Random Graph Theory based Connectivity Analysis
in Wireless Sensor Networks with Rayleigh fading
channels
Jingbo Dong, Qing Chen and Zhisheng Niu
Department of Electronic Engineering, Tsinghua University
Tsinghua National Laboratory for Information Science and Technology , Beijing 100084, P. R. China
Email: [email protected] [email protected] [email protected]
Abstract— Connectivity is an essential merit of wireless sensor
networks. There has been great interest in exploring the minimum density of sensor nodes that is needed to achieve a connected
wireless network. This becomes difficult when uncertain features
increase, such as Rayleigh fading channels. In this paper, we
describe a range-dependent model for sensor networks by using
random graph theory, and study the connectivity problem with
this model. We calculate the probability of an arbitrary node
being isolated, and thus obtain the probability of the whole
network being connected. By giving the required minimum
density, our work can guide in designing of the wireless sensor
networks with fading channels. Moreover, the numerical results
shows that the fading effect would degrade the connectivity of
the wireless sensor networks.
I. I NTRODUCTION
Sensor networks, in which numerous mobile nodes communicate with each others in an ad hoc manner, received
strong interests in recent years [1]. Modeling sensor networks
becomes important but difficult when uncertain features, such
as the presence of shadowing and fading, increase. Rather
than a circle in ideal models, the communication range in a
fading environment is time-varying, which brings in the link
uncertainness [2]. In most applications of sensor networks,
such as environment and habitat monitoring or military cases,
instead of being put on the pre-designed positions, small
sensor nodes are often dispersed from air vehicles to the land
surface [3], and thus obey some random spatial distribution.
These uncertain and random features strongly impact the
connectivity, which reveals the nodes’ ability of cooperating
with each others, and thus becomes an important merit to
evaluate a sensor network.
In a sensor network, we say there is a link between two
nodes when packets can be successfully delivered from one
to the other. A sensor network is called connected if for two
arbitrary nodes, there exits a route, which consists of such
links, from one to the other. Traditional work on connectivity
analysis of sensor networks often focuses on finding a critical
transmission range to keep the network connected. However,
some low-cost sensor nodes may not support power-adaptive
transmissions. On the other hand, changing the transmission
range can be reformulated as changing the density of the
sensor networks, in which each node are using fixed transmit
power. In this paper, we focus on the relationship between
the network connectivity and network density λ by a suitable
scaling [4].
The wireless links are uncertain and unstable, while the
node positions are uncertain and unstable as well in some
cases. When the number of nodes is large, random features
and stochastic arguments become important in modeling the
wireless sensor networks. Recently, random graph theory is
introduced into the modeling of sensor networks with uncertain features. A random graph often can be imagined as
a living organism which evolves with time. By giving a set
of vertices in advance, the edges are generated according to
some randomization rules [8]. The two most important rules
are G(n, M ) and G(n, p). The notations G(n, M ) and G(n, p)
represent the set of the graphs generated by the rules. In
the former one, the number of edges is fixed at M in any
graph, and each graph appears with the equal probability. In
the latter one, each single edge are chosen independently with
probability p.
However, the above two basic rules cannot be directly used
to model wireless sensor networks for the following reasons.
First, the two rules both ignore the positions of nodes, which
are important information in sensor networks. Second, it is not
reasonable to assume that the probability of a link occurring
is independent with the distance between two terminal nodes.
In [5], the author proposed a new random graphs model, in
which nodes are fixed in the lattice, and the edge between two
arbitrary neighbors is generated with the same probability p
independently. The numerical results are presented in [5] to
show some examples of a phenomenon of phase transition,
which means the probability of a network being connected
suddenly changes from 0 to 1 when p increase across a
critical value pc . However, the lattice model does not consider
the spatial random distribution of nodes. The author in [4]
proposed Poisson Boolean model, in which the nodes are
Poisson distributed but the link connectivity is deterministic.
The phase transition is observed in this paper too, while the
parameter becomes λ. The author in [6] shows that the real
deployments have a transitional region, and illustrates the
impact of the transitional region, which is actually caused
by the randomness of link connectivity, never decrease by
changing the power. In the Figure 1, the simulation is taken for
shadowing environments, the y-axis represents the probability
1
Deterministic model
Rayleigh fading
0.8
f(s)
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
s
Fig. 1.
Fig. 2.
Transitional region
of link connectivity.
In this paper, by taking the randomness of both the node
positions and the link connectivity into account, we proposed a
more novel range-dependent random graph model for sensor
networks, in which the edges are chosen with a probability
that is dependent whit the distance between end nodes. Being
inspired by that of Bettstetter et al. [7], who analyze the
connectivity of an ad hoc network in the presence of lognormal
shadowing but does not give a closed form, we present an
analytical procedure for the computation of the probability of
a network being connected by using our model. We obtain
the closed form of expressions of the connectivity probability
in both the deterministic transmission range model and the
rayleigh fading model for the fist time. Additionally we show
that by using our model, the phase transition no longer
appears: There is no critical phase that the nodes changing
from ”very likely separated” to ”very likely connected”. In
other words, if the graph-theoretical change parallels phase
transitions of matters, it is more like liquid colophony to solid
colophony rather than water to ice.
II. S YSTEM M ODEL
We use a vertex to represent a node in a sensor network
and an edge represents a link between the two end nodes. The
set V = {vk }, k = 1, 2, ..., n includes all vertices. The nodes
obey the Poisson point distribution. In further analysis in this
paper, we assume the number of nodes is large enough. For an
arbitrary region A with the area A, the probability that there
are n nodes in it is,
(λA)n −λA
e
.
n!
The probability of a edge appearing is range-dependent. In
other words, it can be expressed as a function of the distance
s between its two end vertices in the fading environment.
We here generally use the function f (s) to represent the
edge appearing probability vs the distant between two nodes.
f (s) = P(an edge exists between two nodes|the distance
between these two nodes is s). This function, though could
vary with different environments(e.g. indoor or outdoor, urban
P (N = n) =
f(s) vs s
or suburb), has some common properties. It is a non-increasing
with the following property: when s → 0, f (s) → 1 and
s → ∞, f (s) → 0 . Two examples of f (s) are given as
following:
1) Deterministic Path-Loss: In the case of a deterministic
channel model, there exist a critical range s0 so that a node is
able to communicate with all the nodes lying within it. This
means, the link probability function f (s) can be expressed as
following:
(
0, s ≥ s0
f (s) =
1, s < s0
2) Rayleigh Fading: Then let us consider a case in
which the channel model presents a random component. In
a Rayleigh Fading environment, the squared magnitude is
exponentially distributed with probability density
1
x
exp(− 2 ), x ≥ 0,
σ2
σ
which can be viewed as the p.d.f. of the received power. Then
take the path-loss into account, we get:
p(x) =
f (s) = P(Pr ≥ Pth |s)
s
= P(Ps0 ( )n ≥ Pth )
s0
= e−
( s )n
s0
σ2
,
in which Ps0 is the power at the distance s0 which is the
reference point. Pth is the power threshold. The third equality
holds when we set s0 is the same value in the pure path
loss model for fair issues. Assume Ps0 = Pth to assure s0
has the same value in the pure path loss model. Figure 2
shows f (s)vss in the deterministic model and the model with
Rayleigh fading channel. We assume s0 = 1 and σ = 1 during
the simulation.
We denote the set of random graphs generated through the
previous progress as G(λ, f ). The edge generation rule of our
model is range-dependent, so we call it a Range-dependent
Random Graphical model. This model addresses the random
features of both the nodes positions and the link connectivity
between them.
III. C ONNECTIVITY A NALYSIS
Hence, using Theorem 1, the Piso is given by:
2
A. Isolation Probability for single node
Piso = e−πλs0 ,
A concept of the isolation probability for a single node
is highly related with the probability of a network being
connected(We use the donation Pcon throughout this paper).
We define a node is isolated when there exists no links between
it and any other nodes in G(λ, f ), we denote the probability of
an arbitrary node being isolated by Piso . . In fact, we show in
the following subsection that, there is a tight lower bound of
Pcon is a function of Piso . So we first calculate the probability
of an arbitrary vertex to be isolated using our model G(λ, f ).
Consider the vertex and the region A. We denote A as the
area of this region, which is a ring with the inside radius s
and the outside radius s + ∆s. We have
A(s) = π((s + ∆s)2 − s2 ).
Set ∆s ¿ s, we get the following theorem:
Theorem 1: Denote the Isolation Probability as Piso , then
R∞
Piso = e−2πλ 0 sf (s)ds .
Proof: Recall the Poisson point distribution, set
(λA(s))n −λA(s)
e
,
n!
which is the probability that there are n vertices in the region
A
Since ∆s ¿ s, we have f (s + ∆s) ≈ f (s), and A(s) =
π((s + ∆s)2 − s2 ) ≈ 2πs∆s.
Going throughout any ring with the center of this vertex,
we can calculate the overall Isolation Probability as follows:
Pn (s) =
Piso = lim
∆s→0
= lim
∆s→0
= lim
∆s→0
∞ X
∞
Y
−2πλ
Piso = e
e
e
−λA(k∆s)f (k∆s)
e
P (connected|N = n) ≥ (1 − Piso )n
1
= ((1 − Piso ) Piso )nPiso
≈ e−nPiso .
Going throughout any non-negative n, we finally get the
probability of the whole network being connected as the
following theorem:
Theorem 2:
P (connected) ≥ e−λA(1−e
log Piso = log lim
∆s→0
e
∞
X
∆s→0
= −2πλ
−λA(k∆s)f (k∆s)
)
.
n=0
∞
X
P (N = n)P (connected|N = n)
(λA)n −λA −nPiso
e
e
n!
Piso
= e−λA(1−e
)
.
k=0
= −2πλ lim
Z
≥
∞
X
n=0
∞
Y
Piso
Proof:
k=0
Take logarithm at both sides of the equation, we get
.
P (connected|N = n) ≥ (1 − Piso )n ≈ e−nPiso .
Proof:
P (connected) =
.
4
2
n
s2
0 σ Γ( n )
n
B. Network Connectivity
To obtain Pcon , we then turn to calculate the conditional
probability of the network being fully connected by introducing the following lamma.
Lemma 1: When n is large enough, one lower bound of
the conditional probability of the network being connected
can be given as:
Pn (k∆s)(1 − f (k∆s))n
k=0 n=0
∞
Y
−λA(k∆s) λA(k∆s)(1−f (k∆s))
k=0
∞
Y
which is the same with that in [7].
In the Rayleigh fading environment, the integral:
4
Z ∞
Z ∞
( s )n
s0
s2 σ n Γ( n2 )
sf (s)ds =
se− σ2 ds = 0
.
n
0
0
Applying Theorem 1, we obtain:
(k∆s)f (k∆s)∆s
k=0
∞
sf (s)ds.
0
Finally, we obtain the Isolation Probability as
Piso = e−2πλ
R∞
0
sf (s)ds
.
According to Theorem
R ∞1 ,to obtain Piso , we first calculate
the following integral: 0 sf (s)ds.
In the deterministic model,
Z ∞
Z s0
1
sf (s)ds =
sds = s20 .
2
0
0
Theorem2 gives us a lower bound of the network connectivity probability.
The authors in [5] and [4] show that under their models,
there exists a phase transition of network probability from 0
to 1 when the parameters, either p in [5] or λ in [4], increases.
In other words the connectivity probability is a step function.
Since Piso is a function of f (d), and thus a function of d,
the network connectivity probability is also a function of d.
That means, when d changes continuously, this probability
also changes continuously. Moreover, Pcon is a non-negative
value. We could not find a step function that is closed to Pcon
everywhere when λ increases. In other words, the phenomena
of the phase transition no long appears in the model with both
the nodes positions and the link connectivity random.
1
1
0.9
0.9
m=1
0.8
0.8
m=2
Deterministic Model
Raylei fading
m=3
0.7
0.7
m=4
m=5
0.6
m=0
P5
P(m)
con
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-3
10
-2
-1
10
10
0
10
0
0
0.5
1
1.5
2
λ
Fig. 3.
Network connectivity probability for n = 4, σ = 1, and
s0 = 1
IV. N UMERICAL R ESULTS
For fair issues, we set the path loss parameter in both the
deterministic model and Rayleigh fading model to be the same
value 4. To keep a fair comparison between the two models,
the Rayleigh fading parameter σ is set to be 1. Without loss
of generality, s0 is set to be 1, and the area A is set to be 100.
The density λ raises from 0 to 5.
The curve on top of Figure 3 represents the probability of
a network being connected Pcon vs the network density λ
in a Rayleigh Fading environment. It shows a problem that
when λ is small, Pcon reaches a significant positive value,
which is supposed to be 0 in common sense. This phenomenon
occurs because in our model we treat a network with no node
in it as a connected one, which actually is meaningless in
most applications. And when λ is small, there is a significant
probability that no node is in a given region. To adjust it, we
cut off the first m items in the sum of the proof of Theorem
2, which means we treat the network is never connected when
the nodes number is below or equal to m. Denote the adjusted
(m)
value by Pcon :
(m)
Pcon
= Pcon −
m−1
X
P (N = n)P (connected|N = n)
n=0
Figure 3 shows the adjusted function with m = 0, 1, . . . , 5
in the environments with Rayleigh fading channels. Figure 4
shows the adjusted curves in both the deterministic model and
Rayleigh fading model. It can be seen, the value Pcon without
Rayleigh fading is always greater than that with Rayleigh
fading when the density increases, which means Rayleigh
fading plays a negative role to connectivity in wireless sensor
networks.
V. C ONCLUSIONS
Connectivity is an important merit for QoS Support in
Wireless sensor networks, especially in fading environments.
We use random graphs to build a range-dependent model
for wireless sensor networks, which takes the randomness of
2.5
λ
3
3.5
4
4.5
5
Fig. 4. Network connectivity probability in Deterministic Model and
Rayleigh Fading Environment
both the nodes positions and and the link connectivity. We
calculate the probability of an arbitrary node being isolated,
and then obtain a lower bound of the probability of the
networks being fully connected. Our results are useful for
designing a wireless sensor network, which consists many
nodes distributed randomly in a particular large area. For a
given area to be covered by sensor nodes and a given channel
environment, we now can determine the minimum density λ
that is needed to achieve a connected sensor network. The
numerical result of this paper shows that the phenomena of
phase transition, which is mentioned in most former works,
no longer appears when both the nodes position and the link
connectivity are random. Moreover, the fading effect leads
to lower network connectivity comparing with the non-fading
environments.
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