Communications and Network, 2013, 5, 414-420
http://dx.doi.org/10.4236/cn.2013.53B2076 Published Online September 2013 (http://www.scirp.org/journal/cn)
Copyright © 2013 SciRes. CN
Shannon Entropy in Distributed Scientific Calculations on
Mobiles Ad-Hoc Networks (MANETs)
Pablo José Iuliano, Luís Marrone
LINTI, Facultad de Informatica UNLP, La Plata, Argentina
Email: piuliano@info.unlp.edu.ar, lmarrone@linti.unlp.edu.ar
Received June 2013
ABSTRACT
This paper addresses the problem of giving a formal metric to estimate uncertainty at the moment of starting a distri-
buted scientific calculation on clients working over mobile ad-hoc networks (MANETs). Measuring the uncertainty
related to the successful completion of a distributed computation on the aforementioned network infrastructure is based
on the Dempster-Shafer Theory of Evidence (DST). Shannon Entropy will be the formal mechanism by which the con-
flict in the scenarios proposed in this paper will be estimated. This paper will begin with a description of the procedure
by which connectivity probability is to b e obtained and will continue by presenting the mobility model most appropriate
for the performed simulations. Finally, simulations will be performed to calculate the Shannon Entropy, after which the
corres po nding conclusio ns will be pre s ented.
Keywords: MANETs; Shannon; Uncertain; Simulation; Distributed
1. Introduction
Mobile computing has been established as the de facto
standard for Web access, owing to users preferring it to
other connection alternatives. Mobile ad-hoc networks,
or MANETs, are currently the focus of attention in mo-
bile computing, as they are the most flexible and adapta-
ble network technology in existence today [1]. These
qualities are particularly desirable in the development of
applications meant for this kind of infrastructure—a
number of American government projects, such as the
military investment in resources for the development of
this technology, bear witness to this fact.
As previously mentioned, ad-hoc mobile networks are
the most flexible and adaptable communication architec-
ture currently in existence. These wireless networks are
comprised of interconnected autonomous nodes. They are
self-organized and self-generated, which eliminates the
need for a centralized infrastructure.
The use of this type of networks as a new alternative
for the implementation of distributed computing systems
is closely related to the capability to begin calculation,
assign parts and collect results once computation is fi-
nished. Due to the intrinsic nature of this kind of network,
there is no certainty that all the stages involved in this
kind of calculation can be completed, which makes esti-
mating the uncertainty in t hese scenarios, a vital capabil-
ity.
2. Measuring the Problem
The movement patterns of the autonomous nodes, and
consequently their interaction, will have a significant im-
pact in the success or fa ilure in collecting the results of a
distributed computation. In order to incorporate the no-
tion of connectivity among the nodes, a development will
now be presented that shows a formalization of the Con-
nectivity Probability among all the nodes that make up a
MANET, that is, the probability that there is a path be-
tween one node and an y of the rest.
Afterwards, we will take on the task of characterizing
the mobility of the nodes, particularly their median speed
and direction, the range of their communication signal
and the size of the surface on which they circulate. Fi-
nally, another section will detail how to estimate Shan-
non Entropy.
2.1. Defining Connectivity Probability
Let D be the domain bounded by the Euclidean plane
2
{,}R xy=
, within D there are n nodes. At initial time
t=0, the nodes are somehow located and moving. Let
(, )
i ii
r xy=
be the radius vector of node i. Thus, we as-
sume that each node has a communication capacity in the
range r: if the distance between two nodes is greater than
r, then they cannot establish communication. Nodes can
transmit information using multihop connections.
Therefore, we can define a network as connected if
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
415
each pair of nodes has a path between them. Connectivity
Probability quantifies the likelihood of obtaining a con-
nected network from a set of nodes. Clearly, in scenarios
where nodes maintain fixed positions, the connectivity
will depend on node density and connection range. Typ-
ically the simulations of static scenarios that attempt to
determine the connection probability of a number of nodes
located randomly in the simulation area introduce a ran-
dom variable that equals 1 when the network is connected
and 0 otherwise. Thus, the average of the said variable
over the number of trials gives the Connectivity Proba-
bility [2].
For nodes with mobility, time interval divisions are in-
troduced and defined thus:
123
,,,
τττ
±±±
(1)
where
()
kk
ττ
+−
denotes a time interval during which the
network is connected (unconnected). The following func-
tion can then be introduced :
1 si
()
0 si
k
k
t
ft
t
τ
τ
+
+
=
(2)
time intervals can be considered to be randomly distri-
buted, whereby the previously presented function turns
into a stochastic process. Consequently, in dynamical
environments, Connectivity Probability is defined as fol-
lows:
[ ]
()P Eft
++
=
(3)
where
[ ]
.E
is the expected value, as long as it exists. It
can be seen that
+
is time-dependent:
()P Pt
++
=
. For
stationary stochastic processes
P const
+
=. If the statio-
nary process is ergodic, then (3) can be substituted by:
0
1
lim( )ft dtP
τ
τ
τ
++
→∞
=
(4)
This equality is equivalent to:
[ ]
1
lim(0,)
Pmes T
τ
τ
τ
+
+→∞
=
(5)
where
k
T
τ
++
=
and the mes function are used to
measure the total length of the
[ ]
0,T
τ
+
interval. The
problem of whether the network is connected is thus re-
duced to determining the existence and estimation of the
expected value (3), and if the mobility model is statio-
nary and ergodic, (5) can be used to estimate connectivi-
ty [2].
2.2. Dynamical Systems and Stochastic Processes
In a homogeneous network system where node capacity
and properties are equal among all, it can be reasonably
assumed that it can be described by a single system of
differential equations, both for a single node and for all
of them. If some form of randomness is introduced to
node movement, a differential stochastic process will be
needed. If, moreover, the stochastic process is considered
to be stationary, a system of autonomous differential eq-
uations can be used where the right side of the equation
does not explicitly depend on time and where nodes dif-
fer only from their initial conditions [2]. In dynamical
systems theory, a phase flow is defined as a group of
changes along the trajectory during a time interval. Dy-
namical systems are generated by phase flows and can be
described by differential equations as follows:
(), xgx x= ∈Π
(6)
where
is the phase space, x is a set of coordinates in
(usually position and speed) and the dot indicates that
time is the differential. Let n be a number of nodes and
1
,,
n
xx
its phase coordinates, then these coordinates
satisfy the following differential equation :
() ()
(), 1,,
kk
xgx kn== …
(7)
Thus the dynamic of the n nodes is completely defined
by dynamical system (7), which is the direct product of
the n copies of the original dynamical system, (6). Its
phase space
ˆ
πππ(π)n
= ×…× =
is a direct product of
the n copies of the initial phase space and phase coordi-
nates
1
ˆ(, ,)
n
xx x= …
are a set of coordinates of individu-
al nodes. If system (6) has an invariant measure
in
π
,
system (7) will also have an invariant measure in
ˆ
π
and
the direct product will be
1
ˆ
n
µµ µ
= ×…×
. In the con-
nectivity problem, phase space
ˆ
π
can be divided into two
domains D and D' =
ˆ
π\D thus: when
ˆ
xD
, all the
nodes out of the existing n can communicate with each
other. And when
ˆ
xD
, some nodes cannot be reached
by some others. Following the approach from dynamical
systems, the connectivity probability can be estimated as
a time interval when
ˆ()xt D
.
Estimating the connectivity measure can be signifi-
cantly simplified if dynamical system (7) is ergodic in
ˆ
π
.
By definition, a system is ergodic if the measur e of some
invariant sub domain of the phase space equals zero or
the measure of the entire space.
Let
ˆ
()fx
be a measurable and integrable function in
ˆ
π
, for all the solutions of ergodic system (6) there is:
ˆ
0π
ˆˆ
1 ()
ˆ
lim(( )),
ˆ
π
T
T
f xd
fx tdt
T mes
µ
→∞
=
∫∫
(8 )
where
ˆ
π
ˆˆˆˆ
π(π)mes d
µµ
= =
(9)
is the measure of the entire phase space [2].
Let f be a function characteristic of a measurable do-
main D:
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
416
ˆ
1 si
ˆ
()
ˆ
0 si
xD
fx
xD
=
(10)
Since f is limited and D is measurable,
ˆ
:πfR
is
integrable. In this case, the left side of (12) is equivalent
to the time interval
0tT≤≤
when
ˆ()
xt resides in the
domain D.
Thus the Connectiv ity Probability of an ad-hoc mobile
network will be equivalent to the right side of (12):
ˆ
π
mes D
mes
(11)
This approximation can be interpreted in terms of the
theory of stochastic processes in phase space ˆ
π. The
probability for a system in measurable domain D is de-
termined by formula (11). Let
()fx
be a function cha-
racteristic of domain D and
()
xt the solution of system
(6). Thus the function (())f xt can be interpreted as a
stochastic process. Let E[f(t)] be the expected value of
the function f(t) at time t. If the right side of Equation (6)
is not time-dependent, then the stochastic process is sta-
tionary. In particular, this means that E[f(t)] does not
depend on t. If the system is also ergodic, the expected
value can be calcul a t e d using formula (11):
0
1
ˆ
lim(( ))[]ˆ
π
T
T
mes D
fx tdtEf
T mes
→∞ = =
(12)
Therefore, the problem of calculating expected value
(3) is reduced to a geometric problem in which we must
determine the volume of the domains in a phase space if
the process is ergodic [2].
2.3. Shannon Entropy: Measuring Uncertainty
Uncertainty, in particular the amount of conflict in the
system, will be measured using the Dempster-Shafer
Theory of Evidence (DST). Functions for estimating the
conflict in a system using a probability distribution must
fulfill certain axiomatic requirements [3], namely:
Let
c
f be the estimator of the amount of conflict and
12
,,,
n
p ppp
=〈 …〉
the probability distribution, c
f
must fulfill:
Expansibility: adding a 0 component to the proba-
bility distribution does not modify the value of the un-
certainty measure.
Symmetry: the calculated uncertainty does not vary
in relation to the permutation of the arguments.
Continuity: function
c
f is continuous for all
12
,,,
n
p ppp
=〈 …〉
.
Subadditivity: the uncertainty of the joint probabil-
ity distribution is less than or equal to the uncertainties of
the marginal distributions.
Additivity: for any pair of marginal probability dis-
tributions that are non-interactive, the uncertainty of the
associated joint distribution must be equal to the sum of
the uncertainties of the marginal distributions.
Monotonicity: uncertainty must increase if the num-
ber of elements increases.
Branching: Let 12
,,,
n
p ppp=〈 …〉
over
{ }
12
,,,
n
X xxx= …
. If two partitions are generated from
X,
{ }
1
,,
s
Ax x= …
and
{ }
1, ,
sn
Bx x= +…
, then
12
(, ,,)
cn
fpp p
…=
Normalization: to ensure uncertainty can be meas-
ured in bits, it is required that:
12
(, )(,,,)
12
(,,, )
s
cAB c
AA A
ss n
c
BB B
p
pp
fpp fpp p
pp p
fpp p
+…
++
+…
11
(,)1
22
c
f= (13)
Shannon Entropy will be the formal mechanism by
which the conflict will be estimated in this document.
This measure of uncertainty stems from a probability dis-
tribution obtained from observing the results of an expe-
riment or any other research mechanism. Probability dis-
tribution p has the form
()ppx xX=〈 ∈〉
where X is
the domain of discourse. Additionally, a decreasing func-
tion in relation to incid ence probability is de fined, called
anticipatory uncertainty, which must have a decreasing
monotonous continuous mapping, and be additive and
normalized. This yields that the anticipatory uncertainty
of an x result is:
2
log( )px.
Thus, Shannon Entropy, which provides the expected
value of the anticipatory uncertainties for each element
of the domain of discourse [3], takes the following form:
2
()()()log ()
xX xX
Sppxpx px
∈∈
== −
∑∑
(14)
The normalized version of (14) takes the following
form:
2
2
1
()()log ()
log (||)
xX
Sppx px
X
=−
(15)
and is the one used to calculate uncertainty in the simula-
tions per formed.
3. Simulation
Following, we present an adjustment to the previously
obtained theoretical results, in order to reach a simulation
method that is consistent with them. A description of
scenarios posed and results obtained will follow.
3.1. Adjustment of mes
ˆ
π
and mes D
It is considered that the area where the computational
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
417
model proposed for distributed calculations on ad-hoc
mobile networks operate will be small—the work surface
will be comparable to that of a university campus, go-
vernmental building or office [4]. This results in
ˆ
π be-
ing the total simulation surface and mes D, the area
where the nodes are in positions that keep the network
connected. However, calculating mes
ˆ
π
and mes D as
previously proposed is an extremely complicated and
laborious task [2]. For this reason, an alternative method
is presented to determine first the Co nnectivity Probabil-
ity and later the uncertainty involved.
With the goal of validating the scenario put forth in
previous sections, the simulation will take place using a
modified version of the Monte-Carlo method, where the
nodes will be initially located in random positions in
such a way that they will form a connected network.
Their position will be updated in each instance of the
simulation, in accordance with the specifications of the
RWMM model [5], and afterwards the network connec-
tivity will be verified. Thus, with the calculation framed
within the aforementioned simulation process, the Con-
nectivity Probability will be obtained by means of the
M/N quotient, where N is the total number of simulations
and M is the number of simulations in which the network
was connected where N was great enough. Thus, mes ˆ
π
is N and mes D is M [2].
3.2. Results of the Simulation
Different ad-hoc mobile network topologies will have
different Connectivity Probability values, and, therefore,
the Shannon Entropy will vary.
In RWMM [5], the nodes for each simulation stage
will select a direction in which they will move randomly
between (0,
2π
], and the speed at which they will move
will be the expected value uniformly distributed between
the speeds of 1 m/s and 10 m/s—the rates at which we
move by foot—which will equal to 3.90 m/s. When a
node reaches the edge of the simulation area, it will ro-
tate 180 degrees and will be placed again within the area,
after which the process will continue.
All the simulations begin in a connected network to-
pology and a fixed connectivity radius within which a
node can be connected to another. Then, stage after stage,
the following operations will take place:
1) For each node
i
node
of the ad-hoc mobile net-
work, a direction
i
dir
is randomly chosen between (0,
2π
], and its position
(, )
i ii
posxy= is updated in ac-
cordance with
()
i ii
newposposdir Vdt=+××
where
i
newpos
is the new position of
i
node
.
2) Once all the node positions have been updated, for
each node
i
node
in position
i
pos
it is verified wheth-
er it can establish a connection with another node
j
node
located within its connection radius. This verification is
performed by means of the calculation of the Euclidean
distance
,ij
Dist
between the two nodes, later checking
whether
,
2
ij
Dist RADIUS<=
is fulfilled, where RA-
DIUS is the connection radius between the two nodes.
3) If each of the nodes can establish a connection with
all other nodes in the network, the resulting network is
still connected and the connected topology incidence coun-
ter M is increased by one unit.
4) Once the N stages are comple ted, with
5
10N=
(this
number is sufficient to achieve at least 99:999999995%
confidence and a distance between the empirical and the
real at most 0.01 using by the Dvoretzky-Kiefer- W olf o-
witz inequality [6] and obtain adequate statistical guar-
antees), Connectivity Probability and Shannon Entropy
emerge as a result of the simulation process. Two con-
siderations must be emphasized regarding the calculation
of these two measures:
a) A s previously mentioned, because the calculation of
Connectivity Probability is framed within the simulation
process, it can be obtained by means of the M/N quotient,
where N is the total number of simulation stages and M is
the number of times when the network was connected.
b) Shannon Entropy is calculated by means of the fol-
lowing f ormula:
22
()
(log()(1)log(1))si01
0si01
Sp
pp ppp
pp
−+− −<<
=
=∨=
(16)
As shown, S(p) is zero when p reaches the value zero
or one, theref o re , there will always be uncertain about the
result. Another interesting fact is that the formula of
Shannon Entropy which results from this is normalized,
since:
22
1 11
()()() ()
log (||)log (2)1
SpSpSp Sp
X=== (17)
The results of the simulation scenarios are shown in
Figure 1.
3.3. Correlation between Connectivity
Probability and Shannon Entropy
The theoretical development of the first lines of this doc-
ument and the results of the simulations will allow the
reader to sense the existence of a relationship between
Connectivity Probability and uncertainty in a system. For
this reason, the following section will analyze this rela-
tionship formally. We can begin to study the relationship
between Shannon Entropy and Connectivity Probability
by observing what happens when the first of the two
magnitudes reach its limit values, i.e., its minimum and
maximum. The first of these values, equal to zero, is reg-
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
418
Figure 1. Simulation Results.
istered when the Connectivity Probability has reached
either one or zero. This shows that when there is no pos-
sibility to maintain connectivity or when the poss ibility is
absolute, uncertainty disappears. The maximum value of
entropy is reached when the probability takes medium
values, which means that there is a state of total uncer-
tainty.
In both cases, the relationship between Shannon En-
tropy and Connectivity Prob ability is evident, but for th e
analysis of the other cases, correlation coefficients must
be used.
The correlation coefficient that is most widely used is
Pearson’s coefficient (r):
1
i
i
pp
n
=
(18)
1
i
i
ss
n
=
(19)
1()
pi
i
S pp
n
= −
(20)
1()
si
i
S ss
n
= −
(21)
1()( )
psi i
i
Spps s
n
= −−
(2 2)
ps
ps
S
rSS
=
(23)
and the extreme values of its possible results are: 0 (no
relationship) and
1±
(maximal relationship) [7]. The
variables analyzed by means of this method, p and s,
must fulfill certain requirements. The two that are the
most relevant to this study are as follows:
1) Variables p and s must be continuous.
2) The relationship between p and s must be linear.
Of the two previous conditions, the most relevant to
our observations, or restrictive of them, as the reader
prefers, is 2, since it would require the relationship be-
tween Connectivity Probability and Shannon Entropy to
be linear, which is not the case. Therefore, it is evident
that Pearson’s Coefficient cannot be used as proposed,
since the relationship under analysis is curvilinear. For
this reason, this behavior will be analyzed dividing vari-
able p into two segments, which will result in two study
groups: the first, we will call 1
gwhere (0,0,5)p, and
the second,
2
g, where [0, 5,1)p. Correlation will
therefore be calculated in separate groups.
The results obtained and detailed in Table 1, show
that:
1) 1
g exhibits direct (positive) dependence between
p and s, i.e., for large values of p there will be large val-
ues of s.
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
419
Table 1. Connectivity P/Shannon Entropy Correlation by
Groups.
Sup. N. Grp. 1 C. Pearson
Grp. 1 N. Grp 2 C. Pearson
Grp 2
50 m × 50 m - - [2, 100] 0.993
100 m × 100 m [2, 11] 0.975 [12, 100] 0.965
150 m × 150 m [2, 28] 0.958 [29, 100] 0.971
2) 2
g shows that the relationship between p and s is
inverse (negative) dependence, i.e., for large value s p the
values of s will be small.
Based on these results, we can conclude the following
for MANETs that operate on surfaces of:
1) 50 m × 50 m: uncertainty will decrease as the
amount of network nodes increases, due to greater Con-
nectivity Probability.
2) 100 m × 100 m: first, if the amount of nodes varies
between two and eleven, starting from two and taking
eleven as a maximum, as more nodes are added to the
network, uncertainty and probability will increase together.
Once the twelve node threshold is reached, uncertainty
will begin to decrease, while probability increases.
3) 1 50 m × 150 m: this case is similar to the last, with
the exception that node intervals are displacedwhen
the amount of nodes varies between two and twenty-
eight, uncertainty and probability will increase as nodes
increase, and if the amount varies between [29, 100],
uncertainty will decrease while probability still incre as e s .
It is clear that the most interesting results are those
registered in
2
g
for all surfaces, as it is there that com-
putations will have the greatest probability to succeed
with less uncertainty. However, the question remains as
to what amount of nodes and Connectivity Probability
will bring a success certainty high enough to begin com-
putation. One valid criterion is to detect value
[0.5,1]
i
v
where Connectivity Probability and uncertainty are equal
or close enough and operate on the uncertainty interval
between
[0, ]
i
v
. Values
i
v
for the performed simulations
are detailed in Table 2. Therefore, fo r surfaces of 50 m ×
50 m, 100 m × 100 m and 150 m × 150 m, distributed
calculations will begin when 2, 14 and 34 nodes have
been reached, respectively.
4. Conclusion
The development of this work duly evidenced and do-
cumented that the uncertainty existing at the beginning of
a distributed computation on a MANET will depend di-
rectly on the amount of nodes participating in it and on
the surface involved. This statement is based on the re-
sults obtained from the simulations detailed in this doc-
ument, which allowed us to conclude that uncertainty be-
gins to decrease once node density has reached a certain
threshold, and that this threshold takes different values
Table 2. Values of vi.
Sup. Nodes Connectivity Probability Value vi
50 m × 50 m 2 0.779 0.760
100 m × 100 m 14 0.738 0.828
150 m × 150 m 34 0.824 0.671
for differe nt su rfa ce s.
Works oriented towards correctly identifying the amount
of uncertainty existing at the time the results of a distri-
buted calculation on ad-hoc mobile networks are collected
bring the potential benefit that they can be used to de-
velop more intelligent workload distribution strategies
that take into accoun t the amount of uncertainty they w ill
have to deal with, which will necessarily results in more
efficient computations. In this sense and based on the
latest studies oriented towards providing more certain
mechanisms as to the conservation of power in the de-
vices that comprise a MANET [8] or on equally relevant
studies focusing on achieving the greatest cooperation
possible between the nodes of an ad hoc mobile network
[9], thus mitigating their intrinsic egotism, the results of
having an uncertainty measure that would either indicate
that there is no certainty to achieve calculation comple-
tion or ensure its success will be twofold. In the first of
the aforementioned two fields of study, preventing work-
load distribution in situations where calculation concr e-
tion is not ensured will have a direct repercussion in the
conservation of power in devices, which will result in
longer operational periods which unable to identify the
aforementioned scen arios. The secon d research field seeks
to maximize cooperation among the nodes. With this in
mind, in scenarios where completion certainty is medium
or low, one possible distribution strategy oriented toward
collaboration could be assigning workload only to the
most collaborative nodes, to avoid the risk of assigning
load to un-collaborative nodes, which, in the event of
result collection failure, may take a more selfish or con-
servationist attitude tow ard their resources (such as pow-
er) and leave the MANET. In a scheme of mobile distri-
buted calculation where all participants offer their colla-
boration to find the answer to a common interest problem,
such as the SETI@Home program [10], measuring un-
certainty can be used as a function to grant credit to col-
laborators—when a participant is notified that there is a
medium to high level of uncertainty regarding computa-
tion success and they decide to participate nonetheless,
more credits can be granted than in scenarios where total
certainty of success exists. If more credits mean more
benefits for the participant in some way, for example,
publicity of the most committed participant in the calcu-
lation environment, then we would have a psychological
mechanism of positive reinforcement that would promote
node collaboration, which would enable a network con-
P. J. IULIANO, L. MARRONE
Copyright © 2013 SciRes. CN
420
formed by more collaborative and satisfied participants
that are less egotistic.
All potential strategies of distributed computing over
MANETs presented in this document and others that can
emerge from an intelligent use of uncertainty measures
will bring with them new types of applications that will
seize all the power of the underlying network infrastruc-
ture.
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