generating Social Networks: Insights from Social Distance Attachment Model

Real-world social networks often exhibit high levels of clustering, positive degree assortativity, short average path lengths (small-world property) and right-skewed but rarely power law degree distributions. On the other hand homophily, defined as the propensity of similar agents to connect to each other, is one of the most fundamental social processes observed in many human and animal societies. In this paper we examine the extent to which homophily is sufficient to produce the typical structural properties of social networks. To do so, we conduct a simulation study based on the Social Distance Attachment (SDA) model, a particular kind of Random Geometric Graph (RGG), in which nodes are embedded in a social space and connection probabilities depend functionally on distances between nodes. We derive the form of the model from first principles based on existing analytical results and argue that the mathematical construction of RGGs corresponds directly to the homophily principle, so they provide a good model for it. We find that homophily, especially when combined with a random edge rewiring, is sufficient to reproduce many of the characteristic features of social networks. Additionally, we devise a hybrid model combining SDA with the configuration model that allows generating homophilic networks with arbitrary degree sequences and we use it to study interactions of homophily with processes imposing constraints on degree distributions. We show that the effects of homophily on clustering are robust with respect to distribution constraints, while degree assortativity can be highly dependent on the particular kind of enforced degree sequence.

Limit theory for isolated and extreme points in hyperbolic random geometric graphs

Given $\alpha \in (0, \infty )$ and $r \in (0, \infty )$, let ${\mathcal {D}}_{r, \alpha }$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha ^{2}$. Consider the Poisson point process having uniform intensity density on ${\mathcal {D}}_{R, \alpha }$, with $R = 2 \log (n/ \nu )$, $n \in \mathbb {N}$, and $\nu < n$ a fixed constant. The points are projected onto ${\mathcal {D}}_{R, 1}$, preserving polar coordinates, yielding a Poisson point process ${\mathcal {P}}_{\alpha , n}$ on ${\mathcal {D}}_{R, 1}$. The hyperbolic geometric graph ${\mathcal {G}}_{\alpha , n}$ on ${\mathcal {P}}_{\alpha , n}$ puts an edge between pairs of points of ${\mathcal {P}}_{\alpha , n}$ which are distant at most $R$. This model has been used to express fundamental features of complex networks in terms of an underlying hyperbolic geometry. For $\alpha \in (1/2, \infty )$ we establish expectation and variance asymptotics as well as asymptotic normality for the number of isolated and extreme points in ${\mathcal {G}}_{\alpha , n}$ as $n \to \infty $. The limit theory and renormalization for the number of isolated points are highly sensitive on the curvature parameter. In particular, for $\alpha \in (1/2, 1)$, the variance is super-linear, for $\alpha = 1$ the variance is linear with a logarithmic correction, whereas for $\alpha \in (1, \infty )$ the variance is linear. The central limit theorem fails for $\alpha \in (1/2, 1)$ but it holds for $\alpha \in (1, \infty )$.

An Analytic Latency Model for a Next-Hop Data-Ferrying Swarm on Random Geometric Graphs

Next-hop data-ferrying is a data-driven approach to ferrying data between graph components of a disconnected network. In contrast to pre-planned routing methods like the Traveling Salesman tour, only the next-hop is planned, allowing the ferry to be reactive to the data flows within the network. When multiple ferries are used, explicit coordination between them is difficult due to the data-driven approach of the algorithm, and the distributed and disconnected nature of the problem. However, coordinated behavior can still be produced through swarm intelligent means and while this is a useful pragmatic result, mathematical models describing system properties are difficult to develop. This generally results in algorithm correctness being demonstrated through simulation. In this article, we describe the development of a mathematical model of such a next-hop ferrying swarm. We show that the closed-form expression for user data latency is a close match to simulation results for Random Geometric Graphs under the assumption of a zero expected degree. In non-zero degree graphs, a near constant offset between the model and simulation is observed.

Plane and planarity thresholds for random geometric graphs

A random geometric graph, [Formula: see text], is formed by choosing [Formula: see text] points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most [Formula: see text]. For a given constant [Formula: see text], we show that [Formula: see text] is a distance threshold function for [Formula: see text] to have a connected subgraph on [Formula: see text] points. Based on this, we show that [Formula: see text] is a distance threshold for [Formula: see text] to be plane, and [Formula: see text] is a distance threshold to be planar. We also investigate distance thresholds for [Formula: see text] to have a non-crossing edge, a clique of a given size, and an independent set of a given size.

Gaussian fluctuations for edge counts in high-dimensional random geometric graphs

Consider a stationary Poisson point process in Rd and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have midpoint in the d-dimensional unit ball. A quantitative central limit theorem for this counting statistic is derived, as the space dimension d and the intensity of the Poisson point process tend to infinity simultaneously.

Shape of shortest paths in random spatial networks.

In the classic model of first-passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order L^{χ}, while the transversal fluctuations, known as wandering, grow as L^{ξ}. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ=3/5 and χ=1/5, or ξ=7/10 and χ=2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel-time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first-passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbor random geometric graphs, where via Monte Carlo simulations we find ξ=0.60±0.01 and χ=0.20±0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β skeletons and the Delaunay triangulation and are characterized by the values ξ=0.70±0.01 and χ=0.40±0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the so-called Kardar-Parisi-Zhang (KPZ) relation χ=2ξ-1 is satisfied for all these models. These results shed some light on the Euclidean first-passage process but also raise some theoretical questions about the scaling laws and the derivation of the exponent values and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

A local search algorithm with reinforcement learning based repair procedure for minimum weight independent dominating set

The minimum weight independent dominating set problem (MWIDS) is a famous NP-hard combinatorial optimization problem. We herein propose a local search algorithm with reinforcement-learning-based repair procedure (LSRR). The proposed algorithm combines local search with repair procedure based on the mind of reinforcement learning. This algorithm iterates through three procedures: the greedy procedure to improve the initial solution, the local search procedure to further improve the solution, and the repair procedure to destroy the initial solution and then reconstruct a new solution. In addition, because of the particularity of the weight functions in all benchmarks, we propose three novel scoring functions. Experiments are performed on two types of graphs including random graphs and random geometric graphs. Experimental results display that LSRR outperforms the previous MWIDS algorithms significantly.

Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold $${\mathcal {M}}$$ in $$\mathbb {R}^d$$ as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of $$O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )$$ to the eigenvalues and eigenfunctions of the weighted Laplace–Beltrami operator of $${\mathcal {M}}$$ . No information on the submanifold $${\mathcal {M}}$$ is needed in the construction of the graph or the “out-of-sample extension” of the eigenvectors. Of independent interest is a generalization of the rate of convergence of empirical measures on submanifolds in $$\mathbb {R}^d$$ in infinity transportation distance.

Lossy asymptotic equipartition property for geometric networked data structures

This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation principles for properly defined empirical measures on CGRGs. As a motivation for this article, we apply our results to some data from Wireless Sensor Network for Monitoring Water Quality from a Lake.

Approximation Theory for Connectivity of Ad Hoc Wireless Networks With Node Faults

In this letter, we propose an approximation theory for connectivity of general random geometric graphs (RGGs) with stochastic node faults, which is regarded as an abstract model of ad hoc wireless sensor networks (WSNs) with unreliable nodes. In the model, each node in an RGG is independently at fault and the disconnection probability of a resulting survival network is evaluated. The proposed approximation formula for the probability is asymptotically correct and reveals the phase transition phenomena of connectivity in the model. In addition, we obtain phase transition thresholds of a practical Rayleigh single-input single-output (SISO) model and Rayleigh multiple-input multiple-output (MIMO) model, which shows the effectiveness of the MIMO system against attenuation and random node faults. The numerical results in Rayleigh MIMO model show that the approximation formula well estimates the connection probability even for finite graphs and that it provides useful information for design of WSN immune to node faults.

Improved Bounds for Max Consensus in Wireless Networks

In consensus problems, the goal is for the nodes of a network to converge to a certain quantity or a function of their values using local communications. In the maximum value consensus problem, the objective of these communications is for all the nodes to converge to the maximum of their initial values. There are two existing algorithms for the maximum value consensus problem in asynchronous networks: Random-Pairwise-Max and Random-Broadcast-Max for which the bounds on the mean convergence time have been derived in the literature. In this paper, we derive tighter bounds on the expected convergence time of these two algorithms when run on grid networks and random geometric graphs, respectively—two models commonly used to capture salient properties of wireless networks. We show that Random-Pairwise-Max run on a 2-D grid graph with $n$ nodes converges in expectation after $\mathcal {O}(n^{3/2})$ iterations, and Random-Broadcast-Max run on a random geometric graph with $n$ nodes converges in expectation after $\mathcal {O}((n / \log n)^{3/2})$ iterations. These bounds improve over the previous best-known upper bounds by factors of $\sqrt{n}\log n$ and $\log n + \log ^2 n$ , respectively. Experiments illustrate that the proposed bounds can be up to $95\%$ tighter than the previous state-of-the-art bounds. Furthermore, we enhance the proposed bounds by introducing probabilistic network link failures, e.g., to model packet drops in wireless networks.