The Asymptotic Connectivity of Random
Cognitive Radio Networks
Zhiqing Wei
†
, Jianwei Liu
∗
, Zhiyong Feng
∗
Wireless Technology Innovation Institute
Beijing University of Posts and Telecommunications
Beijing 100876, P. R. China
Email:
†
zhiqingwei@gmail.com,
∗
{liujianwei, fengzy}@bupt.edu.cn
Wei Li, and T. Aaron Gulliver
Department of Electrical and Computer Engineering
University of Victoria
Victoria, BC, Canada V8W 3P6
Email: weili@ieee.org, a.gullive@ece.uvic.ca
Abstract—In this paper, we investigate the connectivity of
random cognitive radio networks with different routing schemes.
Two coexisting ad hoc networks are considered with m primary
users (PUs) and n secondary users (SUs) randomly distributed
in a unit area. The relation between n and m is assumed
to be n = m
β
. We show that with the HDP-VDP routing
scheme, which is widely employed in the analysis of throughput
scaling laws of ad hoc networks, the connectivity of a single
SU can be guaranteed when β > 1, and the connectivity of a
single secondary path can be guaranteed when β > 2. While
circumventing routing can improve the connectivity of cognitive
radio ad hoc network (CRAHN), we verify that the connectivity of
a single SU as well as a single secondary path can be guaranteed
when β > 1. Thus to achieve the connectivity of secondary
networks, the density of SUs should be larger (asymptotically)
than that of the PUs. A smart routing scheme can also improve
the connectivity of CRAHN. Our results serve as a guide to
deployment and routing design for cognitive radio networks.
Index Terms—Asymptotic Connectivity; Cognitive Radio Net-
works; Routing
I. INTRODUCTION
Cognitive radio (CR) [1] is one of the most promising
technologies for efficient spectrum utilization. Through flexi-
ble and comprehensive use of the available spectrum [2], CR
enables optimization of radio resource utilization by exploring
and exploiting spectrum holes. Spectrum holes are the spec-
trum available for secondary users (SUs) and may exist in
temporal or spatial dimensions.
The asymptotic connectivity of large-scale wireless net-
works has been widely explored since the seminal work of
Gupta and Kumar [3]. Assuming n ad-hoc nodes are randomly
distributed in a a disc of unit area, they showed that if each
node transmits at a power level so as to cover an area of
πr
2
=
log n+c(n)
n
, then the resulting network is can achieve
1-connectivity if and only if c(n) → ∞ [3], i.e., there exists a
path between any pair of nodes with high probability (w.h.p.).
Zhang et al. in [4] considered the k-connectivity, i.e., at least
k disjoint paths exist between any pair of nodes. Due to the
This work is supported by the National Basic Research Program (973
Program) of China (No. 2009CB320400), the National Natural Science
Foundation of China (61227801, 61121001, 61201152), the National Key
Technology R&D Program of China (2012ZX03003006, 2012ZX03006003-
003), and the Program for New Century Excellent Talents in University
(NCET-01-0259).
interaction between primary users (PUs) and secondary users
(SUs), the connectivity of a cognitive radio ad hoc network
(CRAHN) differs from that of traditional homogeneous ad
hoc networks. In cognitive radio network, the channels are
dynamic because of the spatial and temporal dynamics of the
primary traffic [6]. Ren et al. in [6] determined the delay
and connectivity scaling behavior in ad hoc cognitive radio
networks using the theories of continuum percolation and
ergodicity. Ao et al. in [7] investigated the connectivity of
cooperative cognitive radio network from a percolation-based
perspective. In [8], Abbagnale et al. proposed a Laplacian
matrix based method to measure the connectivity of CRAHNs.
However, most of the existing literature on CRAHN connec-
tivity does not consider the routing scheme in the connectivity
analysis. Investigating connectivity without considering the
routing is not practical, since the routing schemes in most
networks are relatively simple and can not find the complex
path. In this paper, we show that the routing scheme has a
significant influence on connectivity of CRAHN. Assuming
two coexisting ad hoc networks with m primary users and n
secondary users randomly distributed in a unit area. And the
relation between n and m is n = m
β
. With the HDP-VDP
routing scheme, which is widely adopted in the analysis of
throughput scaling laws of ad hoc networks, when β > 1, we
can guarantee the connectivity of a single SU, when β > 2,
we can guarantee the connectivity of a single secondary path.
While circumventing routing can improve the connectivity of
CRAHN, we show that when β > 1, we can guarantee the
connectivity of a single SU as well as the connectivity of
a single secondary path. Thus to achieve the connectivity in
secondary networks, the density of SUs must be bigger than
that of PUs asymptotically and a smart routing scheme is
essential. Our research may guide the deployment and routing
design of cognitive radio networks.
II. NETWORK PROTOCOL AND DEFINITIONS OF
CONNECTIVITY
m PUs and n SUs are uniformly distributed in a unit square,
which share the same space and spectrum. The PUs act as
if the SUs do not exist while the SUs need to mitigate the
interference to PUs. We assume n and m have relation n =
m
β
, where β is a positive number.
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