Random Geometric Graphs Research Articles

Concentration for Poisson functionals: Component counts in random geometric graphs

Upper bounds for the probabilities P(F≥EF+r) and P(F≤EF−r) are proved, where F is a certain component count associated with a random geometric graph built over a Poisson point process on Rd. The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay.For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in settings where the underlying Poisson process has infinite intensity measure.

Consensus dynamics on random rectangular graphs

A random rectangular graph (RRG) is a generalization of the random geometric graph (RGG) in which the nodes are embedded into a rectangle with side lengths a and b=1/a, instead of on a unit square [0,1]2. Two nodes are then connected if and only if they are separated at a Euclidean distance smaller than or equal to a certain threshold radius r. When a=1 the RRG is identical to the RGG. Here we apply the consensus dynamics model to the RRG. Our main result is a lower bound for the time of consensus, i.e., the time at which the network reaches a global consensus state. To prove this result we need first to find an upper bound for the algebraic connectivity of the RRG, i.e., the second smallest eigenvalue of the combinatorial Laplacian of the graph. This bound is based on a tight lower bound found for the graph diameter. Our results prove that as the rectangle in which the nodes are embedded becomes more elongated, the RRG becomes a ’large-world’, i.e., the diameter grows to infinity, and a poorly-connected graph, i.e., the algebraic connectivity decays to zero. The main consequence of these findings is the proof that the time of consensus in RRGs grows to infinity as the rectangle becomes more elongated. In closing, consensus dynamics in RRGs strongly depend on the geometric characteristics of the embedding space, and reaching the consensus state becomes more difficult as the rectangle is more elongated.

Probability, Trees and Algorithms

The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures.

Stochastic Geometry: Boolean model and random geometric graphs

This paper collects the four contributions which were presented during the session devoted to Stochastic Geometry at the journees MAS 2014 . It is focused in particular on several questions related to the transmission of information in a general sense in different random media. The underlying models include the Boolean model, simplicial complexes or geometric random graphs induced by a point process.

The coverage holes of the largest component of random geometric graph

In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.

Random geometric graph description of connectedness percolation in rod systems.

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.

Hierarchical routing over dynamic wireless networks

The topology of a mobile wireless network changes over time. Maintaining routes between all nodes requires the continuous transmission of control information, which consumes precious power and bandwidth resources. Many routing protocols have been developed, trading off control overhead and route quality. In this paper, we ask whether there exist low-overhead schemes that produce low-stretch routes, even in large networks where all the nodes are mobile. We present a scheme that maintains a hierarchical structure within which constant-stretch routes can be efficiently computed between every pair of nodes. The scheme rebuilds each level of the hierarchy periodically, at a rate that decreases exponentially with the level of the hierarchy. We prove that this scheme achieves constant stretch under a mild smoothness condition on the nodal mobility processes. Furthermore, we prove tight bounds for the network-wide control overhead under the additional assumption of the connectivity graph forming a doubling metric space. Specifically, we show that for a connectivity model combining the random geometric graph with obstacles, constant-stretch routes can be maintained with a total overhead of nlog2n bits of control information per time unit. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 47, 669-709, 2015

Bootstrap percolation and the geometry of complex networks

On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having N vertices, a dependent version of the Chung–Lu model. The process starts with infection rate p=p(N). Each uninfected vertex with at least r≥1 infected neighbors becomes infected, remaining so forever. We identify a function pc(N)=o(1) such that a.a.s. when p≫pc(N) the infection spreads to a positive fraction of vertices, whereas when p≪pc(N) the process cannot evolve. Moreover, this behavior is “robust” under random deletions of edges.

Asymptotic properties of Euclidean shortest-path trees in random geometric graphs

We consider asymptotic properties of two functionals on Euclidean shortest-path trees appearing in random geometric graphs in R2 which can be used, for example, as models for fixed-access telecommunication networks. First, we determine the asymptotic bivariate distribution of the two backbone lengths inside a certain class of typical Cox–Voronoi cells as the size of this cell grows unboundedly. The corresponding Voronoi tessellation is generated by a stationary Cox process which is concentrated on the edges of the random geometric graph and whose intensity tends to 0. The limiting random vector can be represented as a simple geometric functional of a decomposition of a typical Poisson–Voronoi cell induced by an independent random sector. Using similar methods, we consider the asymptotic bivariate distribution of the total lengths of the two subtrees inside the Cox–Voronoi cell.

Synchronizability of random rectangular graphs.

Random rectangular graphs (RRGs) represent a generalization of the random geometric graphs in which the nodes are embedded into hyperrectangles instead of on hypercubes. The synchronizability of RRG model is studied. Both upper and lower bounds of the eigenratio of the network Laplacian matrix are determined analytically. It is proven that as the rectangular network is more elongated, the network becomes harder to synchronize. The synchronization processing behavior of a RRG network of chaotic Lorenz system nodes is numerically investigated, showing complete consistence with the theoretical results.

Chasing robbers on percolated random geometric graphs

In this paper, we study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber on the vertices of a graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. We focus on $\G(n,r,p)$, a percolated random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the game of Cops and Robber played on $\G(n,r,p)$ for a wide range of $p=p(n)$ and $r=r(n)$.

Hyperbolic graph generator

Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks can be adequately modeled as random geometric graphs in hyperbolic spaces. In this paper, we present a computer program to generate such graphs. Besides real-world-like networks, the program can generate random graphs from other well-known graph ensembles, such as the soft configuration model, random geometric graphs on a circle, or Erdős–Rényi random graphs. The simulations show a good match between the expected values of different network structural properties and the corresponding empirical values measured in generated graphs, confirming the accurate behavior of the program. Program summaryProgram title: Hyperbolic graph generatorCatalogue identifier: AEXC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 101190No. of bytes in distributed program, including test data, etc.: 771660Distribution format: tar.gzProgramming language: C++.Computer: Any.Operating system: Any.Classification: 6.3, 4.13, 23.Nature of problem: Generation of graphs in hyperbolic spaces.Solution method: Implementation based on analytical equations.Additional comments: Can be used as a command-line tool or installed as a library to support more complex software.Running time: Depends on the number of nodes. A few seconds for the graph in the example provided.

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

Phase Transitions for Random Geometric Preferential Attachment Graphs

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Phase Transitions for Random Geometric Preferential Attachment Graphs

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Metapopulation persistence in random fragmented landscapes.

Habitat destruction and land use change are making the world in which natural populations live increasingly fragmented, often leading to local extinctions. Although local populations might undergo extinction, a metapopulation may still be viable as long as patches of suitable habitat are connected by dispersal, so that empty patches can be recolonized. Thus far, metapopulations models have either taken a mean-field approach, or have modeled empirically-based, realistic landscapes. Here we show that an intermediate level of complexity between these two extremes is to consider random landscapes, in which the patches of suitable habitat are randomly arranged in an area (or volume). Using methods borrowed from the mathematics of Random Geometric Graphs and Euclidean Random Matrices, we derive a simple, analytic criterion for the persistence of the metapopulation in random fragmented landscapes. Our results show how the density of patches, the variability in their value, the shape of the dispersal kernel, and the dimensionality of the landscape all contribute to determining the fate of the metapopulation. Using this framework, we derive sufficient conditions for the population to be spatially localized, such that spatially confined clusters of patches act as a source of dispersal for the whole landscape. Finally, we show that a regular arrangement of the patches is always detrimental for persistence, compared to the random arrangement of the patches. Given the strong parallel between metapopulation models and contact processes, our results are also applicable to models of disease spread on spatial networks.

A random geometric graph built on a time-varying Riemannian manifold

The theory of random geometric graph enables the study of complex networks through geometry. To analyze evolutionary networks, time-varying geometries are needed. Solutions of the generalized hyperbolic geometric flow are such geometries. Here we propose a scale-free network model, which is a random geometric graph on a two dimensional disc. The metric of the disc is a Ricci flat solution of the flow. The model is used to physically simulate the growth and aggregation of a type of cancer cell.

Local limit theorems via Landau–Kolmogorov inequalities

In this article, we prove new inequalities between some common probability metrics. Using these inequalities, we obtain novel local limit theorems for the magnetization in the Curie–Weiss model at high temperature, the number of triangles and isolated vertices in Erdős–Rényi random graphs, as well as the independence number in a geometric random graph. We also give upper bounds on the rates of convergence for these local limit theorems and also for some other probability metrics. Our proofs are based on the Landau–Kolmogorov inequalities and new smoothing techniques.

Statistical mechanics of random geometric graphs: Geometry-induced first-order phase transition.

Random geometric graphs (RGGs) can be formalized as hidden-variables models where the hidden variables are the coordinates of the nodes. Here we develop a general approach to extract the typical configurations of a generic hidden-variables model and apply the resulting equations to RGGs. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a nontrivial satisfaction problem: Given N nodes, a domain D, and a desired average connectivity 〈k〉, find, if any, the distribution of nodes having support in D and average connectivity 〈k〉. We find out that, in the thermodynamic limit, nodes are either uniformly distributed or highly condensed in a small region, the two regimes being separated by a first-order phase transition characterized by a O(N) jump of 〈k〉. Other intermediate values of 〈k〉 correspond to very rare graph realizations. The phase transition is observed as a function of a parameter a∈[0,1] that tunes the underlying geometry. In particular, a=1 indicates a rigid geometry where only close nodes are connected, while a=0 indicates a rigid antigeometry where only distant nodes are connected. Consistently, when a=1/2 there is no geometry and no phase transition. After discussing the numerical analysis, we provide a combinatorial argument to fully explain the mechanism inducing this phase transition and recognize it as an easy-hard-easy transition. Our result shows that, in general, ad hoc optimized networks can hardly be designed, unless to rely to specific heterogeneous constructions, not necessarily scale free.