Topology Maintenance in
Asynchronous Sensor Networks
Reuven Cohen
Boris Kapchits
2
Outline
Neighbor Discovery in Sensor Networks
Estimating the in

segment Degree of a
Hidden Neighbor.
An Efficient Topology Maintenance
Algorithm.
Simulation Study.
3
Sensors and sensor networks
Sensor is a cheap tiny device that is able to detect local events and report
them to a centralized gateway using wireless comm.
A sensor network consists of many sensors and a gateway
–
The sensors perform some common task, like smoke detection or temperature
measurement, and report to the gateway.
Since not all the sensors are in the transmission range of the gateway,
their messages should be forwarded by other sensors
Most often the network structure cannot be pre

engineered, since the
sensors are placed randomly in the covered area.
4
Energy saving
Since the sensor network should operate unattended for a long
time, its most scarce resource is energy.
Energy is consumed when the sensor is transmitting or
receiving.
–
If the receiving/transmitting unit is powered off, energy
consumption is minimized.
the sensor is said to be in “slipping mode”
–
In order to save energy, each sensor will switch between “sleeping
mode” and “active mode”. Sensor will spend in “active mode”
about
2
% of its lifetime.
–
However, in order to communicate, two sensors should be in
“active mode” simultaneously.
asleep
active
asleep
active
asleep
5
Neighbor Discovery
The sensor nodes are placed randomly over the area of
interest, and their first step is to detect their immediate
neighbors, i.e., the nodes with which they have direct wireless
communication.
Although sensor nodes are static, topology changes still take
place, due to the following factors:
Loss of local synchronization due to accumulated clock drifts.
Disruption of wireless connectivity between adjacent nodes by
a temporary event, such as a passing car or animal, a dust
storm, rain or fog.
The ongoing addition of new nodes in some networks to
compensate for nodes that do not function any more
We distinguish between the neighbor discovery in the initial
phase and the ongoing topology maintenance at the operation
phase.
6
Neighbor discovery (cont.)
a
b
c
•
Every time a node wakes up, it
broadcasts HELLO. Only a node that is
also awake can hear this message and
establish connections.
•
Node
a
receives HELLO from node
c.
Hence, they discover each other.
•
However,
a
and
b
do not know about b
and vise versa.
a
and
c
discover
each other
b
sends HELLO
message, which is
heard by no one
7
The differences between neighbor
discovery and topology management
For neighbor discovery, an aggressive protocol, one which
requires the sensor to stay in active mode and expend a lot of
energy until detection, is usually acceptable.
Neighbor discovery is performed when the sensor has no clue
about the structure of its immediate surroundings. In particular,
the sensor is unable to perform any useful task. Hence, energy
consumption in this state is less of an issue.
When the sensor performs topology maintenance, it can
perform topology maintenance together with these neighbors in
order to consume less energy.
Neighbor
discovery
Topology
maintenance
8
New approach to neighbor
discovery
As far as we know, this paper is the first to distinguish between
neighbor discovery and topology maintenance in sensor network, and
to explicitly address the topology maintenance problem.
The figure below summarizes this idea:
–
When node
u
is initialized, it performs neighbor discovery.
–
After a certain time period in the
neighbor discovery
state, during which
the node is expected, with high probability, to find most of its neighbors,
the node moves to the
topology maintenance
state.
The main idea behind the topology maintenance scheme is that once
most of the neighbors know each other, the task of finding a new
neighbor a part of the topology maintenance is divided among all the
nodes in its vicinity.
Connectivity lost
Neighbor
discovery
Topology
maintenance
Time elapsed, or enough
neighbors are detected
9
An example for topology
maintenance
b
c
a
c
and
a
discover each
other
c
tells
b
to
wake up at
certain time,
a
and
b
discover
each other
During topology maintenance
phase most of the nodes are
organized into clusters.
Thus, as soon as a new node is
discovered by one of the cluster
members, it can be almost
immediately discovered by the rest.
10
Outline
Neighbor Discovery in Sensor Networks
Estimating the in

segment Degree of a
Hidden Neighbor.
An Efficient Topology Maintenance
Algorithm.
Simulation Study.
11
Hidden node degree estimation
In order to calculate the HELLO message frequency node
v
should estimate the number of in

segment stations that
participate in the discovery of a hidden node
u
.
This is equal to
deg
S
(u),
namely, the number of neighbors
that the hidden node u has inside the connected segment
deg
S
(u)=
4
v
u
deg
S
(v)=
3
12
The estimation methods
Three methods are considered:
1.
The average in

segment degree of the segment's nodes is
used as an estimate of the in

segment degree of
u
.
2.
The number of
v
’s in

segment neighbors,
deg
S
(v)
, is used
as an estimate of
deg
S
(u)
.
3.
A linear combination of
1
and
2
.
deg
S
(u)=
4
v
u
deg
S
(v)=
3
13
Estimation methods analysis
Let
X
be a random variable that indicates the degree
deg
S
(v)
of
v
, a node in the segment
S
.
Let
Y
be a random variable that indicates the degree
deg
S
(u)
of
u
, a hidden neighbor of
v
, which we want to estimate. Note that
u
itself is also not aware of the value of
Y
.
Let
Y'
be the estimated value of
Y
. Clearly, we want
Y'
to be as
close as possible to
Y
.
We use the mean square error measure,
MSE,
to decide how
good our estimate is.
–
The
MSE
is defined as
E((Y

Y')
2
)
.
We assume that
X
and
Y
have the same distribution.
We also assume that the nodes, both in

segment and hidden,
are distributed uniformly on the plane.
14
Theorem
1
Let u, v and w be nodes in a geometric graph with
the same transmission range, where nodes are
distributed uniformly. If u is a neighbor of v and v is a
neighbor of w, then the probability that u is also a
neighbor of w is
C
=
1

3
/(
4
π
)√
3
~
0.586503
.
The proof consists of some geometrical and probabilistic calculations.
This Theorem is used further, in the analysis of
Method
2
and
3
.
15
Method
1
analysis
For the first method, where node
v
measures the
average in

segment degree of the segment's nodes
and uses this number as an estimate of the in

segment degree of
u:
Y’=E(X)=E(Y)
The method accuracy:
MSE
1
= E((Y

Y')
2
) = E((Y

E(Y))
2
) =
Var
(Y).
–
Var
(Y)
is the variance of
Y.
16
Method
2
analysis
For the second method, we have
Y'=X
The method accuracy:
MSE
2
= E((Y

X)
2
)=… =
2
(E(X
2
)

E(XY))
Note that
E(XY)=Σ
y
[y P(Y=y) E(X  Y=y)].
But from Theorem
1
:
E(X  Y=y) =
C
y + (
1

C
)E(X).
Hence, substituting into the equation gives:
E(XY)=…=
C
E(Y
2
)+(
1

C
)E(X)E(Y).
And finally:
MSE
2
=
2
(
1

C)Var(Y)~
0.84
Var(Y)
17
Method
3
analysis
We estimate Y in the following way:
We can divide the neighbors of
u
into two subsets:
1)
Those that are also neighbors of
v
2)
Those that are not.
By
Theorem
1
,
the average size of the first subset
is
C
X
. The average size of the second subset is
(
1

C
)E(X)
.
Therefore, we estimate the degree of
u
by
Y'=CX+(
1

C)E(Y).
18
Method
3
analysis (cont.)
MSE
3
=E((
C
X+ (
1

C
)E(Y)

Y)
2
) =
Σ
x
Σ
y
(
C
x+(
1

C
)E(Y)

y)
2
P(X=x,Y=y) =
C
2
E(X
2
)+(
1

C
)
2
E(Y)
2
+E(Y
2
)+
2
(
1

C
)
C
E(X)E(Y)

2
C
E(XY)

2
(
1

C
)E(Y)
2
Since E(X)=E(Y) and E(X
2
)=E(Y
2
), we get:
MSE
3
=(
1
+
C
2
)E(Y
2
)+(
2
C

1

C
2
)E(Y)
2

2
C
E(XY) =
(
1

C
2
)
Var
(Y)~
0.67
Var
(Y)
19
Which of the three methods is the
best?
MSE
1
=
Var
(Y).
MSE
2
=
2
(
1

C
)
Var
(Y)~
0.84
Var
(Y)
MSE
3
= (
1

C
2
)
Var
(Y)~
0.67
Var
(Y)
Hence,
MSE
1
>
MSE
2
>
MSE
3

The third method yields the best (smaller)
MSE
.

However, note that this method requires some
global knowledge of the network topology, while the
second method requires only local knowledge.
20
Outline
Neighbor Discovery in Sensor Networks
Estimating the in

segment Degree of a
Hidden Neighbor.
An Efficient Topology Maintenance
Algorithm.
Simulation Study.
21
A Neighbor Discovery Scheme
Suppose that node
u
is in neighbor discovery state:
It wakes up every
T
I
seconds in average for a period of time
equal to
H
, and broadcasts HELLO messages.
The demand is that the nodes of segment
S
will discover
u
within a time period
T
with probability
P
.
Each node
v
in the segment
S
is in topology maintenance state,
wakes up every
T
N
(v)
seconds for a period of time equal to
H
,
and broadcasts HELLO messages.
Neighbor
discovery
Topology
maintenance
u
segment
S
22
A Neighbor Discovery Scheme (cont.)
In order to discover each other, nodes
u
and
v
should have an
active period that overlaps by at least a portion
δ
,
0
<
δ
<
1
, of
their size
H
.
If node
u
wakes up at time
t
for a period of
H
, node
v
should
wake up between
t

H(
1

δ
)
and
t+H(
1

δ
)
. The length of this
valid time interval is
2
H(
1

δ
)
.
Since the average time interval between two wake

up periods
of
v
is
T
N
(v)
, the probability that
u
and
v
discover each other
during a specific HELLO interval of
u
is
2
H(
1

δ
)/T
N
(v)
.
u
t
t+H
v
v
t

H(
1

δ
)
t+H(
1

δ
)
23
The probability of discovery
Let
n
be the number of in

segment neighbors of
u
.
When
u
wakes up and sends HELLO messages, the probability
that at least one of its
n
neighbors is awake during a sufficiently
long time interval is
1

(
1

2
H(
1

δ
)/T
N
(v))
n
.
Consider a division of the time axis of
u
into time slots of length
H
.
The probability that
u
is awake in a given time slot is
H/T
I
, and
the probability that
u
is discovered during this time slot is
P
1
= H/T
I
(
1

(
1

2
H(
1

δ
)/ T
N
(v))
n
)
.
Denote by
D
the value of
T/H
. Then, the probability that
u
is
discovered within at most
D
slots is
P
2
=
1

(
1

P
1
)
D
.
Therefore, we seek the value of
T
N
(v)
that satisfies the following
equation:
P
2
=
1

(
1

P
1
)
D
≥ P
24
The sufficient value of
T
N
(v)
The last expression can also be stated as
P
1
≥
1

(
1

P)
1
/D
Hence
H/T
I
(
1

(
1

2
H(
1

δ
)/ T
N
(v))
n
) ≥
1

(
1

P)
1
/D
and therefore
T
N
(v)≤
2
H(
1

δ)
/(
1

(
1

T
I
/H(
1

(
1

P)
1
/D
)
)
)
1
/n
Since
v
does not know the exact value of
n
,
v
can estimate it
using the schemes presented earlier.
25
T
N
(v)
as a function of maximum
tolerated delay T.
The estimated number of
participating nodes is
n
=
10
.
We see the value of
TN(v
)
as a
function of the desired discovery
time
T
for
P
=
0.5
,
0.8
and
0.95
.
P
is set to
0.8
and
n
varies between
5
and
50
.
TN(v)
is calculated as a function of
the desired discovery time.
26
Outline
Neighbor Discovery in Sensor Networks
Estimating the in

segment Degree of a
Hidden Neighbor.
An Efficient Topology Maintenance
Algorithm.
Simulation Study.
27
Simulation setup.
2
,
000
sensor nodes, randomly placed over a
10
,
000
x
10
,
000
grid.
The transmission range is set to
r=
300
units. Any two nodes whose
Euclidean distance is not greater than
r
are considered to have
wireless connectivity.
A half of the nodes are randomly selected to be hidden.
We require that every hidden node will be detected with probability
P
,
which ranges between
0.3
and
0.7
within a predetermined period of
time
T
=
100
time units.
The hidden nodes are assumed to be in the neighbor discovery state,
where they are supposed to wake up randomly, every
T
I
=
20
time units
on the average.
A non

hidden node
v
is assumed to be in the topology maintenance
state, where it wakes up randomly, every
T
N
(v)
time units on the
average, in order to discover hidden nodes.
When a node is detected, it joins the segment.
A hidden node that detects another hidden node remains in the
neighbor discovery state.
28
Ratio of hidden nodes
The initial ratio is
0.5
. After
100
time units, this ratio decreases to:
•
0.35
for P=
0.3
•
0.25
for P=
0.5
•
0.15
for P=
0.7
After
200
time units:
•
93
% of the nodes are detected for
P=
0.7
•
75
% for
P=
0.3
.
29
Average frequency of HELLO intervals
of the in

segment nodes
We can see that for the smaller
value of
P
(the lower curve), the
frequency is almost
75
% lower
than the frequency for the larger
value of
P
. We can also see that
for a given value of
P
, the average
frequency of HELLO periods
decreases with the time. This is
because the segment grows, and
more nodes participate in the
discovery process.
32
A self

contained algorithm vs. our
algorithm.
Figure shows the ratio of hidden nodes after
T
time units for
networks with different transmission ranges. This simulation
compares our protocol with one that uses a trivial hidden
neighbor detection algorithm. The simulation starts with
50
%
hidden nodes, and P=
0.5
.
Our protocol guarantees that after
T
time units the number of
hidden nodes will decrease by half to
25
%. In contrast,
The trivial protocol discovers half of the hidden nodes only
when the transmission range is ~
0.06
.
When the transmission range is shorter, this protocol discovers
a smaller fraction of the hidden nodes.
When the transmission range is longer, this protocol discovers
more nodes during a time period of
T
.
33
A self

contained algorithm vs. our
algorithm

the figure
We conclude that our algorithm can self

adjust to invest the minimum
energy needed to guarantee the required discovery rate, whereas
the trivial algorithm cannot.
34
Conclusions
We expose a new problem in wireless sensor networks,
referred to as ongoing topology maintenance.
We argued that ongoing topology maintenance is essential
even if the sensor nodes are static.
We showed that by having the nodes in a connected segment
work together on ongoing topology maintenance, we can
guarantee that
(a)
Hidden nodes will be detected with a certain probability
P
and within a certain time period
T
; and that
(b)
The energy expended by the segment nodes on the
detection of hidden nodes is minimized.
35
Conclusions cont.
We showed that our scheme works well if every node
connected to a segment estimates the in

segment degree of its
potential hidden neighbors.
We proposed three estimation algorithms and analyzed their
mean square errors.
We then presented an ongoing topology maintenance
algorithm.
Using simulations, we analyzed several aspects of our
algorithms.
–
We showed that when the hidden nodes are uniformly
distributed in the area, the simplest estimation method is
good enough.
–
When the hidden nodes are concentrated around some
dead areas, the third method was shown to be the best.
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