Self-organizing Wireless Sensor Networks Based on
Biological Collective Decision Making for treating Information Uncertainty
Saeko Shigaki∗ , Naomi Kuze∗ , Daichi Kominami† , Kenji Kashima‡ and Masayuki Murata∗∗ Graduate School of Information Science and Technology, Osaka University, Osaka, Japan

Email: {s-sigaki,n-kuze,murata}@ist.osaka-u.ac.jp† Graduate School of Economics, Osaka University, Osaka, Japan

Email: d-kominami@econ.osaka-u.ac.jp‡ Graduate School of Informatics, Kyoto University, Kyoto, Japan

Email: kashima@amp.i.kyoto-u.ac.jp

Abstract—Due to the rapid growth in scale and complexityof information networks, self-organizing systems have been
focused on for realizing new network control architectures thathave high scalability, adaptability, and robustness. However,in self-organizing systems, the uncertainty (incompleteness,
ambiguity, and dynamicity) of information observable forcomponents in the system can lead to the slow adaptation
to environmental changes and the lack of a global optimality,
which complicates a practical use of self-organizing systems
in industrial and business fields. In this study, we adopt the
principle of collective decision making, in which a coordinateddecision in a group is achieved through local interactions ofcomponents, in order to realize a network control mechanism
adaptable to such information uncertainty. Specifically, weapply Effective Leadership model, which is a mathematicalmodel of collective decision making, to a self-organizing control
mechanism. In Effective Leadership model, there are two
types of individuals, informed and non-informed ones, andcollective decision is achieved through local interaction of them.
Through simulation experiments, we reveal the advantages andcharacteristics of the network control mechanism based on
Effective Leadership model.
Index Terms—Collective Decision Making, Potential-based
Routing, Information Uncertainty, Bio-inspired Network Controlling

1. Introduction
Because of the rapid growth in scale and complexityof information networks, self-organizing systems have been
paid much attention for realizing new network architecturesthat have high scalability, adaptability, and robustness [1],[2]. In self-organizing systems, each component behavesautonomously only with simple rules and local information.
As a result of local interactions of components, a global
behavior or pattern emerges in a macroscopic level. By
adopting the principle of self-organization, network control
systems can be robust and adaptable to unexpected environmental changes. However, there is a significant problem for apractical use of self-organizing control systems in industrial
and business fields. In information networks, information
observable for each component is uncertain. That is, observable information is•••

dynamic according to changing condition of nodes,
incomplete because of the constraints of sensor
nodes’ capacity,
ambiguous because of estimation/communication error.

Such uncertainty of observable information can lead to theslow adaptation to environmental changes and the lack of a
global optimality of systems. To tackle this problem, we
introduce the principle of collective decision making ofswarms to network control mechanisms. This is because,in swarms of animals such as birds, fish, and insects, individuals can make a coordinated decision through local
interactions among them although the perceptive ability ofeach component is limited (incompleteness, ambiguity) andits condition, e.g., the direction of the move, is changing
dynamically (dynamicity) so that information which it has
is uncertain.
In this study, we focus on Effective Leadership model [3]as a mathematical model of the behavior of collective
decision making in swarms, and adopt it to realizing a
network control mechanism that can work under information uncertainty. In Effective Leadership model, there are
two types of individuals, informed individuals and noninformed individuals. On one hand, informed individuals
are more experienced and well-informed than the others
so that they have pertinent information, i.e., the preferred
directions to move in. They have a role as leaders of thegroup to guide the other individuals to move in their own
preferred directions. On the other hand, non-informed individuals have limited information and decide their direction
to move in following individuals surrounding themselves.
Consequently, informed individuals lead non-informed ones
to move in the preferred decisions through local interactions
among individuals, and, as a result, individuals can make acoordinated decision. In Effective Leadership model, as thegroup size (the number of individuals) becomes larger, theratio of informed individuals to the whole group needed forthe achievement of collective decision to all individuals in
the group is smaller. This indicates that Effective Leadershipmodel has high scalability to the size of groups.
We apply Effective Leadership model to self-organizing
network control mechanisms to conquer information uncertainty. We take potential-based routing with an external
controller proposed in [4] as an example of self-organizing
network control mechanisms, and propose potential-based
routing based on Effective Leadership model. In previous
work [4], the authors introduce an external controller topotential-based routing for facilitating the adaptation speed
to environmental changes. In the mechanism, an external
controller monitors the state of a network and feedbacks
control inputs to partial nodes called controlled nodes.
Through local interactions among nodes, the influence ofcontrol feedback by the external controller expands all overthe network. In other words, controlled nodes have a role
to guide the other nodes to adapt to environmental changes
quickly. In this study, we consider controlled nodes as leader
nodes which guide the other nodes, which we call follower
nodes, to make a coordinated decision. Leader and follower
nodes correspond to informed and non-informed individuals
in Effective Leadership model, respectively.
Through simulation experiments, we reveal the advantages and characteristics of the network control mechanism
based on Effective Leadership model. First, we investigate
the appropriate selection of leaders in network systems.
In Effective Leadership model, individuals move continuously and their relative positions are dynamically changing.
Therefore, we investigate how the selection of leaders affectto control efficiency when introducing to a static topology
and reveal leader selections appropriate for faster adaptation to environmental changes. Second, we evaluate theperformance of our proposed mechanism in wireless sensor
networks. We evaluate our mechanism through simulations
in environment considering constraints of wireless sensor
networks, such as communication delay, packet loss andcongestion, and prove that our mechanism has high scalability and work well under information uncertainty.
The remainder of this paper is as follows. First, in Section 2, we apply Effective Leadership model to network control mechanisms to deal with two information uncertainty
while calculating control feedback. We propose potentialbased routing based on Effective Leadership model. In Section 3, we evaluate our proposed method through simulation
experiments and discuss the results. Finally, in Section 4 we
describe conclusion and future work of this study.

2. Application of Collective Decision Making
to Potential-based Routing
In this section, we take potential-based routing with theexternal controller that is proposed in previous work [4] asan example of self-organizing network control mechanisms.
We apply Effective Leadership model [3], a mathematicalmodel of collective decision making in swarms, to themechanism.

We explain potential-based routing and its improvement,
potential-based routing with the external controller, in Section 2.1 and Effective Leadership model in Section 2.2.
Then, in Section 2.3, we propose potential-based routing
which adopt the principle of Effective Leadership model.

2.1. Potential-based Routing
Potential-based routing is a self-organizing routing
mechanism for wireless sensor networks [4]–[8]. In
potential-based routing, data packets are forwarded in accordance with a potential field, a kind of gradient fields, which
includes routing information. The potential-field emerges asa result of local interactions of nodes.
In potential-based routing, each node is assigned a scalarvalue called “potential.” In general, a potential field isconstructed so that a lower potential value is assigned toa nearer node from the sink node. Therefore, with a simple
forwarding rule, “sending a data packet to a neighboringnode with a lower potential value,” data packets can be delivered to sink nodes. Because potentials of nodes are updated
through local interactions among nodes, it is known that
potential-based routing can work with low communicationand calculation costs even in large-scale networks.
2.1.1. Potential Field Construction. Sheikhattar andKalantari [6] focused on the convergence of potential-based
routing and facilitated the potential convergence speed. They
proposed a potential calculation method based not only on
current potentials but also on last potentials to accelerate potential convergence. In this method, node i’s potential θi (t)at time t is updated by (1).
θi (t + 1) = (α + 1)θi (t) − αθi (t − 1)⎛⎞#
+ βσi ⎝
{θk (t) − θi (t)} + fi (t)⎠ ,(1)k∈Nb (i)where Nb (i) is a set of neighbors of node i. α is a parameterthat determines the weight of current and last potential
values when calculating the next potential value. Larger αmeans that the weight of the last potential value is larger andtherefore the system becomes less subject to temporal noises
and disturbance, while the convergence speed is slower. β isa parameter that determines of the weight of neighbor nodepotentials. In prior work [6], σi is defined as σ0 /|Nb (i)| (σ0is a parameter). In this study, we set σi to constant value σ(0 < σ < 1) since potentials diverge in some situations with
σi = σ0 /|Nb (i)|. fi (t) corresponds to the flow rate of node i
at time t. For sensor node i, flow rate fi (t) is a positive
value, which indicates the data generation rate of node i.
For sink node i, flow rate fi (t) is a negative value, which
implies the amount of data packets&
delivered to node i. If
flow rates of nodes are set to satisfy n∈{1,··· ,N } fn (t) = 0,then a potential field is constructed such that the actual rates
at which data packets are delivered to nodes satisfy the given
flow rates. In this study, we set flow rates of sink nodes so

Figure 2. Overview of Effective Leadership model

Figure 1. Potential-based routing with the external controllerthat the numbers of data packets delivered to each sink nodeare equal for load balancing. In details, we set flow rates ofsink nodes to the same value.
2.1.2. Potential Field Construction with the External
Controller. In previous work [4], the authors introduced an
external controller, which monitors and controls a system,
into potential-based routing proposed in prior work [6] forfacilitating the convergence speed of potentials.
The role of the external controller is to (i) collect potential information of a partial set of nodes (observable nodes),(ii) estimate potential values of the other nodes based on thepotential dynamics model, and (iii) provide control feedback
to another partial set of nodes (controlled nodes) to facilitate
potential convergence as shown in Fig. 1. In potential-based
routing with the external controller, the potential value θi′ (t)of node i at time t is updated by (2).
θi′ (t + 1) = (α + 1)θi′ (t) − αθi′ (t − 1)
⎞
⎛
#
′
′
+ βσi ⎝
{θk (t) − θi (t)} + fi (t)⎠ + ηi (t),(2)

k∈Nb (i)where ηi (t) corresponds to the control input provided tonode i by the external controller. If node i is not controlled
one, ηi (t) is 0. Control inputs are calculated based on H ∞control theory [4], [8].
In this study, we apply Effective Leadership model topotential-based routing with the external controller. We explain the details in Section 2.3.
2.1.3. Data Packet Forwarding. When a sensor node
receives a data packet, it probabilistically determines thenext hop node of it in accordance with potentials of itself
and its neighbors. Probability Pi→n (t) that node i selects
neighboring node n ∈ Nb (i) as the next hop node of thedata packet at time t is calculated by (3).

Pi→n (t)
'
=

!

0,

θi (t)−θn (t)j∈Nlow (i) {θi (t)−θj (t)},if n ∈ Nlow (i)otherwise,(3)

where Nlow (i) is a set of neighboring nodes of node i withlower potential than node i. Nodes with lower potentials arelikely to receive a larger number of data packets.

2.2. Effective Leadership Model
Effective Leadership model [3] is a mathematical modelof collective decision making in swarms. Figure 2 shows
the overview of Effective Leadership model. In Effective
Leadership model, there are two types of individuals, informed individuals and non-informed individuals. Informed
individuals have information about their preferred directions,and decide their directions according to both social interactions with their neighbors and their own preferred directions.
On the other hand, non-informed individuals decide theirdirections in accordance only with social interactions.
Individual i in a group has position vector ci (t) anddirection vector vi (t) at time t. Individuals attempts to maintain a minimum distance distα from other individuals foravoiding collisions. When the distance between individual iand j is lower than distα , individual i moves away fromindividual j . In details, desired direction di (t) of individual i
at time t is determined by (4).

di (t + ∆t) = −#j∈Nb (i,distα )cj (t) − ci (t),|cj (t) − ci (t)|(4)

where Nb (i, distα ) corresponds to a set of individuals whose
distances to individual i are lower than distα .
If there are no other individuals within range distα , anon-informed individual is attracted to individuals within
range distρ . distρ indicates the local interaction range of individuals. Non-informed individual i determines its desired
direction di by (5).

#di (t + ∆t) =j∈Nb (i,distρ )+#cj (t) − ci (t)|cj (t) − ci (t)|j∈Nb (i,distρ )vj (t).
|vj (t)|(5)

The first term of the right side of (5) corresponds tothe vector from the position of individual i to the averaged
position of neighboring individuals. This implies that noninformed individuals attempt to attract to neighboring individuals. The second term of the right side of (5) corresponds
to the averaged direction of neighboring individuals. Thisimplies that non-informed individuals attempt to align theirdirections with neighboring individuals.
On the contrary, informed individuals decide their desired directions based not only on social interactions but
also on their preferred directions. Informed individual i has
information about its preferred direction gi and decides its
desired direction d′i (t) as time t by (6).
d′i (t + ∆t) =d̂i (t + ∆t) + ω0 gi|d̂i (t + ∆t) + ω0 gi |,(6)

where d̂i (t) is the unit vector of di (t), i.e., d̂i (t + ∆t) =di (t+∆t)|di (t+∆t)| . ω0 is a parameter that determines the weight ofpreferred direction gi when the informed individual decides
its desired direction. In prior work [9], ω0 is considered as“assertiveness” of individuals. If ω0 is 0, informed individuals are not influenced by their preferred directions g. The
higher the value of ω0 is, the larger the influence amountof preferred directions g to desired directions of informed
individuals is.

2.3. Potential-based Routing Based on Collective
Decision Making
In this paper, we applied Effective Leadership model
for a process where each node in potential-based routing
coordinates to form a potential field (Table 1). In the proposed mechanism, we consider controlled nodes as leader
nodes that have a role of leaders that guide the other nodes
just like informed individuals in Effective Leadership model.
According to control feedback provided by the external
controller, leader nodes can update their potentials for fasterconvergence of potentials. On the contrary, we consider noncontrolled nodes as follower nodes that behave following
their neighboring nodes just like non-informed individual.
Through local interactions among leader/follower nodes,
leader nodes guide follower nodes, which results in faster
potential field construction. The overview of the proposedmechanism is shown in Fig. 3.
We explain the potential field construction in our proposal mechanism. On one hand, follower nodes update
their potential values in accordance with local information,i.e., potential values of itself and neighbors, using (1). On

Figure 3. Potential-based routing based on Effective Leadership model

the other hand, leader nodes update their potential values
based not only on local information but on control feedback
provided by the external controller. Potential value θi′′ (t) ofleader node i is updated using θi (t) by (7).
θi′′ (t) = (1 − ω)θi (t) + ωgi (t),(7)

gi (t) = θi (t) + ηi (t).

(8)

where gi (t) is the preferred potential value of leader node i.
ω (0 < ω < 1) is a parameter corresponding to ω0 in
Effective Leadership model, which represents the strengthof the tendency of leader individuals to lead the swarmin (6), and it is the weight for the target potential gi (t).
Preferred potential value gi (t) is calculated using control
feedback ηi (t) provided by the external controller by (8)Control feedback by the external controller is calculated
so that the potential field converges to the target potential
field Θ̄ = {θ1 , · · · , θN } in a short time.
When a sensor node receives a data packet, the node
probabilistically selects the next hop node of the data packet
by using (3).

3. Simulation Experiments
In order to find a clue of applying Effective Leadershipmodel to the network and to evaluate the performance ofthe proposed mechanism against information uncertainty, we
performed following two kinds of simulations.
First, in Section 3.1, we quantitatively evaluated therelationship between the performance, and the number andthe positions of leader nodes through numerical simulation
using MATLAB. Through these experiments, we also investigated leader selection that fascinates the potential-field
construction. In previous work [3], the authors consider individuals in a group move dynamically and fluidly changing
their position without static topology, therefore we made aninvestigation into the selection of leaders in a topology.
Second, we evaluated the performance of the proposedmechanism assuming wireless sensor networks where eachnode behaves asynchronously. We averaged the time stepsof 5 trials in the following all simulations. In Section 3.2,
we conducted simulation experiments assuming that theexternal controller can observe potential information of

TABLE 1. T HE C ORRESPONDENCE OF E FFECTIVE L EADERSHIP MODEL AND POTENTIAL - BASED ROUTING
Effective Leadership model
A group of various individuals
with different preferences and abilities
Leader individuals having
more information than the others
Non-informed individuals
Position information c of an individual andits neighboring individuals
Direction information v of an individual
and neighboring individuals
Target direction g
Accuracy of the direction to move inall nodes without any communication delay for revealing
the advantages and properties of the proposed mechanism.
Actually, in real network environment, it is difficult forthe controller to observe all information in the networkand the information include communication delays. As aresult, observable information become uncertain. We then,in Section 3.3, evaluated the performance of the proposedmechanism in the case where observable information of theexternal controller is dynamic and incomplete. Specifically,
we assumed that the external controller can observe potential
information of a partial set of nodes, observable nodes, via
sink nodes with communication delays. As a result, theproposed mechanism proved high scalability from the resultsof above 2 simulations in Section 3.1 and Section 3.2.
Third, in Section 3.3, we took errors in control caused byuncertain information into consideration through simulationthat the controller can acquire information of a bounded
scope. As a result, the proposed mechanism proved robustness against information uncertainty.

3.1. Relationship between the performance and thenumber and the positions of leader nodes
Considering to apply Effective Leadership model to
network control, the number and positions of the leader
nodes become important because in previous work [3] theauthors didn’t consider individuals have static topologies
like networks. From the viewpoint of cost, it is desirable
that the number of leader nodes with higher performancebe smaller, it is also necessary to clarify the selection of
the leader nodes for that purpose. In particular, in Effective
Leadership model [3], it is shown that as the size of theswarm (the number of individuals) is larger, appropriate
action can be taken by the group as a whole by a small
proportion of leader individuals. If a similar tendency isfound when corresponding Effective Leadership model to
network control, we can consider it to be useful for application to a large-scale network. Therefore, in this section, weinvestigated the relationship of the number and positions ofleader nodes with the control performance of the external
controller [4].
In [4], [8], using potential-based routing which is aself-organizing route control method, authors achieved toPotential-based routing
A network composed of various kinds of nodes
with different technical standards and performances
Leader nodes receiving control inputs
from the external controller
Follower nodes
Potential information θ of itself
and neighboring nodes
The changing amount of potential of a node
Target potential values g
Convergence time steps of the potential-field

improve convergence speed to the target potential by introducing an external controller that observes network information and performs control feedback. They revealed thatthe positions and the number of nodes (controlled nodes)
which receive control feedback, affects the convergence
speed (control performance of the controller). In this section,
we regarded controlled nodes as leader nodes, investigated
the relationship between the number and the positions ofleader nodes and control performance of the controller,
and obtained knowledge about the number and positionsof appropriate and efficient leader nodes. In addition, as aresult, the proposed mechanism showed high scalability thatis similar in previous work [3].
The external controller is designed based on H∞ control [10] and it is calculated using MATLAB’s dhinf lmi
function. This function gets control parameters as inputthat are set to achieve the control target according to the
network topology and the positions of the leader nodes andreturns outputs of the optimum H∞ performance γopt andthe controller transfer function G. The transfer function G
as output satisfies the following two conditions:••

The system is stable and even when instantaneous
disturbance is given to a system in equilibrium, thesystem returns to equilibrium again with the lapseof time.
The closed loop norm of the controller ||G||∞ issmaller than γopt .

Here, the closed loop norm of the controller ||G||∞ represents the maximum value of the controller gain (ratio of
output to input), the smaller the value of γopt , the smaller thegain. That is, the ability to suppress input disturbance andconverge the system to equilibrium is high. In the proposedmechanism, the input to the controller is the deviation
between the target potential and the current potential, since
the output from the controller corresponds to the deviation
between the target potential and the potential obtained asa result of the control. The smaller the value of γopt , thehigher the ability to converge the potential to the target value
and the robustness against noise and errors. In other words,
the higher the degree of contribution of the leader nodes tothe improvement of the control performance, the smaller thevalue of γopt . In this research, the value of γopt is used as anindex to measure the control performance of the controller.

250

Glid (3x3)Glid (4x4)Glid (5x5)Glid (6x6)Minimum of Gopt

200

150

100

50

0

0.2

0.4

0.6

0.8

1

Rate of leader node

Figure 4. Relationship between the ratio of leader nodes in the network
and Gopt (γopt )We investigated the relationship with the control performance of the controller by calculating the value of γopt for
all possible number and positions of leader nodes. We usedlattice-shaped topologies that the sizes were 3×3, 4×4, 5×5
and 6×6, and set the ratio of leader nodes to all nodes to
0.025, 0.05, 0.1, 0.2, 1.0. For each network size, leader node
percentage, we calculate γopt for all leader node patterns andthe network size. Figure 4 represents the minimum value ofγopt for each percentage of leader nodes.
From the result of Fig. 4, we can say that the value ofγopt decreases as the proportion of the leader node increases.
From this, it is shown that the control performance of thecontroller increases as the ratio of the leader node increases.
We can consider that the reason is that the control input
propagates faster to the entire potential field. This will beverified later.
On the other hand, compared with the decrease in γopt
when the leader node percentage changes from 0.025 to
0.2, the decrease in γopt when the leader node percentagechanges from 0.2 to 1 is fairly small. From this we cansay the less the ratio of leader nodes to all nodes the larger
the change in control performance. This tendency becomes
more prominent as the number of nodes increases (Fig.4).
Also, when the ratio of leader nodes is around 0.2, thevalue of γopt decreases as the network size increases fromthe network size of 3×3 to 5×5. This result shows the more
the number of nodes, the higher control performance canbe obtained with a smaller proportion of leader nodes. As
shown in [3], when the size (number of individuals) of thegroup becomes larger, it gets more possible for the groupas a whole to take appropriate action by a small proportionof leader individuals, and we achieved incorporating thisbenefit to network control, which leads high scalability.
Figure 5, 6 show the selection of leader nodes of the
maximum (Fig. 6) and minimum (Fig. 5) case of γopt in
the above simulation.
There are multiple position patterns of leader nodes
where γopt is the maximum or minimum, this figure represents one of them. From Fig. 5, in the case where γopt isFigure 5. Leader selection that mim- Figure 6. Leader selection that maximizes γopt
imizes γopt
Average hop length to the nearest leader node

0

3
2.5
2
1.5
1
0.5
0

0

20

40

60

80

100

120

140

160

Gopt

Figure 7. Correlation between the number of hops to the leader node andγopt

the maximum value, that is, where the control performanceis lowest, the leader node is deployed on the edge of the
network. In contrast, in the case where γopt is minimum andcontroller performs best (Fig. 6), leader nodes are deployed
to be spread uniformly throughout the network. Also, Fig. 7
shows the relationship between the number of hops to theleader node and γopt . From this, it is said that the numberof hop count to leader nodes has a great influence on γopt .
On the other hand, when setting the leader node, it
is not cost-effective to determine the selection of leader
nodes by calculate the average hop count of all nodes in the
network because as the total number of nodes increases thecomputational complexity increases greatly. In this research,in order to select leader nodes uniformly across the fieldwith light cost of calculation, we classify the network into
clusters according to the number of leader nodes using the
K-Means method and placed one leader at the center of one
cluster. This solved the trade-off between the computational
complexity and the control performance.

3.2. Properties of the proposed mechanism
In this section, based on the results in Section 3.1,
we conducted evaluation in the wireless sensor network
environment. We used square lattice-shaped topologies andFigure 9. Example of randomly-positioned topology of 64 nodes
TABLE 2. N ETWORK ENVIRONMENT IN THE SIMULATIONEXPERIMENTS

Figure 8. Relationship between the ratio of leader nodes in the networkand the convergence time of potentials

the total number of nodes was set to 64 (8×8), 144 (12×12),
256 (16×16). In this simulation, the communication rangeof the node is set to 50m, and the nodes existing within
the communication range are connected to each other. The
number of sink nodes is set to 4 for the total number ofnodes of 64, 6 for the case of 144, and 8 for the case of
256 total nodes.
In this evaluation, the leader node is set from 1% to
7% as a ratio with respect to the total number of nodes.
We determined the positions of the leader nodes as we
did in Section 3.1. The external controller [4] gives control
feedback to the leader node at 50 second intervals. For simplification, this time, we exclude error in control feedback
by assuming that the external controller can acquire thepotential information of all the nodes without delay.
In this evaluation, immediately after the start of a simulation, potential information of all nodes begins to be updated
and the external controller begins to control. Then, at 20
time steps after the start of the simulation, each sensor node
starts transmitting data packets. The generation rate of thedata packet in the sensor node located in the upper half of thegrid network is 0.75 packet/time step, and the generation rateof data packets in the sensor node located in the lower half
is 0.25 packet/time step. At this time, the external controller
performs control so that the number of data packets received
by each sink node becomes equal. After 200 time steps fromthe start of the simulation, the generation rate of data packets
in each sensor node is changed upside down in the network.
In this evaluation, after the data packet generation ratechanges, the time until the potential field converges again
is evaluated so that the number of data packets received
by each sink node becomes equal. After changing the data
packet generation rate, the generation rate of the data packet
in the sensor node located in the upper half of the lattice
network is 0.25 packet/time step, the generation rate of thedata packet in the sensor node located in the lower half is
0.75 packet/time step. The simulation settings are shown in
Table 2.
Figure 8 shows a graph plotting the relationship be-

Data
The size of a data packet
The size of a control packet
The size of a ID packet
The size of an Ack packet
The number of buffer of a sensor node
The transmission interval of ID packets
The potential update interval (1 step)
The control interval of the external controller

Value
128 bytes
30 bytes
28 bytes
22 bytes
1
1 sec
50 sec50 sectween network size, leader node ratio, and improvement of
convergence speed of potential field. From Fig. 8, it was
shown that as the ratio of the leader node is larger thetime to converge becomes shorter, that is, the convergence
performance improves as a whole. When the number ofnodes is 64 and the ratio of the leader node is 0.047, andwhen the ratio of the leader node is 256 and the ratio of
the leader node is 0.063, this trend is not applicable andthe convergence time is prolonged, but this is caused by
the occurrence of the oscillation due to the delay occurring
when the control feedback propagates. However, although
the convergence time itself is long, the generated oscillation
is extremely fine without affecting the path control. From
the above, there is a trade-off relationship between theratio of the leader node and the convergence speed, and
it is necessary to properly set the ratio of the leader nodeaccording to the control request.

3.3. Performance evaluation in the case with dynamic and incomplete information
In this section, we examined the proposed mechanism
from a viewpoint of robustness against uncertain information when control. In Section 3.2, the external controller
calculated feedback input using global information without
communication delay to exclude the error from the simulation. On the other hand, considering the real network
environment, it might be better the external controller cancalculate feedback by only uncertain information. In thisevaluation, we investigated the potential convergence timeafter traffic changes in three cases: (i) the external controller
can observe information of a partial set of nodes (nodes
within one hop from sink nodes) with communication delay(observable information is dynamic and incomplete), (ii) theexperiments, we investigated the relationship among the
network size (the number of nodes), the ratio of leader
nodes, and the acceleration of the adaptation speed to environmental changes. Simulation results showed that as thenumber of nodes increases, a lower ratio of leader nodes
is needed to facilitate the adaptation speed. Moreover, we
showed that the acceleration amount of the adaptation speed
deeply depends on the average distance between nodes andtheir own nearest leader nodes.
As future work, we will consider another information
uncertainty that information become ambiguous because oferror or noise in communication. we will investigate the relationship among the network size (the number of nodes), theratio of leader nodes, and the acceleration of the adaptation
speed in cases with more complicated network topologies
such as random networks and small world networks. Then,
we develop a network control mechanism conquering information uncertainty and prove its advantages.
Figure 10. Comparison of convergence time steps

external controller can observe potential information of allnodes with communication delay (observable information
is dynamic), and (iii) the external controller can observe
potential information of all nodes without any communication delay (observable information is certain). Here, weused the network topology shown in Fig. 9, and leader nodes
were selected using the same scheme as the evaluation in
Section 3.2. The other settings, including the generation rateof data packets at sensor nodes and their changes, were
following the evaluation in Section 3.2.
Figure 10 shows the potential convergence time incases (i)–(iii). In Fig. 10, the vertical axis is time steps
spent to potential convergence after traffic changes andhorizontal axis is number of leader nodes in a 64-nodes
randomly-positioned topology like of Fig. 9. The red solidline indicates the potential convergence time in case (i), thegreen dotted line indicates the result of using global information (2-hops range from sink nodes) with communication
delay, and the blue dashed line indicates that in case (ii).
Compared to case (iii), the convergence time in case (ii)is almost same or even less compared to the red solid line
when the number of leader nodes are small (1–4) even with
communication delays. Although the controller of case (i)can obtain information of only about 73% nodes in addition
to the communication delay, the time steps are almost sameor even less when the number of leader nodes is 1–4, orabout 1–6% of all nodes. In the blue dashed line, the time
steps spent to convergence increase when the number ofleader nodes is 10, and the reason why this outlier occurs
is currently under investigation.

5. Acknowledgment
This work was supported by JSPS Grant-in-Aid forYoung Scientists (Start-up) 16H06915.

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