Journal of Communications Vol. 9, No. 2, February 2014

Self-Adjustment Mechanism Guaranteeing Asymptotic
Stability of Clusters Formed by Autonomous Decentralized
Mechanism
Hiroki Takayama, Sota Hatakeyama, and Masaki Aida
Graduate School of System Design, Tokyo Metropolitan University, Tokyo, Japan
Email: hiroki.takayama.tmu@gmail.com; sota-hatakeyama@sd.tmu.ac.jp; aida@sd.tmu.ac.jp

From the above background, we have previously proposed autonomous decentralized clustering mechanism tointroduce hierarchical structures in MANETs [10]. In our
mechanism, each node has values of a certain distributionin advance, and nodes exchange a portion of their value
with the adjacent nodes by obeying a predefined action
rule based only on local information. After that, ourmechanism can form spatial structures in the network,
then cluster structures are decided based on local maximum values of the distribution. We also have verified theeffectiveness of our mechanism by comparing with autonomous decentralized clustering mechanism based on
bio-inspired approach which is well known autonomous
decentralized mechanism [11], [12]. In this verification,
we showed that our mechanism can act about 30 times
faster than the mechanism based on bio-inspired approach[13]. Therefore this suggests the possibility that ourmechanism has the flexibility for dynamic change of
network topology with movement of nodes.
In large scale MANETs, the most drastic changes of
network topology are fusion and division of networks.
The fusion of networks means that a node group encounters the other node groups which are outside of communication range and that these node groups enter to inside ofcommunication range. On the other hand, the division of
networks means that a node group is divided into somenode groups with movement of nodes. In these drasticchanges, cluster structures of different subnetworks needcommon criteria when fusion of subnetworks occurs.
This is because cluster structures cannot be formed basedon the common criteria when the fusion of different subnetworks occurs. Therefore cluster structures need tohave common criteria even in different subnetworks.
However, cluster structures formed by our mechanism do
not have the common criterion. In our mechanism, themagnitude of the amplitude of the distribution in different
subnetworks varies. Here, we define the magnitude of theamplitude of the distribution as the difference between
maximum of the distribution and minimum of the distribution. If subnetworks have different amplitudes, subnetworks cannot create appropriate clustering based on
the common criteria because cluster structures having thehuge amplitude strongly affect clusters formation when
the fusion of subnetworks occurs (Fig. 1). To avoid thisproblem, different subnetworks should have the common

Abstract—Clustering mechanisms are crucial for scalable routing in mobile ad hoc networks (MANETs). We have previously
proposed an autonomous decentralized clustering mechanismthat is based on differential equations. This method adopts both
the diffusion effect for autonomous decentralized control, andthe drift effect for balancing with the diffusion effect. In ourconventional mechanism, we need to set the strength of diffusion effect stronger than the drift effect because fine tuning forbalancing both effects is difficult. However, it results in a decrease of the number of clusters by the diffusion effect, and it isa problem on forming sustainable cluster structures. We need
the solution of this problem for clustering in MANET. In thispaper, we propose a mechanism autonomously adjusting theeffects of both diffusion and drift. We show that our proposedmechanism can form sustainable cluster structures reflecting thecharacteristic of our conventional mechanism. Finally, we evaluate the effectiveness of the proposed mechanism for an environment in which the fusion of subnetworks occurs.
Index Terms—self-adjustment, autonomous decentralizedmechanism, clustering mechanism, structure formation, mobile
ad hoc networkI.

INTRODUCTION

The autonomous decentralized mechanism in which
each node can act autonomously based on local information of itself is an effective control framework for theenvironment in which it is difficult for us to know thestate of a whole system exactly. In particular, mobile ad
hoc networks (MANETs) are one of the systems in which
the mechanism is effective. This is because MANETs arecomposed autonomously of only mobile nodes, and the
topology of MANETs changes frequently with the
movement of nodes [1], [2]. In MANETs, since routing
schemes based on a flat structure have scalability problems, the hierarchical structure is essential to introduce
dynamic routing [3]–[8]. For example, communication
overhead of representative proactive protocol as routing
protocol for a flat structure increases with the square of
number of nodes [9].

Manuscript received October 16, 2013; revised February 4, 2014.
This work was supported by a Grant-in-Aid for Scientific Research(B) No. 23300031 (2011.2013) from the Japan Society for the Promotion of Science, and by National Institute of Information and Communications Technology (NICT).
Corresponding author email: aida@sd.tmu.ac.jp.
doi:10.12720/jcm.9.2.180-187©2014 Engineering and Technology Publishing180

Journal of Communications Vol. 9, No. 2, February 2014

amplitude. The most primitive way to realize the common
amplitude is to set appropriately the values of parametersin the clustering mechanism, in advance. However, [14]clarified that setting the value of parameters is difficult.
Therefore, in this paper, we propose a self-adjustment
mechanism that can adjust the value of parameters autonomously so as to uniform the magnitude of the amplitude.

( ) ( ))⁄ as:

and

(
)

(

)

term
(

) (

Fig. 3. Decision of potential function

where
isfy

and

the

diffusion
(

(

)

term
(1)

) by using back-diffusion

is positive constant, andneeds to sat(2)Fig. 1. Fusion of subnetworks having different amplitude

The effect of the drift term is to emphasize the peaks of
distribution, on the other hand, the effect of the diffusion
term is to smooth the distribution. Our mechanism forms
the spatial structures by balance between the drift effectand the diffusion effect (Fig. 2). The function ( ) is
determined by a potential function ( ) as
( )
(3)
( )

Fig. 2. Summary of our autonomous decentralized clustering mechanism

The drift effect should be opposite effect comparedwith the diffusion effect. Thus, we decide the potentialfunction as (4):This paper is organized as follows: Section II briefly
explains our autonomous decentralized clustering mechanism for introducing hierarchical structures for ad hoc
networks. Section III proposes the self-adjustment mechanism that can adjust the magnitude of amplitude. Section
IV investigates whether the self-adjustment mechanism
can adjust the parameters so as to uniform the magnitudeof the amplitude. Section V verify that the selfadjustment mechanism can maintain cluster structures ofsubnetworks when the fusion of subnetworks occurs. In
section VI, we conclude this paper.

(

II. AUTONOMOUS DECENTRALIZED CLUSTERINGMECHANISM
We briefly explain our autonomous decentralized clustering mechanism in continuous 1-dimensional space [10].
Let the density distribution of a certain quantity at position at time t in space be ( ). The value of initial
distribution ( ) is determined based on a certain metric, and each position exchanges a part of the value of( ) with adjacent positions by obeying predefinedrule. For example, a certain metric is chosen based onbattery energy or node degree. By continuing the exchange, this mechanism can form spatial structures ofclusters that is described by ( ).
Here, we explain the predefined rule for exchanging
the value of ( ). Let ( ) be 1-dimensional vector
representing the magnitude of flow exchanging the valueof ( ) per unit of time. ( ) is expressed by the drift©2014 Engineering and Technology Publishing

181

)

( (

)

(

)

)

(4)

where is positive constant. Here we use
as updateinterval
toupdatethepotential
function.
( )⁄in (4) means time reversal of( )⁄indiffusion phenomenon. That is,(4) sharpens the peaks of the distribution. Then, we makethe distribution upside down (Fig. 3). Hereby, the drift
effect can be opposite to the diffusion effect. From theabove discussion, the action rule is determined by only
local information of adjacent nodes. Since similar
characteristic of local interaction is also true in dimensional space, this mechanism can work as aautonomous decentralized manner for arbitrary network
topology.
Next, we show the action rule for an arbitrary network[10]. We define the set of nodes that are adjacent to nodeas
. We also define discretized time as. The() at next time of the ( ) at time ()
of node is as following.
(

)

( )

∑(

( )

( ))

(5)

( ) and( ) are the variations of drift
whereeffect and diffusion effect within unit of time. Here, we( ) and( ) as following.
defineJournal of Communications Vol. 9, No. 2, February 2014

( )

{
( )

( )

(

( )

)

( )

(

( )

)

( (

( )

)

(

( ( )

the distribution. By basing on this idea, a cluster is defined by nodes reaching the same peak, and each point
having maximal value is a cluster-head.

(6)

))

(7)

( ))

(8)

III. SELF-ADJUSTMENT MECHANISM
The above mentioned mechanism can form cluster
structures by basing on the predefined local rule of (1). In
this procedure, we need to adjust the balance between thedrift effect and the diffusion effect, in advance. Here, we
consider about how to set parameters of these effects (Fig.
4). If we set parameters with the drift effect stronger thanthe diffusion effect, some peaks are over-emphasized by
the drift effect. Therefore spacial structures are notformed when we set parameters with the drift effectstronger than the diffusion effect. On the other hand, if
we set parameters with the diffusion effect stronger thanthe drift effect, we can get spacial structures with a certain spatial size. Thus we have ever set parameters withthe diffusion effect stronger than the drift effect in simulation studies. However, if we set parameters like that, themagnitude of the amplitude of the distribution decays tozero exponentially. This is because the diffusion strongly
affects the amplitude of the distribution. Of course cluster
structures are maintained even if time progresses because
the amplitude of the distribution has no relation with cluster structures. However, when drastic change like thefusion of subnetworks occurs, they have different amplitudes, in general. This causes that cluster structures in a
subnetwork having small magnitude of the amplitude
cannot maintain when the fusion of different subnetworks
occurs because a subnetwork having large magnitude of
the amplitude has a huge effect on the cluster structures.
Although one solution for the problem is to become thediffusion effect equaling to the drift effect by setting parameters, it is difficult [14].

We explain about the above equation. Because the drift
effect emphasizes peaks of the distribution, a part of the
distribution moves to from (to from ) in the case of( )). That is, a part of distribution( ( )moves to nodes having the smaller potential value fromnodes having the larger potential value. Here, equation( ) holds. On the other hand, because the( )diffusion effect flats the distribution, a part of the distribution moves to from (to from ) in the case of( )).

( )( ( )( )Next, we explain how to determine the potential func)tion ( ). The value of the potential function (of node at time
is as follows:
(

)

( ( )

( )

∑(

( )))
(9)

( ) is variation created by the backwhere thediffusion of ( ) to from within unit time. Here, we( ) berequire attention for deciding the variation( ) has possibilities to act
cause the variation
against our purpose. For example, nodes which are notpeaks of the distribution have possibilities to be emphasized as peaks of the distribution when the variation( ) gives values to multi-nodes. Thus we define the( ) for networks as transferring the value
variation
towards a node having largest value among adjacent( )nodes. From the above discussion, the variation
is given as follows:
( )

{

( )

( ) (
(

( )
)

( ))
(10)

( )

(

( ( )

Also, the value of parameters

( ))

)

(11)

needs to satisfy
Fig. 4. Balance between drift effect and diffusion effect(12)
We can consider two approaches to solve this problem.
One is an approach to stop the smoothing efect of the
distribution. This approach is discussed in [14]. In thisapproach, nodes record a value of the distribution of eachtime, and each node uses a value of the distribution of thesame time even if in different subnetworks. Thereby, thisapproach can uniformize the magnitude of the amplitude
between different subnetworks. Another is an approach to
keep a balance between drift and diffusion effects adap-

whereis the maximum degree of nodes in the network. This is because if the value of the parameters
are larger than that of (12), the values of some nodes become negative value by the diffusion effect.
Next we explain how to decide clusters. Clusters are
based on local maximal value. That is, when each node
tracks along the direction of the steepest ascent of the
distribution, each node reaches local maximal value of©2014 Engineering and Technology Publishing182

Journal of Communications Vol. 9, No. 2, February 2014

The actual form of the function ( ) should be determined according to the following three conditions.
( ) should be monotonically increasing function.
( ) should have the upper-bound.
 In order to keep the amplitude of distribution at an
appropriate value, ( ) should have large gradientaround the corresponding value of | ( )|tively, by self-adjustment. In this approach, we do notneed special configuration like a memory to record thevalue of distributions required in a prior approach. Thus,
this paper discusses the later approach.
To consider about the approach based on selfadjustment, we need to consider relative strength between
the drift effect and the diffusion effect. In conventional
mechanism, since the relative strength decided by setting
parameters does not change even if time progresses, theamplitude of the distribution continues to decay in thecase of the diffusion effect stronger than the drift effect.
Hence we consider a mechanism that adjusts the relative
strength autonomously even if time progresses in order tobecome the diffusion effect equaling to the drift effect.
The concept of the mechanism is as following: If the drift
effect is stronger than the diffusion effect, the mechanism
changes autonomously the diffusion effect to strong effect. On the other hand, if the diffusion effect is stronger
than the drift effect, the mechanism changes autonomously the diffusion effect to weak effect.

Fig. 6. Typical example of function (|conditions()|) satisfying the three

The first condition is required for stabilizing the amplitude of the distribution. Since the larger value of| ( )| means stronger drift effect, stronger diffusion
effect is required. Therefore, (| ( )|) should become large. Conversely, for small value of | ( )| ,(| ( )|) should become small. The second condition
concerns the limitation (14) of the maximum value ofdiffusion coefficient. In the case of (13), the parametersneed to satisfy(|

Fig. 5. Concept of self-adjustment mechanism

Thereby, the mechanism can equalize the two effects
autonomously.
By basing on the above concept, we introduce a function changing the relative strength to the diffusion term of(1). Here, to introduce the function, we use the fact thatthe gradient of the distribution becomes likewise larger
when the magnitude of the amplitude becomes larger.
Also, to use the fact is suitable for autonomous decentralized control because the gradient of distribution is decided by only local information. From the above discussion,
we use the function as a diffusion coefficient which depends on the absolute value of the gradient of distribution,and try to govern the strength of diffusion effect adaptively. Let us introduce the extended local action rule ( )as follows:
(

)

(

) (

)

(|

(

)|)

(

)
(13)

where (| ( )|) is a function to adjust the strength ofthe diffusion effect.

©2014 Engineering and Technology Publishing

(

)|)

(14)

Then if the function does not have the upper-bound,
we cannot set the value of the parameters. Conversely, if has the upper-bound, we can set the parameters
to satisfy the condition (14). The last condition concerns the position of the equilibrium in self-adjustment
mechanism. Let us consider that appropriate magnitudeof the amplitude is featured by the corresponding gradient| ( )|. Then, should be the equilibrium. Thismeans that the gradient of (| ( )|)should have larger value around | ( )|.
From the above discussion, we can choose
(|

(

)|)

(|

(

)|

)

(15)

as a typical example of the function (| ( )|) satisfying the above three conditions. This function is the logistic curve shown in Fig. 6. Here in Fig. 6, the verticalaxis of the results expresses the value of the function ,and horizontal axis of the results expresses the magnitudeof the amplitude. In this example, parameter
need tosatisfy the limitation
(16)

183

Journal of Communications Vol. 9, No. 2, February 2014tional mechanism. In the former evaluation, we consider
a way to evaluate as follows. First we estimate a convergence value for the time progress of the amplitude. Second we subtract the estimated value from the amplitude.
Finally we plot the differences in semi-log scale, then if
the differences decayed to zero exponentially, we consider the magnitude of the amplitude as converging on a
finite constant value. In later evaluation, we verify thatthe mechanism can form cluster structures based on initial distribution by using the patterned structure. Here, if
cluster structures formed by self-adjustment mechanism
are same structures with cluster structures formed by ourconventional mechanism, we assume our mechanism canmaintain the characteristic of our conventional mechanism.

because the upper-bound of is 1. This is the same condition as (12) in the previous mechanism.

Fig. 7. Initial distribution
TABLE I: PARAMETER SETTING (1)randomized structure

patterned structure

B. Evaluation

IV. EVALUATION FOR ASYMPTOTIC STABILITY OF
CLUSTERSIn this section, we verify that our mechanism can converge the amplitude of the distribution on a finite constantvalue, through simulation.
A. Simulation Methodology
In this subsection, we explain simulation methodologyof the verification. In this verification, we use a unit disk
graph having torus boundary to exclude the influence ofthe boundary. We can obtain the unit disk graph throughfollowing procedure. We arrange some nodes randomly,
and then we set a circle of certain length around eachnode. If nodes are inside the circle, we set links for thesenodes. Here, since we can assume the circles as wirelesscommunication range, the unit disk graph is one of themodels which can express the topology of MANETs.
Also, we set 1,600 nodes on a plane of 1 km × 1 km, anduse 60 m as communication range. We also use two initial distributions as Fig. 7 in which the value of distributions is described by using color map. The left figure in
Fig. 7 expresses randomized structure. Here, the value ofthe left figure in Fig. 7 is set to random value using theuniform distribution between [0, 10]. The right figure in
Fig. 7 expresses patterned structure. Here, the smaller
valued area of the right figure in Fig. 7 is set to random
value using the uniform distribution between [0, 1], andthe larger valued area of the right figure in Fig. 7 is set torandom value using the uniform distribution between [0,10]. For example, we can assume the value of the randomized structure as battery energy. We also can assume
the patterned structure as an environment having powersupply. We show parameters used in the evaluation in
Table I.
We use two evaluation indexes. One is that the amplitude of the distribution can converge on a finite constantvalue from two initial conditions. The other is that themechanism can maintain the characteristic of our conven©2014 Engineering and Technology Publishing

Fig. 8. Temporal variation of amplitude for randomized structure

Fig. 9. Temporal variation of amplitude for patterned structure

In this subsection, first we evaluate whether the magnitude of the amplitude converges on a finite positive constant. Fig. 8 and Fig. 9 show the results in which we subtract the estimated value from time progress of amplitude
and plotted in semi-log scale. Here, the vertical axis of
the results expresses the magnitude of the amplitude, andhorizontal axis of the results expresses the time. From theresult of Fig. 8 and Fig. 9, we can see that the mechanism
can converge on a finite constant. Here, we estimated thevalue of convergence as 0.2 in the randomized structure,
we also estimated the value as 1.4 in the patterned structure. Next we verify that the advantage of our conven184

Journal of Communications Vol. 9, No. 2, February 2014

gradient around the corresponding value of | ( )| .
Hereby, we can consider that drift effect and diffusion
effect cannot balance because the function does not havelarge gradient. From the result of Fig. 12 that describes
cluster structures, we can see that cluster structures do notdepend properly on characteristic in networks’ initialconditions as compared to result of the prior mechanism
in Fig. 11. Thus we can expect that the conditions areneeded in order to achieve the self-adjustment mechanism.

tional mechanism is inherited to the self-adjustment
mechanism. The advantage of the prior mechanism is toform cluster structures based on initial condition. Hence
if cluster structures formed by our mechanism are samestructures with our conventional mechanism, we can confirm that our mechanism can maintain the characteristicof our conventional mechanism. The left panels of Fig. 10
and Fig. 11 describe the value of the distribution and theright panels of the Fig. 10 and Fig. 11 describe cluster
map that different clusters are denoted by different colors.
Also, Fig. 10 is the result of self-adjustment mechanism.
On the other hand, Fig. 11 is the result of our conventional mechanism. By comparison with cluster structures
formed by prior mechanism, we can see that our mechanism has the same characteristic with prior mechanism.
From the above evaluation, we verified that our mechanism can maintain the amplitude of distribution, and canmaintain the characteristic of our conventional mechanism.

Fig. 12. Cluster structures formed in the case of different function

V. EVALIATION FOR THE FUSION OF SUBNETWORKSIn this section, we verify that our mechanism canmaintain cluster structures based on initial distribution
even if the fusion of subnetworks occurs.
A. Simulation Methodology

Fig. 10. Cluster structures formed by self-adjustment mechanism

Fig. 11. Cluster structures formed by conventional mechanism
TABLE II: PARAMETER SETTING (2)parameter

value

Fig. 13. Summary of the simulation model

Next, we consider the different model for comparison.
Let us consider the case that a function g does not satisfy
the abovementioned conditions. As an example of thecases, we set
(|

(

)|)

|

(

)|

To assume the environment in which the fusion of different subnetworks occurs, we consider a model as following in our simulation (Fig. 13). First we use the unit
disk graph as network topology, and we set a partition
which can interrupt communication as Fig. 13 in order todivide a network into two subnetworks. Next we change
only the time of one of their subnetworks, and then weremove the partition. Here if we use the conventional
mechanism, the amplitudes of subnetworks exhibit differ-(17)

We show parameters used in the evaluation in TABLE
II. Here, since is an unbound function, parameterisstrongly restricted. Also the function does not have large©2014 Engineering and Technology Publishing185

Journal of Communications Vol. 9, No. 2, February 2014ent behaviors. Finally we change time, that is, we create afusion of subnetworks. Then we verify which mechanism
can effectively maintain cluster structures based on initial
distribution.
TABLE III: PARAMETER SETTING (3)conventional mechanism

proposed mechanism

Fig. 15. Fusion of subnetworks in the case of self-adjustment mechanism

VI. CONCLUSIONSIn this verification, we use a unit disk graph having torus boundary to exclude the influence of the boundary.
We set 1,600 nodes on a plane of 1 km × 1 km, and use
60 m as communication range. We remove a partition attime 1,000, and then change the time till 2,000. As initialcondition, we use the patterned structure in Fig. 7. We
use this initial condition as the index of this evaluation.
This is because we can verify easily whether cluster
structures are based on initial condition even if the fusionof subnetworks occurs.

In this paper, we proposed the self-adjustment mechanism that adjusts the magnitude of the amplitude to become the finite magnitude. In order to evaluate the effectiveness of the self-adjustment mechanism, we verified
whether the magnitude of amplitude converges on a finite
positive constant by using two initial conditions that areimportant metrics ensuring the effectiveness of the mechanism. We also verified whether our mechanism can form
cluster structures reflecting the characteristic of our conventional mechanism. Finally we evaluated the effectiveness of self-adjustment mechanism for an environment inwhich the fusion of subnetworks occurs. Future studies
will investigate for topology change with movement of
nodes.

B. Evaluation
Before the verification of our mechanism, we need toverify that the conventional mechanism cannot maintaincluster structures based on initial distribution when thefusion of subnetworks occurs. From the result of left panel in Fig. 14, we can confirm that the magnitude of theamplitude between subnetworks varies. From the result ofright panel in Fig. 14, we can also confirm that cluster
structures are not based on initial distribution. This isbecause cluster structures having the huge amplitude
strongly affect clusters formation. From the results, we
confirmed that cluster structures formed by prior mechanism are not formed based on initial distribution when thefusion of subnetworks occurs.

ACKNOWLEDGMENT
This work was supported by a Grant-in-Aid for Scientific Research (B) No. 23300031 (2011–2013) from theJapan Society for the Promotion of Science, and by National Institute of Information and Communications
Technology (NICT).
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Next we verify the effectiveness of our mechanism.
rom the result of left panel in Fig. 15, we can recognizethat the magnitude of amplitude between subnetworks
varies but it is smaller than the result of Fig .14. Also,from the result of right panel in Fig. 15, we can recognizethat our mechanism can form cluster structures based on
initial distribution by comparison to cluster structures
formed by the conventional mechanism in Fig. 11. From
the results, we confirmed that our mechanism can form
cluster structures based on initial condition even if thefusion of subnetworks occurred.
©2014 Engineering and Technology Publishing

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Journal of Communications Vol. 9, No. 2, February 2014[10] C. Takano, M. Aida, M. Murata, and M. Imase, “Proposal for

Sota Hatakeyama received the B.E. degreefrom Tokyo Metropolitan University, Japan, in
2012. He is currently a graduate student toward
the M.E. degree at the Graduate School of
System Design, Tokyo Metropolitan University.
His research interests are in traffic control in
information networks.

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Hiroki Takayama received the B.S. degreefrom Tokyo Metropolitan University, Japan, in
2011. He is currently a graduate student toward
the M.E. degree at the Graduate School of the
System Design, Tokyo Metropolitan University.
His primary interests include ad hoc networks.

©2014 Engineering and Technology Publishing187

Masaki Aida received his B.S. and M.S. degrees in Theoretical Physics from St. Paul’s
University, Tokyo, Japan, in 1987 and 1989,
respectively, and received the Ph.D. in Telecommunications Engineering from the University of Tokyo, Japan, in 1999. In April 1989,
he joined NTT Laboratories. From April
2005to March 2007, he was an Associate Professor at the Faculty of System Design, Tokyo
Metropolitan University. He has been a Professor of the Graduate
School of System Design, Tokyo Metropolitan University since April
2007. He received the Best Tutorial Paper Award from the IEICE
Communications Society in 2013. His current interests include traffic
issues in computer communication networks. He is a member of theIEEE, IEICE and the Operations Research Society of Japan.