10 research outputs found
On the relation between graph distance and Euclidean distance in random geometric graphs
Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=¿(vlogn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).Peer ReviewedPostprint (author's final draft
On the relation between graph distance and Euclidean distance in random geometric graphs
International audienceGiven any two vertices u, v of a random geometric graph G(n, r), denote by d_E(u, v) their Euclidean distance and by d_G(u, v) their graph distance. The problem of finding upper bounds on d_G(u, v) conditional on d_E(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper, we improve the known upper bounds for values of r = \omega\sqrt(log n)) (i.e. for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on d_G(u, v) conditional on d_E(u, v)
Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model
We compute the expected number of cars that have at least one two-hop path to
a fixed roadside unit in a one-dimensional vehicular ad hoc network in which
other cars can be used as relays to reach a roadside unit when they do not have
a reliable direct link. The pairwise channels between cars experience Rayleigh
fading in the random connection model, and so exist, with probability function
of the mutual distance between the cars, or between the cars and the roadside
unit. We derive exact equivalents for this expected number of cars when the car
density tends to zero and to infinity, and determine its behaviour using
an infinite oscillating power series in , which is accurate for all
regimes. We also corroborate those findings to a realistic situation, using
snapshots of actual traffic data. Finally, a normal approximation is discussed
for the probability mass function of the number of cars with a two-hop
connection to the origin. The probability mass function appears to be well
fitted by a Gaussian approximation with mean equal to the expected number of
cars with two hops to the origin.Comment: 21 pages, 7 figure
Localization Game for Random Geometric Graphs
The localization game is a two player combinatorial game played on a graph
. The cops choose a set of vertices with .
The robber then chooses a vertex 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 . If this information is sufficient to locate
the robber, the cops win immediately; otherwise the cops choose another set of
vertices with , and the robber may move to a
neighbouring vertex. The new distances are presented to the robber, 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 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 slightly above
the connectivity threshold
Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds
Curvature is a fundamental geometric characteristic of smooth spaces. In
recent years different notions of curvature have been developed for
combinatorial discrete objects such as graphs. However, the connections between
such discrete notions of curvature and their smooth counterparts remain lurking
and moot. In particular, it is not rigorously known if any notion of graph
curvature converges to any traditional notion of curvature of smooth space.
Here we prove that in proper settings the Ollivier-Ricci curvature of random
geometric graphs in Riemannian manifolds converges to the Ricci curvature of
the manifold. This is the first rigorous result linking curvature of random
graphs to curvature of smooth spaces. Our results hold for different notions of
graph distances, including the rescaled shortest path distance, and for
different graph densities. With the scaling of the average degree, as a
function of the graph size, ranging from nearly logarithmic to nearly linear
On the relation between graph distance and Euclidean distance in random geometric graphs
Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=¿(vlogn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).Peer Reviewe