A Group of sensors is a collection of overlapped sensors in a single area. Let us define the degree of overloaded sensors by the maximum number of sensors overlapped in the same area, here we denote to the maximum coverage degree of an area by
where 
where 
is an area notation called r, 
are the overlapped sensors, and 
the number of overlapped sensors. The overlapped sensors that create a degree of an area 
Create a group of sensors denoted by 
we call 
the coverage degree. 
, and
. The maximum degree of sensor 
and senor 
is 
that occurs in the area of intersection, which means that there is one and only one area covered by two sensors and that is the maximal overlapping that could be produced, so sensor 
and 
create a group of two sensors denoted by
. Sake of convenience, we denote to the group of sensors that build up the WSN by
which we call it the mother network group or simply the mother group.
Here we start by asking, how many sub-areas are generated if 
unit disk sensors are partially overlapped? Assuming there is no fully overlapping between sensors, and all sensors are homogenous (sensors have the same sensing range). Say 
is the function to count the number of sub-areas, definitely
.
, 
, 
, 
(see 
?
.
Theorem 1: the sub-areas number of a sensors group
is
Proof: suppose we have a group of sensors 
as shown in 
to
, the number of areas for the top sensor 
is seven (red areas namely 1, 2, 3, 4, 5, 6, 7). The number of areas inside the sensor’s range 
are seven, namely (4, 5, 6, 7, 8, 9 10), but the red colored areas, 2, 3, 4, 5, already counted in
, so there are only 3 blue colored areas inside sensor’s range
, namely 8, 9, 10. For the sensor
, it has seven areas inside its sensing range, the 3, 5, 6, 7 are red areas already counted in sensor’s range
, the area 10 is blue area already counted in sensor’s range
, thus still two black areas in sensor’s range
, namely 11, 12. For sensing range of
, there are seven areas inside it, three are red areas (4, 5, 6), two are blue areas (9, 10), and one area is black (11), so there is still one area only in 
(13). Therefore, the number of areas from top to down is 7 + 3 + 2 + 1 = 13. Generally, counting the sub-areas from top to down, the top sensor contains
areas, the second sensor inside the group
contains 
areas, the third sensor contains 
areas… the last sensor contains one area
. Totally, there are
of areas. Generally, there count of areas is
In addition, we can proof theorem 1 by counting the areas of a group basing on the degree of coverage, if an area covered by k sensors then it called k-covered area. For a group 
there is only one area is k-covered (the maximum degree of coverage), in the remainder areas, there are k areas are1-covered, k areas are 2-coverd, k areas are 3-coverd… k areas are k-1 covered. Let 
be the number of areas that j-covered inside a group of sensors
. For example, 
means, there are five areas 1-coverd in
. In 
, the number of 1-coverd areas is five. We can count the sum of areas of sensors group 
as below:
Lemma 1: for sensor range belongs to a group
, there are only one area k-covered, k area 1-coverd, k areas are 2-coverd, k areas are 3-coverd… and k areas k-1 covered.
Lemma 2: for a group
, all sensors have the same characteristics, for example, the number of areas, the degree of coverage for each area, the number of intersection points located on the border of the sensor, and the number of intersection points located inside sensor’s range.
Counting the intersection points of k-overlapped sensors is an easy combination problem. Before proving, here we denote to the number of intersection points by
, clearly
, 
, 
, 
, 
,
… then what is the
? 
Theorem 2: the number of intersection points of sensors group 
is
Proof: Assuming that there are k sensors and each sensor has two intersection points with each neighbor sensor, since each sensor has 
intersection points with others, applying this method to all sensors, we get the total amount of intersection points as
. However, while calculating, every single point has been repeatedly counted twice, thus the right answer in regards to the intersection points quantity should be
.
We can use top down approach to calculate the number of intersection points, as shown in the 
. 
is the top sensor; 
is the bottom sensor of the group. The number of intersection points inside (internal) and on the border of sensing range of the top sensor 
is
. In the second node
, there are k-2 of intersection points. In the third node
, there are k-3 of intersection points. In the fourth node
, there are k-4 of intersection points, and there is 0 intersection points in the bottom sensor.
Totally the number of intersection points is
Another method to count the intersection points of a group, we can imagine that the number of intersection points as the number of 2-permutation of k sensors, for example, S is a set of overlapped sensors 
. The 2-permutation of S is:
,
We can use the Recurrence relation to find the number of intersection points of a group of sensors. We can find the recurrence relation of the number of intersection points as below:
which can be easily solved using generation function [
, so
.
In
Theorem 3: for a group of sensors
, the number of intersection points within the sensing range of sensor is
Proof: it is easy to realize that the number of intersection points (internal and external) of the sensor is satisfying the recursive relation:
So finding the solution to this recursive relation is the proof of the theorem.
Suppose the generation function is
In addition, suppose that 
Then
The external intersection points of sensors are the points located on the border of a sensor. However, the internal points are those points located inside the sensors but not on the border.
Lemma 3 (the number of external points): the number of intersection points located on the border of a sensor, which belongs to a group of sensors 
is 
.
Proof: from
We can solve this relation using generation function as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem
.
Lemma 4 (the number of internal points): the number of intersection points located inside a sensor (not including the points located on the border) is
Proof: from
We can solve this relation using generation function as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem.
From lemma 3, and lemma 4, we get the number of intersection points of a sensor that belongs to a group of sensors 
by counting the intersection points located on the border of the sensor (external points) and the intersection points located inside the sensor (internal points).
Theorem 4: The number of areas inside the sensor 
that belongs to a group of sensors 
is
Proof: it is easy to realize that the number of areas inside the sensor is satisfying the recursive relation:
We can solve this relation using generation function as in the proof of theorem 3. Therefore, the solution to this recursive relation is the proof of this theorem
In
Our goal is to count the number of areas inside the sensor 
that associated to multiple groups. The groups to which 
belongs can be defined as following:
Say that 
then we can define the mother group of 
as:
. 
, where a, b, c are the positive integers numbers that represent the degree of coverage.
As shown in 
, 
, 
, 
, 
, 
and 
as below:
Since 
only, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
Since
, then the mother group of sensor 
is 
It is clearly that the mother group of sensors of the whole network is equal to the union of mother groups of all sensors as shown below:
Let us now count the number of areas inside a sensor; these areas are generated by intersection of multiple groups of sensors. For facilitate, let us define 
as the number of areas inside a sensor
, which are generated by overlapping of a group of sensors
. As explained above, the mother group of 
is 
, according to theorem the number of areas which created inside
is 
(indicated in 
, is
.
The total number of areas inside a sensor
, which, associated to multiple groups is denoted by
. these areas are created by intersection of sensors belong the mother groups
.form the first glance, the 
seems like
However, this form is not correct, because 
is an element belongs to every sup-group of 
this means that there is one area will be counted 
times. Let us denote the length of mother group by 
which indicates the number of sub-groups inside the mother group of the sensor. So the corrected count of areas inside
, which belongs to
, is:
Below we can count the number of areas of sensors of
One of our aims is to find the number of distributed messages that will be generated during communications of sensor 
associated to mother group
. Let us define the number of messages by
.For ease, let 
be the order of
.
Indicates the number of sensors that belong to every sub-group inside the mother group
, but not including
, with no repetition, (some sensors might belong to more than one sub-group). For example 
is
; the order of 
is
To generalize this idea, we can write the equation further. We have 
associated to the mother group 
, the order of mother group of 
is as the equation below
Here the integer number 
is the count of subgroups of
. In addition, c is the repetition.
It is clear that 
since the degree of sub-group 
is one and the there is only one sub-group. Applying this calculation to mother group of sensor 
, the order
.
Theorem 4: The number of distributed messages sent form
, associated to a mother group
, is 
Proof: The number of messages depends on the degree of overlapped sensors. The more the degrees of coverage are, the more the areas will be generated. Therefore, the more messages will be generated. When a target moves within the range of a sensor
, it will send notification messages to all neighbors but certainly not to itself. Since the sensor contains a certain number of intersection areas and a certain number of sensors cover these areas, the sensor will send a notification message to all the sensors that cover the same area. 
is the number of areas inside 
which belongs to
, and 
is the order of
, then
.
The number of messages of the network in
