On the optimal size of distributed cooperative systems

O THE OPTIMAL SIZE OF
DISTRIBUTED COOPERATIVE SYSTEMS





KAM MU LOOG






ATIOAL UIVERSITY OF SIGAPORE
2009
O THE OPTIMAL SIZE OF
DISTRIBUTED COOPERATIVE SYSTEMS





KAM MU LOOG





A THESIS SUBMITTED FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY OF EGIEERIG

DEPARTMET OF MECHAICAL EGIEERIG
ATIOAL UIVERSITY OF SIGAPORE
2009
Acknowledgements
I am very thankful and deeply indebted to my supervisor, A.P. Gerard Leng for all the
hints, suggestions and critics he has been providing throughout this research. Every time I
felt like I was lost and not sure how to proceed, a short discussion with Gerard would
bring my focus back.

I am also grateful to have got to know a bunch of friends in the CosyLab: Tze Liang, Wei
Lit, Han Yong and Azfar. The fun we had together made up for some of the frustrating
moments.

Last but no least, I would like to thank my family and Shiau Chien for keeping faith in
me. The moral support and love you have been showering me with is indeed more than
what words can describe.

Summary
The thesis aims to answer the question “What is the optimal size of a distributed
system?”. A distributed system is broadly defined as a group of identical units executing
a common task in a given operating area. Examples include swarm systems, multi-robot
systems and multi agent systems. While the optimal system size is dependent on the task,
operating area and chosen performance metrics, there are fundamental properties which
are important for distributed systems. In the thesis, the optimal system size is studied
from the perspective of four properties i.e.1) characteristics of the operating area, 2)
connectivity of the units, 3) mutual interference effects between the units and 4)
robustness of the system against noisy information.
The first contribution of this research is the study on the effect of a concave
operating environment on the connectivity of a distributed system. A novel concavity
measure, termed as the blockage, is proposed and used to quantify the relative complexity

of 2D concave shapes. A computer algorithm has been developed to evaluate this
measure for complicated shapes. The deficiencies of a few other existing concavity
measures will be contrasted and compared with the blockage measure. A general
relationship relating the number of units required to ensure a certain connectivity
probability will be given.
The second major study proposes the existence of a power-law relationship
between the communication range and the number of units from the connectivity
perspective. Based on extensive simulation results, this power-law is shown to hold for
distributed systems under a wide variety of mobility models and, most importantly, for
small system size.
In the third part of this research, the optimal swarm size is studied under the
interference effect and robustness requirement. The discussion will start with the proposal
of the use of the failure probability to measure the effect of hazards on a swarm. Then,
another measure for the degradation in the performance as a function of swarm size is
proposed. A team of imperfect swarming robots carrying out surveillance task was
simulated and used to demonstrate the key ideas of this study.
The fourth study aims at studying the advantage of larger system size in reducing
the noise effect on the information being sensed. A stochastic model for a swarm using
Master Equation is derived. With the key results of this study, the system size required to
achieve a desired level of noise suppression can be predicted.
Lastly, a short case study will be shown to demonstrate the applications of the
results from the four major studies in this research. Insight to potential new research
directions will be drawn as well.

Table of Contents
1. Introduction
1.1 Overview 1
1.2 The Research 4
1.3 General Claims 4
1.4 Hypotheses 8

1.5 Structure of the Thesis 9
1.6 Contributions 11

2. Research Review
2.1 Artificial Intelligence 14
2.2 Distributed Intelligence 15
2.3 Swarm Intelligence and Swarm Robotics 16
2.4 Connectivity 18
2.5 Swarm Robustness 21
2.6 Stochastic Modelling 23

3. Connectivity of Distributed Systems in Concave Operating
Environment
3.1 Introduction 24
3.2 Blockage - a New Measure of Concavity 29
3.3 Analytical Evaluation of Blockage 31
3.4 Computer Algorithm to Compute Blockage 34
3.5 Blockage and Connectivity Probability 40

4. On the Power Law Relationship of the Critical Transmitting Range and
the Number of Nodes of Ad Hoc Networks
4.1 Introduction 51
4.2 Related Works 53
4.3 Rationale behind the Power Law 55
4.4 Empirical Evidence of the Power Law 59
4.5 Conclusion 73

5. On the Optimal Size of a Robust Swarm
5.1 Introduction 75
5.2 The Contradicting Effects of Large Swarm Size 77

5.3 Robust Swarm Size as an Optimization Problem 80
5.4 Theoretical vs Simulation Results 91
5.5 Conclusion 99

6. Stochastic Swarm Modelling
6.1 Experiment, Simulation and Mathematical Modelling 101
6.2 Swarm Modelling with Stochastic Master Equation 102
6.3 The Advantage of a Swarm in a System with Noisy Global Information 104
6.4 Relationship between System Size and Noise Magnitude 115
6.5 Simulation Results 126

7. Conclusion – General Framework for the Optimal Size
139


References
147

Appendix
154

1
CHAPTER 1
Introduction


1.1 Overview
Robotics research has gone through several phases since the first industrial robot
being introduced by Unimate in the early 1960s. The study of the kinematics, dynamics
and control of a robotic arm was hotly pursued at that time due to its industrial

applications. The main focus was being placed at designing algorithms to achieve precise
control such that tasks could be completed in a structured environment and under the
supervision of an operator. However, it was soon realized that the ability to carry out a
task in an unsupervised manner was the real potential of a robotic system.
Autonomy was the answer to the need of unsupervised operation of robots. In
order to make robots more versatile and adaptive to complicated tasks, the Artificial
Intelligence and Robotics research fields have worked closely in search of true autonomy.
While Robotics research focuses on the relationship between sensory inputs and actions,
Artificial Intelligence research aims at understanding the ways to make computers exhibit
intelligence [Ota, 2006]. Humanoid robots, unmanned ground/air vehicles (UAV, UGV)
are some of the most exciting research fields which could potentially change the way
humans live and work in the near future.
There is a paradigm shift in the field of robotics. Instead of focusing on the
development of the intelligence of a single robot, the pros and cons of multi-agent robot
2
systems are being explored. The basic characteristics of a multi-agent system or
equivalently, distributed autonomous system, are: 1) inability to solve the problem by
each agent alone, 2) fully decentralized control, 3) decentralized data and 4)
asynchronous computation. A multi-agent robot system has been studied for a wide range
of tasks. These tasks can be classified into several categories according to the objectives
and nature of the tasks, see [Parker, 2008] for a review of the common classifications of
the different mission types. Searching for an object, covering a given area, transporting
an object cooperatively from one location to another are some of the more common tasks
being studied. In their research work of Simultaneous Localization and Mapping (SLAM)
[Durrant-Whyte and Bailey, 2006] using multi-robot system, Durrant-Whyte has
demonstrated that it is possible to explore an unknown environment and build a map at
the same time by using an EKF (Extended Kalman Filter).
The traditional multi-robot system research focuses on the motion-planning
algorithms and exploration algorithms. These studies tend to increase the complexity of
each robot in order to acquire more accurate sensory inputs from the environment for the

more sophisticated algorithms. On the other hand, swarm intelligence, a relatively new
research direction, is getting more attention of late. Inspired by the biological swarms
found in the nature, e.g. bees and ants, swarm intelligence aims to reproduce the
collective intelligence exhibited by the biological swarms. A robotic swarm system is
similar to the multi-agent robot system, with the key difference being the emphasis on the
system size and unit simplicity. Some researchers treat swarm size (number of units) as
the distinguishing factor between multi-agent system (small scale) and a swarm system
(large scale). The other key defining characteristic of a swarm is the simplicity of each
3
robotic unit. “Ants aren’t smart. Ant colonies are.” This simple quote from Deborah M.
Gordon, a biologist at Stanford University, highlights the interesting property that the
swarm intelligence is an emergent characteristic, resulting from the collective actions of a
group of simple agents.
Although the multi-agent system is promising, there are significant challenges to
be overcome. The major challenges of multi-agent systems are design related. How does
one decide on the number of agents/units to be deployed for a particular task? How do the
different sizes of a multi-agent system affect the performance of the team as a whole?
Should a multi-agent system be homogenous or heterogenous? Some of these questions
have been answered in parts. In [Mei et al, 2004], Mei et al looked at this problem from
the perspectives of energy constraints, i.e. they examined the relationships between the
optimal number of nodes required to serve random requests and the given energy
constraints. Hayes [Hayes 2002] defined a cost function which relates the number of
robots (sensors), time taken to complete the search task and the moving speed of each
robot. After that, he optimized the number of robots required for a search task by
determining the minimum point of the cost function. The interference effect among the
agents has also been studied using foraging, searching mission setup [Lerman and
Galystyan, 2002]. As expected, agents will compete for space within a confined region,
which increases the time spent by the agents in avoiding each other, rather than doing
useful work.
As emphasized above, a multi-agent system acquires its “intelligence” from the

collective actions of all the agents. This emergent characteristic is elegant, but from the
design perspective, it is hard to “reverse-engineer” a particular desired behaviour of the
4
multi-agent system by tweaking the behaviours of individual agent. While the link
between the global and local behaviours is apparent and strong, it remains an open
research problem which, in the opinion of the author, holds the key for multi-agent or
distributed system to be utilized in more diverse applications.


1.2 The Research
This work aims at answering the question of the optimal swarm size. A swarm
system which is discussed in this research work is broadly defined as a multi-robot
system consists of multiple identical mobile platforms. Swarm system, multi-robot
system and distributed system are used interchangeably throughout this research. The
optimal swarm size is a tricky term, since the optimality itself is a very subjective
concept. A swarm system performing under one particular set of task constraints could be
deemed as both good and bad, depending on the performance metrics so chosen.
However, it is the author’s belief that there are a few fundamental properties which are
inherent in a swarm system that affect its performance. Specifically, the swarm size has
been optimized from different viewpoints, e.g. connectivity, concavity of operating
environment and robustness. The robustness of a multi-robot system against noisy
information based on probabilistic mathematical model has also been studied.

1.3 General Claims
This research focuses on different ways to evaluate:
a. Swarm performance: As mentioned, there is, as yet, no universally agreed upon
consensus in how to evaluate the performance of a swarm. However, focusing on
5
searching task and a few types of basic swarms, the performance metrics can be
defined in a relatively straight forward way.

b. Optimal swarm size: Having defined the swarm performance metrics, optimal
swarm sizes can be found through analytical or numerical methods.

1.3.1 Connectivity
A mobile and information sharing multi-agent robotic system is equivalent to a
Distributed Network. A Distributed Network is concerned with deploying multiple
sensors to operate and gather information in an unknown, cluttered and possibly
hazardous environment to achieve a common goal. Information is shared among the
agents and it maximizes the utilization of the information gathered by each agent. Thus, it
can be argued that a multi-agent robotic system needs to be optimized such that the
connectivity of agents is maintained at high probability. This optimization problem is
central to the ad-hoc networks [Royer and Toh, 1999] research community. In the ad-hoc
network connectivity problem domain, the interest is in finding out how easily
connections can be established among the individual nodes of an ad-hoc network. On the
other hand, for multi-agent robotic system problem, the mobile robots with Random
Direction model mobility [Madsen et al, 2005], can be treated as an ad-hoc network
operating within a given operating environment and the interest is in the optimal number
of nodes required to maintain high connectivity probability.

6
1.3.2 Concave Environment
Most of the ad-hoc or sensor network research has been focusing on nodes or
robots being scattered around in a 2D convex environment. A 2D shape is convex if a line
connecting any two points on the boundary of the shape does not cut the boundary at a
third point. On the other hand, a 2D shape is concave if there are certain points on the
boundary when connected with a straight line, will cut the boundary at least one more
time. The rationale of studying a concave environment is that any realistic operating
environment (e.g. a housing unit or a street with buildings around) for a multi-agent
robotic system is likely to contain features which block the field of view of the sensors or
disrupt the line-of-sight based communication devices of the robots.

To understand the extra challenge that a concave environment poses to the multi-
agent systems, a concavity measure is needed. Currently, there are many different shape
measures studied in the literature [Costa and Cesar 2000]. However, for the connectivity
problem inside a concave region, it is argued that there is a more natural measure of
concavity which has an intuitively physical meaning (see Chapter 3).

1.3.3 Robustness
In the multi-robot research community, it is frequently claimed that its
characteristic of having more than one identical platform carrying out the same task is
one of the major benefits. This redundancy in number distinguishes a multi-robot system
(distributed system) to a single platform system (centralized system) by preventing single
point of failure. The deeper question that arises from this argument is: how much
redundancy should be in place for a swarm system to be sufficiently robust. In other
7
words, what is the number of platforms that need to be deployed for a swarm to exhibit
enough robustness? How does one measure this robustness?
In the second part of this research, a new perspective in looking at the optimal
swarm size is discussed. In this study, imperfect swarm units (i.e. each unit has a non-
zero probability to fail) and hazardous missions are considered. The major difference
between these swarms and those that have been studied [Mei et al 2004, Hayes 2002] is
that each unit in these imperfect swarms has a chance to fail (become inactive), reducing
the number of active units. Secondly, it is argued that the interference among swarm units
needs to be taken into account when the robustness of a swarm is considered. Inspired
mainly by the work of Winfield and Nembrini [Winfield and Nembrini, 2006] which
discussed the relationship between robot failures and swarm size, it is the aim of this
work to extend their model and show that a robust swarm should also weigh in the
deterioration in performance due to the interference among swarm units (see Chatper 5).


1.3.4 Probabilistic Modelling

A robotic swarm is usually an ensemble of many identical robots. Each robot acts
on its own (fully decentralized system), and some of the simpler swarm unit control
algorithms usually include uncertainty or randomness in the decision making process. For
example, a simple obstacle avoidance algorithm could be a rule which makes a robot turn
for a random angle, then move forward. Some may argue that a swarm unit can also be
constructed with fully deterministic control logic, such that the randomness as seen in the
previous case would not be present. Although this is true, the randomness and uncertainty
has manifested itself in many other forms. Sensor and actuator error, communication
8
error and random noise/interference from the environment are some of the more pertinent
source of randomness.
In view of the inherent randomness present in a swarm system, it can be said that
swarm system is stochastic in nature. In other words, given the same initial conditions
(starting locations of each robot, identical operating environment etc), a swarm system
may not end up at an identical final state. Such is the complexity of a swarm system that
any model which tries to describe the actions/states of all the swarm units over time will
become intractable and unmanageable! However, one is not always interested in the
individual behaviour of each robot at a particular time. It is the expected behaviour of a
swarm system which is of interests. Instead of answering the question of “where is robot
XYZ at time t”, a more relevant and interesting question to be answered would be “on the
average, how many robots from the swarm is doing useful work at steady state”. A
probabilistic model seeks to provide a satisfactory answer to the latter question.

1.4 Hypothesis
The hypotheses of this work are:
Major Hypothesis:
For a simple swarm working in a confined area, it is possible to quantify the
performance of the swarm with respect to different constraints and determine the
optimal swarm sizes accordingly. These constraints include concavity of
operating environment (Chapter 3), connectivity of swarm under different

mobility models (Chapter 4) and robustness of the swarm (Chapter 5).

9
Minor Hypothesis:
A probabilistic model based on the Master Equation can be used to study the
relationship between the system size and the reduction of noise effect. (Chapter
6).

1.5 Structure of the Thesis
There are a total of 7 chapters in this thesis, as outlined below:

Chapter 2 Research Review

This chapter presents an overview of a few key related research areas. Fundamental
concepts and issues will be raised and discussed.

Chapter 3 Connectivity of Multi Agent Systems in Concave Operating
Environment
This chapter discusses the effect of a concave operating environment on the connectivity
of a swarm or multi-agent system. A novel concavity measure, termed as the blockage, is
proposed and used to quantify the relative complexity of 2D concave shapes. A computer
algorithm has been developed to ease evaluation of this measure for complicated shapes.
The deficiencies of a few other existing concavity measures will be contrasted and
compared with the blockage measure. A general relationship relating the number of
swarm units required to ensure a certain connectivity probability will be given.


10
Chapter 4 Power Law Relationship of the Critical Transmitting Range and
the Number of Nodes

This chapter argues the existence of a possible power-law relationship between the
communication range and the number of swarm units from the connectivity perspective.
This power-law is conjectured to hold for swarms under a wide variety of mobility
models and, most importantly, for small swarm size.

Chapter 5 Optimal Size of a Robust Swarm
This chapter looks at the optimal swarm size from both the interference effect and
robustness requirement.

Chapter 6 Noise Robustness Study with Stochastic Swarm Modelling
This chapter aims at studying the advantage of larger system size in reducing the noise
effect on the information being sensed. A stochastic model for a swarm using Master
Equation is derived.

Chapter 7 Conclusion
The last chapter concludes the findings from this research and draws insight to potential
new research directions.


11
1.6 Contributions
1.6.1 Connectivity Property
i. It has been found that a simple polynomial relationship exists between the swarm
size and the complexity of a concave area, measured by the blockage. (Chapter
3.5)
ii. A computer algorithm has been developed to facilitate the calculation of the
blockage measure which is proposed to quantify the complexity of a concave 2D
shape. (Chapter 3.4)
iii. A power law relationship of the form
r 

β
α
−
=
is hypothesized and supported
with extensive simulation results to hold for the communication range (r) and the
number of nodes (), with
α
and
β
being functions of the connectivity
probability of an ad hoc network. (Chapter 4.3)
iv. The power law relationship has been shown through extensive simulation results
to describe various different distributed networks better than the currently
accepted log-law relationship. (Chapter 4.4)
v. The parameters
α
and
β
of the power law
r 
β
α
−
=
can be used to quantify and
study the effects of the different settings of a distributed network, e.g. pause time.
(Chapter 4.5)
vi. The size of a distributed network can be optimized with respect to the
connectivity property, having known the empirical relationship as found from

Contributions 1 and 3.

1.6.2 Robustness Property

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i. A novel measure has been proposed to quantify the concept of “robustness” for a
swarm or distributed system. (Chapter 5.3)
ii. A systematic way to arrive at the optimal swarm size under the constraints of
robustness and interference effects has been studied and supported by extensive
simulations. (Chapter 5.4)

1.6.3 Noise Robustness Study with Stochastic Swarm Modelling
i. The stochastic Master Equation has been applied to the swarm systems in a noisy
environment. More specifically, a noisy environment is one which the sensed
information of each robot is contaminated by the presence of white noise.
(Chapter 6)
ii. In the presence of noise (which is modelled as a random signal of zero mean and
finite variance) , a group of robots which pass information around to nearby units
(local information) can improve the overall information sensed and will be able to
move towards a target location (and clutter) with high probability. (Chapter 6.3)
iii. Quantitative results which relate the system size and the steady state dispersion of
the swarm system, measured by the standard deviation of the steady state
distribution, are derived and shown to hold for both 1D and 2D search missions.
(Chapter 6.4-6.5)

1.6.4 A General Framework to Study Optimal System Size
i. A general fitness function which takes into account all the different aspects of a
multi-robot system studied in Chapter 3-6 is defined.

13

ii. The optimal size of a distributed system can be determined by optimizing the
fitness function.

14
Chapter 2
Research Review


2.1 Artificial Intelligence
There is no universally accepted definition for intelligence. It is as much an
abstract concept as it is a subjective one. The ability to read, learn, infer, listen, react to
stimuli are just some of the common traits intelligence is associated with. Artificial
intelligence (AI) researchers aim at studying intelligence and applying it to improve the
way humans live. Following George F. Luger, AI may be defined as a branch of
computer science that is concerned with the automation of intelligence behaviour
[George F. Luger, 2008].
The research of AI has significant impact to the robotics field. Autonomous
robots can only be achieved when there is a certain level of “intelligence” being built-in.
Traditional symbolic AI focuses on the knowledge representation, search methods and
machine learning. All these sub-fields make autonomous robot realizable through
providing the means for solving robot motion planning, control, learning and other
pertinent problems.
Although there have been many successful applications of the traditional AI in
various areas, there are also many criticisms being levelled at the field. The symbol
grounding problem has been plaguing the AI and cognitive science researchers. The core
of the grounding issue is about how symbols get their meanings. Facing with these

15
controversies, the AI researchers have explored other ways of representing intelligence
such as neural computing and other biologically inspired computing models. These

“new” AI methods [Munakata 2008] usually solve a problem by the collective actions of
many individual units (e.g. neurons of neural network, individuals of genetic algorithm).
These works further spur the interests in studying the emergence of intelligence from a
group of simple individual units, which is the main attractiveness of using a distributed
system.

2.2 Distributed Intelligence
Distributed Artificial Intelligence (DAI) is concerned with formulating ways to
model action and knowledge in collaborative enterprises [Gasser, 1991]. According
Gasser and Bond, the two main areas of DAI research are distributed problem solving
and multi-agent system [Bond and Gasser, 1988]. The main advocates of DAI usually
cite that a multi-agent system outperforms a centralized problem solver by having
multiple advantages: solving problems faster through leveraging on parallelism; local
communication to neighbouring units saves transmitting cost compared to global
communication to a central unit; increased flexibility and reliability by having multiple
homogenous or heterogeneous units to take over the tasks of failed agents [Durfee et al,
1989]. The main problems of designing and implementing a distributed system have also
been mentioned in [Gasser, 1991]. O’Hare and Jennings provided an excellent
introduction to the field of DAI in [O’Hare and Jennings, 1996].


16
2.3 Swarm Intelligence and Swarm Robotics
The complex behaviours exhibited by the natural swarms have been attracting the
interests of researchers from different areas. A colony of ants organizes themselves into
different roles such as workers, soldiers to keep the whole swarm function effectively.
Fishes find out that schooling helps them to spot predator and increases their chance of
survival. These are just some of the fascinating behaviours demonstrated by the natural
swarms which often emerge without a leader (central controller) giving out instructions to
everyone.

Inspired by the natural counterpart, research interest in robotic swarms has been
growing. A robotic swarm is basically made up of many simple, identical units. Each unit
is usually inefficient and/or insufficient to solve a particular problem by itself. A swarm
unit works by sharing information locally and reacting to local environmental stimuli.
Through simple interacting rules, these fully decentralized units work around the lack of
global information and exhibit collective intelligence in solving a particular problem.
Being a multi-agent system, a swarm is associated with the advantages of
flexibility, robustness and scalability [Sahin, 2004]. These advantages are directly related
to the size of a swarm system, and can be said to be common features of multi-agent or
distributed system. Beni further clarifies the difference between swarm robotics and
distributed robotics [Beni, 2005] regarding the size of the system:
the use of labels such as “swarm robotics” or “collective robotics/
distributed robotics” should not be in principle a function of the number
of units used in the system. The principles underlying the multi-robot
system coordination are the essential factor.


17
Following Beni’s basic idea about a swarm, the terms “swarm” and “multi-agent
system” and “distributed system” will be used interchangeably throughout this thesis
unless stated otherwise. In addition, the simulations and experiments being carried out to
verify the main results are mainly for swarm size which is smaller than fifty.
Although swarm systems hold much promise, there are considerable challenges to
be overcome in this research area. The effects of communication, homogeneity and
system size are just some of the more important issues in determining the effectiveness of
a swarm system. Besides, due to the advances in MEMS and nanotechnologies, the
research in swarm system is predicted to focus more on swarms consisted of micro-robots
and nano-robots [Trianni 2008]. This miniaturization poses constraints on the
complexities and capabilities of each swarm unit.
In this research work, the primary problem being tackled is the optimal size of a

swarm system. Optimization is the process of finding the “right values” given
performance requirements and/or constraints. Thus, studying the optimal swarm size
would also help providing insights to the ways of quantify and characterize the
performance of a swarm system. In this research, the most fundamental properties of a
swarm system have been chosen as the key characteristics which the swarm size should
be optimized against. These properties are the connectivity of swarm units and the
robustness of the swarm system. Although the results being presented here do not answer
all the six questions listed by Gasser [Gasser 1991], a systematic way of arriving at the
optimal size would ease the design and implementation of swarm or distributed system
significantly.


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2.4 Connectivity
A multi-agent system or distributed network is said to be connected if there exists
at least one communication path between all the units or nodes. The connectivity of a
distributed network is measured by the connectivity probability. This is defined as a
measure of the total time that a network remains connected. In other words, connectivity
probability indicates the likelihood that a network will be in a fully connected status at
any randomly chosen point in time [Madsen et al 2005].
Why is connectivity important to a distributed system? As noted earlier, the
strength of a distributed system lies in the collective actions of all the individual units. To
enhance the performance of the team as a whole, information flow among the units is
vital. In order to facilitate the exchange of information, the distributed system needs to be
in connected status most of the time, i.e. high connectivity probability is desired.
There are a lot of factors affecting the connectivity property of a network. The
number of nodes, the communication range and the mobility are all important factors
which ultimately determine the connectivity property of a network. Xue and Kumar
worked out that the number of neighbours required for a wireless network to be
asymptotically connected (as number of nodes tend to infinity) was of the order of log n,

where n is the number of nodes [Xue & Kumar, 2004]. In another series of work, Santi
studied the relationship between the critical transmitting range (communication range)
and the connectivity property of a wireless network (both sparse and dense) [Santi 2005a]
[Santi and Blough 2003]. Santi found out that when the transmitting range is greater than
a particular value (depending on the network settings), the network will be connected
with high probability. The effect of different node mobility models is another aspect