Abstract-This paper examines the probability of connectivity of ad hoc wireless networks in which nodes are uniformly distributed on the circumference of a circle. We derive the exact probability that the network is composed of at most C clusters. The probability of connectivity is obtained by considering the case of C equal to unity. We also consider the distribution of nodes over a circular disk. The probability of connectivity for this case is found by fitting a log-logistic function using Minimum Mean Square Error (MMSE) criterion

Ankit Misra

Gaurav Teltia

Senior Member, IEEE A K Chaturvedi

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IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 10, OCTOBER 2008 717
On the Connectivity of Circularly Distributed Nodes in
Ad Hoc Wireless Networks
Ankit Misra, Gaurav Teltia, and A. K. Chaturvedi, Senior Member, IEEE
Abstract—This paper examines the probability of connectivity
of ad hoc wireless networks in which nodes are uniformly
distributed on the circumference of a circle. We derive the exact
probability that the network is composed of at most C clusters.
The probability of connectivity is obtained by considering the
case of C equal to unity. We also consider the distribution of
nodes over a circular disk. The probability of connectivity for
this case is found by fitting a log-logistic function using Minimum
Mean Square Error (MMSE) criterion.
Index Terms—Ad hoc networks, connectivity, wireless net-
works.
I. INTRODUCTION
THE probability of connectivity (Pcon) of wireless ad-hoc networks has been an area of keen interest. Al-
though many definitions of connectivity have been given in
the literature [3], the most popular is the one in which a
network is called connected if every pair of nodes in the
network is connected. An exact formula for the probability
of connectivity for a uniform distribution of nodes for a one-
dimensional network is given in [4]. More recently, an exact
expression for the probability that a uniformly distributed one-
dimensional network is composed of at most C clusters was
derived in [1]. The probability of connectivity corresponds to
the special case when C is unity. The probability of connec-
tivity of non-identically distributed nodes is investigated in
[2] using a regression based approach. All these studies have
assumed a random distribution of nodes on a straight line.
Exact results for probability of connectivity are not known for
any other setting. In the case of two-dimensional networks,
only asymptotic connectivity for infinitely many nodes has
been analyzed in [5]-[7].
In this paper we consider the case when the nodes are
uniformly distributed on the circumference of a circle and
derive an exact expression for the probability that such a
network is composed of at most C clusters. This is the first
exact analysis for a setting beyond the straight line. Further,
we derive a regression based expression for the probability of
connectivity when the nodes are uniformly distributed over
a circular disk. The expression has been found to be in
agreement with the asymptotic analysis of [5].
The paper is organized as follows. In Section II we state the
main result for the case of nodes distributed on a circle while
in Section VI we provide its proof. Monte Carlo simulation
results are given in Section III. Nodes distributed on a disk
are considered in Section IV where we give a minimum mean
Manuscript received April 14, 2008. The associate editor coordinating the
review of this letter and approving it for publication was C. Douligeris.
The authors are with the Department of EE, IIT Kanpur, 208016, India
(e-mail: akc@iitk.ac.in).
Digital Object Identifier 10.1109/LCOMM.2008.080586
Fig. 1. Probability of atmost C clusters when 100 nodes are distributed on
the circumference.
square error (MMSE) based expression for the probability of
connectivity. The paper is concluded in Section V.
II. Pcon FOR NODES ON THE CIRCUMFERENCE
Theorem 1: Assume that n nodes with a radio range of ro
are distributed uniformly and independently on the circum-
ference of a circle of radius R. From simple geometry, the
maximum angular separation between any two nodes on this
circle for them to be in range of each other is θc = 2 sin−1 ro2R .
Then the probability P (C), that the network is composed of
at most C clusters is given by:
P (C) = 1 −
m∑
i=C
(−1)i−C
(
i − 1
C − 1
)(
n − 1
i
)[
(1 +
(i + 1)θc
2π
)n
+
nθc
2π
(1 − (i + 1)θc
2π
)n−1
]
+ (−1)i−C
(
i − 1
C
)(
n − 1
i
)
pΔi
where m = min(n − 1, [ 2π
θc
] − 1),
(
i
C
)
= 0 when i < C, [x]
represents the greatest integer that is less than x and
pΔi =
⎧⎨
⎩
(1 − i θc
2π
)
n θc
2π
> 1
i+1
[(1 − i θc
2π
)
n − n θc
2π
(1 − (i + 1) θc
2π
)
n−1 θc
2π
≤ 1
i+1
−(1 − (i + 1) θc
2π
)
n
]
As already mentioned, Pcon corresponds to the special case
when C is equal to unity. The theorem is proved in Section
VI.
III. SIMULATIONS
In Fig. 1. we plot the Monte Carlo simulation results
and the exact expression for P (C) as a function of the
normalized range roR for 100 nodes on the circumference of a
1089-7798/08$25.00 c© 2008 IEEE
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718 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 10, OCTOBER 2008
Fig. 2. MMSE estimation of Pcon for 200 nodes distributed on a disk.
circle. Each simulation point was obtained by simulating 5000
random topologies. Further, the probability that the network
is composed of at most C clusters is depicted for C = 1, 2, 3.
It may be noted that there is complete agreement between the
simulation and theoretical results.
Similar exercise was also done for 10, 20, . . .300 nodes and
in all the cases the simulation results matched the theoretical
results.
IV. Pcon FOR NODES ON A CIRCULAR DISK
The case of nodes distributed on the circumference of
a circle is fundamentally different from that of a circular
disk. As shown in the previous section, the probability of
connectivity for the former can be computed by using the
angular separation between adjacent nodes as a sufficient
statistic when they are ordered according to their locations.
In the case of a disk this approach seems intractable because
in the absence of a natural ordering, angular separation does
not suffice as a sufficient statistic.
The probability of connectivity results reported in the
literature for two dimensional networks [5]-[7] are mainly
asymptotic in nature as the nodes increase without bound.
In this section we consider the same problem but for the
practical case when the number of nodes is finite. We aim
to find the probability of connectivity of nodes distributed
uniformly in both R and θ. Since an exact analysis is difficult
we take recourse to regression using a log-logistic function
(1). The log-logistic function has been earlier used in [2] to
fit a probability of connectivity (Pcon) curve:
Pcon =
1
1 + exp
−
(
log(ro/R)−α
β
) (1)
Firstly, we obtained a Pcon versus roR curve for a disk by
doing sufficiently large number of Monte Carlo simulations
for n nodes uniformly distributed in both R and θ.
We fit the obtained curve to a log-logistic function by
estimating the parameters α and β in (1) using the MMSE
method. The estimated curve using these parameters as well
as the simulated points for n = 200 are plotted in Fig. 2.
We repeated this exercise for n = 10, 20, . . .300
and found that α(n) closely fits a log n curve whereas
β(n) closely fits a log nn curve. The functions α(n)
and β(n) were estimated using the MMSE criterion
again. The obtained α(n) and β(n) are as follows:
α(n) = a1 + a2 log n a1 = 2.043 a2 = −0.7424
β(n) = b1 + b2 log nn b1 = 0.07879 b2 = 0.1964
These functions can be directly substituted in (1), thus
giving us a relation between Pcon, n and roR . This relation
can also be expressed in terms of radio range ro(n) required
to meet a desired probability of connectivity in a network
with n nodes:
ro(n) = Reα(n)eβ(n)log
Pcon
1−Pcon (2)
For a disk of unit area, i.e. πR2 = 1, it is shown in [5] that
if:
πr2o(n) =
log n + c(n)
n
(3)
then the network is asymptotically connected with probability
1 if and only if c(n) → ∞. Interestingly, the expressions for
Pcon and ro(n) obtained for a finite number of nodes are in
agreement with the asymptotic analysis of [5] i.e. if ro(n)
from (3) is substituted in (1), Pcon → 1 as c(n) → ∞ when
n → ∞. Also for any Pcon arbitrarily close to 1, c(n) → ∞
as n → ∞.
V. CONCLUSIONS
We derived an exact expression for the probability of at most
C clusters for nodes uniformly distributed on the circumfer-
ence of a circle. The probability of connectivity corresponds
to the case when C is equal to unity. The probability of
connectivity for nodes on a circular disk was also estimated
using a log-logistic function. The estimated expression was
found to be in agreement with the asymptotic analysis of [5].
REFERENCES
[1] A. Ghasemi and S. Nader-Esfahani, “Exact probability of connectivity in
one-dimensional ad hoc wireless networks,” IEEE Commun. Lett., vol.
10, no. 4, Apr. 2006.
[2] J. Li, L. L. H. Andrew, C. H. Foh, M. Zukerman, and M. F. Neuts,
“Meeting connectivity requirements in a wireless multihop network,”
IEEE Commun. Lett., vol. 10, no. 1, Jan. 2006.
[3] C. Bettstetter, “On the connectivity of ad hoc networks,” The Computer
J., special issue on mobile and pervasive computing, Oxford University
Press, vol. 47, pp. 432–447, July 2004.
[4] M. Desai and D. Manjunath, “On the connectivity in finite ad hoc
networks,” IEEE Commun. Lett., vol. 6, no. 10, Oct. 2002.
[5] P. Gupta and P. R. Kumar, “Critical power for asymptotic connectivity
in wireless networks,” Stochastic Analysis, Control, Optimization and
Applications: A Volume in Honor of W. H. Fleming, W. M. McEneany,
G. Yin, and Q. Zhang, eds. Boston, MA: Birkhauser, pp. 547–566, 1998.
[6] C. Bettstetter, “On the minimum node degree and connectivity of a
wireless multihop network,” in Proc. ACM MobiHoc, pp. 80–91, June
2002.
[7] P. Santi and D. M. Blough, “The critical transmitting range for connectiv-
ity in sparse wireless ad hoc networks,” IEEE Trans. Mobile Computing,
vol. 2, pp. 25–39, Jan.-Mar. 2003.
[8] H. A. David, Order Statistics. John Wiley & Sons, 1970.
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MISRA et al.: ON THE CONNECTIVITY OF CIRCULARLY DISTRIBUTED NODES IN AD HOC WIRELESS NETWORKS 719
VI. PROOF FOR THEOREM 1
The location of ith node on the circumference is specified by θi where θi’s are i.i.d U [0, 2π]. Let us define random variables
(RVs) {Δi}ni=1 as a linear combination of the ordered RVs {θ(i)}ni=1:
Δi = θ(i+1) − θ(i) i = 0, 1 . . . n − 1, where θ(0) = 0 (4)
We obtain the joint pdf of the RVs {Δi}n−1i=0 by performing a Jacobian transformation and then integrating over Δ0 to get the
joint pdf of the RVs {Δi}n−1i=1 .
fΔ1...Δn−1(δ1 . . . δn−1) =
n!(2π−δ1−...δn−1)
(2π)n if
n−1∑
i=1
δi ≤ 2π
fΔ1...Δn−1|A(δ1 . . . δn−1) =
n!(2π−δ1−...δn−1)
(2π)nP (A) if
n−1∑
i=1
δi ≤ 2π − θc
fΔ1...Δn−1|Ā(δ1 . . . δn−1) =
n!(2π−δ1−...δn−1)
(2π)n(1−P (A)) if 2π − θc ≤
n−1∑
i=1
δi ≤ 2π , θc = 2 sin−1 ro2R
(5)
where A denotes the event that the first node and the last node are not within radio range of each other i.e. θ(n)−θ(1) ≤ 2π−θc
and θc is the maximum angular distance between two nodes for them to be in radio range of each other. P (C) can be obtained
by using the total probability formula and the principle of inclusion and exclusion [8]. Thus:
P (C) = P (C|A)P (A) + P (C|Ā)P (Ā)
P (C|A) = 1 −
m∑
r=C
(−1)r−C
(
r − 1
C − 1
)(
n − 1
r
)
P (Δ1 ≥ θc . . . Δr ≥ θc|A)
P (C|Ā) = 1 −
m∑
r=C+1
(−1)r−C
(
r − 1
C − 1
)(
n − 1
r
)
P (Δ1 ≥ θc . . . Δr ≥ θc|Ā)) , m = min(n − 1, [ 2πθc ] − 1)
(6)
We need fΔ1...Δr|A(δ1 . . . δr) for calculating (6), hence we integrate fΔ1...Δn−1|A(δ1 . . . δn−1) over δn−1 . . . δr+1∫ 2π−θc−δ1−...δr
0 . . .
∫ 2π−θc−δ1−...δn−2
0 fΔ1...Δn−1|A(δ1 . . . δn−1)dδn−1 . . . dδr+1
fΔ1...Δr|A(δ1 . . . δr) =
n!(2π−θc−δ1−...δn−1)n−r
(2π)nP (A)(n−r)! +
n!θc(2π−θc−δ1−...δn−1)n−r−1
(2π)nP (A)(n−r−1)! ,
r−1∑
i=1
δi ≤ 2π − θc (7)
Similarly we integrate fΔ1...Δn−1|Ā(δ1 . . . δn−1) over δn−1 . . . δr+1∫ 2π−δ1−...δr
max(2π−θc−δ1−...δr ,0) . . .
∫ 2π−δ1−...δn−2
max(2π−θc−δ1−...δn−2,0) fΔ1...Δn−1|Ā(δ1 . . . δn−1)dδn−1 . . . dδr+1
fΔ1...Δr|Ā(δ1 . . . δr) =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
n!(2π−δ1−...δn−1)n−r
(2π)n(1−P (A))(n−r)! 2π − θc ≤
r−1∑
i=1
δi ≤ 2π ;
n![(2π−δ1−...δn−1)n−r−(n−r)θc(2π−θc−δ1−...δn−1)n−r−1−(2π−θc−δ1−...δn−1)n−r]
(2π)n(1−P (A))(n−r)! 0 ≤
r−1∑
i=1
δi ≤ 2π − θc
(8)
The expression for P (Δ1 ≥ θc . . .Δr ≥ θc|A) is obtained by integrating (7) with these limits:∫ min(2π−θc,2π−(r−1)θc)
θc
− . . . ∫ min(2π−θc,2π−δ1−...δr−1)θc fΔ1...Δr|A(δ1 . . . δr)dδr . . . dδ1 (9)
and the expression for P (Δ1 ≥ θc . . . Δr ≥ θc|Ā) is obtained by integrating (8) with the following limits:∫ 2π−δ1−...δr−1
max(2π−θc−δ1−...δr−1,θc) . . .
∫ 2π−δ1−...δn−2
max(2π−θc−δ1−...δn−2,θc) fΔ1...Δr|Ā(δ1 . . . δr)dδr . . . dδ1 (10)
Substituting the values of above integrals in (6) we get the stated theorem.
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On the connectivity of circularly distributed nodes in ad hoc wireless networks

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On the connectivity of circularly distributed nodes in ad hoc wireless networks

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