ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 1, pp. 73–83.
Pleiades Publishing, Inc., 2017.
Original Russian Text
S.N. Khoroshenkikh, A.B. Dainiak, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 1, pp. 79–91.
Model of a Random Geometric Graph
with Attachment to the Coverage Area
S. N. Khoroshenkikh
and A. B. Dainiak
Moscow Institute of Physics and Technology (State University), Moscow, Russia
Received June 26, 2016
Abstract—We propose a model of random geometric graph with vertices in R
an alternative to existing models of ad-hoc wireless networks. We provide estimates for some
graph invariants in our model in R
1.1. Random Graphs and Large Networks
Modeling and analysis of complex networks is a major topic of computer science (Internet
research in particular), theoretical physics, biology, and other ﬁelds . Complex networks are
often modeled mathematically using random graphs; in this case the analysis of properties of a
large network usually reduces to estimating the asymptotic behavior of related invariants of a
One of the main challenges of modeling a complex network is designing a plausible growth model
(for examples of such models, see, e.g., ). One of the better known examples of growth models
is the preferential attachment model, initially proposed in  for the description of a graph of the
Internet, and later mathematically stated and investigated in  and follow-up papers.
1.2. Ad-hoc Wireless Networks
Ad-hoc wireless networks are decentralized (i.e., having no routers and access points) networks
for data transmission . Cheapening of personal wireless devices stimulates the interest in build-
ing and analyzing ad-hoc wireless networks. Investigation of the structure of these networks is
motivated, among others, by the demand for designing eﬃcient communication algorithms for such
Models for wireless networks have an important feature, which makes the distinction between
them and classical random graph models. Edges (links) of these networks have largely a geometric
nature, which in turn originates from physical laws of signal transmission: the amplitude of a signal
falls with the distance to the source, so direct data transmission is only possible between nodes
that are distanced not too far apart. While in real wireless networks the connectivity is inﬂuenced
by many eﬀects, such as interference and shadowing, in its most basic form we may come up with
the following simple geometric framework.
1. Vertices of the graph are points on the plane;
2. Two vertices are linked with an edge if the distance between them does not exceed some threshold