POLITECHNICA UNIVERSITY OF BUCHAREST FACULTY OF AUTOMATIC CONTROL AND COMPUTERS COMPUTER SCIENCE DEPARTMENT MASTER THESIS The emergence of a leader… [604480]

POLITECHNICA UNIVERSITY OF BUCHAREST
FACULTY OF AUTOMATIC CONTROL AND COMPUTERS
COMPUTER SCIENCE DEPARTMENT

MASTER THESIS

The emergence of a leader in a group

Thesis supervisor :
S.l. dr. ing. Andrei -Horia Mogos

Alexandru -Catalin Ciobanu

BUCHAREST
June 2017

Table of Contents
1. Introduction ………………………….. ………………………….. ………………………….. ………………………….. ……………….. 1
1.1 Problem statement ………………………….. ………………………….. ………………………….. ………………………….. …. 2
1.2 Working methodology ………………………….. ………………………….. ………………………….. ……………………….. 2
2. State of the art ………………………….. ………………………….. ………………………….. ………………………….. ……………. 2
2.1 Swarm formation ………………………….. ………………………….. ………………………….. ………………………….. …… 2
2.2 Swarm intelligence ………………………….. ………………………….. ………………………….. ………………………….. … 5
3. Simple leader types ………………………….. ………………………….. ………………………….. ………………………….. …….. 7
3.1 Collision avoidance leader ………………………….. ………………………….. ………………………….. ………………….. 7
3.1.1 Description ………………………….. ………………………….. ………………………….. ………………………….. …….. 7
3.1.2 Obstacles ………………………….. ………………………….. ………………………….. ………………………….. ……….. 8
3.1.3 Experimental results ………………………….. ………………………….. ………………………….. …………………….. 9
3.2 Motivation leader ………………………….. ………………………….. ………………………….. ………………………….. … 10
3.2.1 Description ………………………….. ………………………….. ………………………….. ………………………….. …… 10
3.2.2 Experimental results ………………………….. ………………………….. ………………………….. …………………… 11
3.3 Independent leader ………………………….. ………………………….. ………………………….. ………………………….. . 15
3.3.1 Description ………………………….. ………………………….. ………………………….. ………………………….. …… 15
3.3.2 Experimental results ………………………….. ………………………….. ………………………….. …………………… 16
3.4 Mixed leader ………………………….. ………………………….. ………………………….. ………………………….. ………. 20
3.4.1 Description ………………………….. ………………………….. ………………………….. ………………………….. …… 20
3.4.2 Experimental results ………………………….. ………………………….. ………………………….. …………………… 21
4. Shape forming leader ………………………….. ………………………….. ………………………….. ………………………….. … 26
4.1 Subleaders ………………………….. ………………………….. ………………………….. ………………………….. ………….. 26
4.2 Algorithms, methods, approach ………………………….. ………………………….. ………………………….. …………. 27
4.3. Experimental results ………………………….. ………………………….. ………………………….. ………………………… 30
5. Conclusions ………………………….. ………………………….. ………………………….. ………………………….. ……………… 32
Appendix A ………………………….. ………………………….. ………………………….. ………………………….. …………………. 34
bibliography ………………………….. ………………………….. ………………………….. ………………………….. ………………… 36

This thesis presents different types of swarm leaders that aid a swarm of individual agents
in forming and maintain a stable structure, in which all members behave approximatel y in the
same way. A leader is a particular agent that has, in addition to the simple agent, new abilities
that allow him to guide the other agents into a particular purpose. A special kind of leader is the
shape forming leader that organizes its followers into forming a predefined shape. The other
types of leaders are the motivation leader, whose abilities increase by the number of followers,
the independent leader whose abilities do not depend on the number of followers and the
collision avoidance leader that helps simple agents avoid obstacles.
1. Introduction
A swarm is a large number of simple agents interacting locally among themselves, and their
environment, with no external control resulting in an interesting behavior. The collective behavior of
thes e individuals is referred to as Swarm Intelligence (SI). These agents may form decentralized, self –
organized, natural or artificial. The agents are based on natural examples of swarms systems such as
ant colonies, animal herds, bird flocks, bacteria, and f ish schools. Each agent follows the same set of
rules, but the local collective behavior, random to a certain degree, leads to the emergence of an
“intelligent” behavior, unknown to the individual agents. The interactions between agents can be
direct or in direct.
A direct contact can be visual or audio, used for example in the waggle dance of the honey
bees. Indirect interaction happens when one agent modifies the environment and the other agents
act based on the new environment. An example of indirect int eractions are pheromone trails of ants
that they leave behind on their path when searching for new food. Research in Swarms and in SI
started in the late 1980s and their applications are in conventional optimization problems,
communications, dynamic contro l, prediction systems, moving objects tracking, heating systems
planning. SI can be applied to a variety of fields such as fundamental research, engineering,
industries, and social sciences.
A subdomain of swarm intelligence is that of swarm formation. Age nts following simple rules
try to organize themselves to form a uniform structure of agents that respect the same rules and
behave almost the same way. Such formations can encounter different obstacles during the
organizational process and must be able to avoid them while maintaining the structure of the entire
group. The first algorithms to simulate this behavior contained only three rules applied to each
individual. These rules include collision avoidance between agents close to each other, maintaining a
certain direction, dictated by the evolving structure and keeping a velocity relative to that of the
closest neighbors. Attaining a certain success, this model encouraged others to suggest
improvements and different behavioral traits for the individuals. O ne such trait is the introduction of
fear, where agents would not collide with one another because of the consequences of such an
action. All the same olfaction was introduced to transit emotion between the members of the group.
Having all these traits ind ividuals could easily form a swarm and maintain it for a period of time.
But, these traits, in certain situations are not sufficient for the successful organization of the
individual agents. For example when avoiding an obstacle the swarm may split into s everal sub –
swarms, each with their different velocity and direction. To reunite them, new behavioral
improvements must be introduced in order to successfully reorganize the swarm into a single stable
structure. For that purpose the idea of a leader in a sw arm was suggested. A leader should be an
agent that has a different set of rules that dictate its behavior and has the ability to stimulate its
neighbors to follow him. This individual is also independent of external influences but must have a
trait that m akes him attractive to the other individuals. Such a leader, or different sub -leaders, could
modify the behavior of a sub -swarm and aid smaller groups into reorganizing into a single stable
structure containing all individuals.

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1.1 Problem statement
An in teresting addition to the model is that of a leader. A leader is an agent that influences the
behavior of its neighbors such that new organizational behaviors emerge. The main difficulty is that
of finding traits for an ind ividual such that he may become a leader and the other simple agents will
adapt to follow his orders while also respecting the pre -established set of rules. Such traits may
include avoiding collision with obstacles, changes in speed or changes in direction. At the same time,
traits must b e found such that in a sim ulation involving multiple leaders, each leader can maintain
his followers for as long as possible while also adopt ing other simple agents.
1.2 Working methodology
The graphical interface was implemented using C++ and the OpenGL graphics library [ 1]
while the main logic and computation is performed using the NVidia CUDA library [ 2]. As such, each
agent’s behavior is simulated on an individual CUDA core. Computations are executed in parallel as
much as possible and global updates are only performed at the end of the current iteration such that
the next iterations start with the newly computed values.
Simulations were performed using a population of 1000 agents on a platform with I5 4400
processor, 12GB Ram and NVIDIA GeForce GTX 1 060 6GB video card. The following scenarios were
studied:
– 1, 2, 4 and 8 leaders
– 1, 2, 4 and 10 obstacles
– Static and dynamic obstacles
– Each of the leader types: motivation, independent, mixed and collision avoiding
– Simple and energy constrained environments .
For each scenario I ran 30 simulations with 5000 iterations each. In every simulation,
regardless of leader type or leader number, leaders appear at the same time. Also, obstacles are
initialized at the beginning of the simulation with random positions and speeds.
The energy consumption metric is used only in scenarios with the mixed leader type as it is
relevant especially to that scenario type.
Results from the total number of agents with a leader and the total number of agents that
enterd the area of influence of an obstacle were centralizes in graphs, in order, to show the modifing
behavior of the population under the influence of a certain leader type and of dynamic and static
obstacles .
2. State of the art
2.1 Swarm formation
A set of algorithms ba sed on swarm behavior are design ed to simulate the formation of
swarms. These algorithms are based on the premise that each individual has a set of rules that must
be followed and no external stimuli influence the decision -making process of the individual. By
following these rules, the individuals can form an entire swarm which can maintain its unity over
time. Interestingly these rules can also lead to the separation of swarms into several sub -swarms
and then to the union of these sub -swarms. The first alg orithms to simulate such behavior were
based on bird flocks in which individual birds interacted with neighbor birds leading to an
organizational process from which a flock of birds that can travel long distances and for a long period
of time emerged.

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In their paper Heppner and Grenader et al. [3 ] proposed a model for Swarm formation based
on the behavior of bird flocks. They found that a coordinated flock might arise as a byproduct of
rules influencing the movement of individual agents. The rules that g overn the individuals are as
follows: each individual must move in the same direction as the flock, each individual must have a
speed relative to the speed of its neighbors, each individual must maintain the same distance
relative to the center of the floc ks, and each individual must maintain the same distance from its
neighbors. The collective behavior simulates the natural behavior of flocking without the existence
of a leader and also it exhibits to a certain degree an unsystematic behavior due to wind, external
disturbances, or other causes. In this model individual can change their position within the flock and
from time to time, the flock can split into two or more “sub flocks”. Initially each bird is assigned
random parameters. With each iteration the flock “recruits” new individuals and after an initial
period of shifting positions a stable flock is formed.
Another Swarm formation algorithm based on the behavior of bird flocks was proposed by
Craig W. Reynolds [ 4]. The suggested model states that eac h individual must follow three rules
stated in the order of precedence: each individual must avoid collisions with nearby neighbors, each
individual attempts to match the velocity with nearby neighbors and it must stay close to nearby
neighbors. Independen t behavior of an individual is modeled by expressed by an acceleration vector
which makes an individual move in a random direction. The behavior of the flock can be modified by
a script such that the flock will move in different direction at different mome nts in time simulating a
natural flock. Avoiding environmental obstacles conflicts with flock centering and each individual
must decide which has a higher importance. The model implements two types of environmental
collision avoidance. The first is based o n the force field concept and the latter called steer -to-avoid
seem closer to the natural mechanism.
Another proposed algorithm for swarm formation and motion was suggested by K. H. Tan
and M. A. Lewis [ 5]. The algorithm addresses the problem of mainta ining the geometric
configuration of a swarm during movement by introducing the concept of virtual structure. A virtual
structure is a collection of agents which maintain a semi -rigid geometric relationship with each
other, in other words a semi -swarm. In this structure the positions of the agents are not entirely
fixed, each agent having a certain amount of liberty to move. The algorithm works in a bi -directional
manner in which new agents adapt their behavior to the virtual structure and the virtual struc ture
adapts its behavior in order to adopt the new agent. In summary the algorithm works in 4 steps. In
the first step the virtual structure is aligned with the new agents. In the second step the virtual
structure is moved to permit the adoption of the new agents. In the third step the individual agents
compute trajectories to intercept the desired virtual structure. And in step four velocities are
adjusted to follow the desired trajectory. The process is repeated until only one virtual structure or
swarm r emains.
T. Balch and R. C. Arkin [ 6] describe several algorithm for swarm formation on robotic teams
based on behavior. These algorithms are useful in formations of robots used for military purposes.
Formation maintenance is accomplished in two steps. Th e first step, named “detect -formation –
position” determines the robots position in the formation based on data from the environment. The
second process, named “maintain -formation” generates commands to the robots to go to the
correct location to maintain th e formation. Three techniques for formation position determinations
have been identified: “unit -centered -referenced” in which a unit -center is computed independently
by each robot and each of them determines its formation position relative to that center, “leader –
referenced” in which each robot determines its position relative to that of the leader and “neighbor –
referenced” in which each robot maintains a position relative to one other predetermined robot.
Each robot determines the position of its neighbors by direct perception of the other robots or by
using world coordinates obtained from global position systems (GPS).

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In their paper J. Cheng, W. Cheng Harvard and Radhika Nagpal propose the swarm for mation
algorithm “ShapeBugs” [7 ]. The algorithm is used to organize agents such that they form a certain
shape. Initially each agent has a map of the form which must be formed. This map is later overlaid
on the agent’s coordinate system to check if the constructed form is correct. At each step an agent
computes its new coordinates via a proximity sensor and a wireless link to nearby neighbors. The
algorithm is composed of two processes that run continuously. In the first process the agent
confronts its learned coordinates with those of its neighbors and in the s econd process controls the
agent’s movement. The algorithm for calculating the new coordinates is based on the gradient –
descent algorithm if the agent thinks it is inside the shape otherwise it calculates a new starting
position by choosing three random ne ighbor agents and intersecting their coordinates. The rules for
movement depend on whether the agent thinks it is inside the shape or not. In the first case it must
make sure that it does not exit the shape which will lead to the dissolution of the formati on and in
the second case it must help the formation of the required shape.
G. St uder and I. Harvey [8 ] extend the “ShapeBugs” formation algorithm such that is enables
agents to agree on a consensus coordinate system starting from no coordinate agreement . This
extension imposes a new lifecycle on each agent that is composed of four primary states: wait,
sense, compute and move. The wait state is the first and final state in the lifecycle. Between the two
states the agent has moved. In the sense state the agent determines the coordinate systems of its
neighbors. In the compute state the agent calculates its new coordinates and in the move state it
moves to the newly computed coordinates. The algorithm is basically an iteration over these states
in the follo wing order: wait, sense, compute, move, sense, compute and wait. The final sense and
compute are necessary such that computational errors are avoided. Also, constraints can be
imposed such as the number of neighbors to take into account when computing the new
coordinates or the agent density within the region of the shape. This algorithm works only on
connected shapes and for very thin shapes some agents which are lost may form shapes with other
lost agents.
C. C. Lin, K. C. Chen, P. Yu. Hsiao and W. J. Ch uang [ 9] describe an algorithm for motion
planning of swarms in order to avoid obstacles and maintain the integrity of the swarm. The
approach consists of two main planners, a global planner and a motion planner. The global planner
generates a trajectory for the agents using a Voronoi diagram. This trajectory is sent as input to the
local motion planner which decides the new position of the agents using a potential -based genetic
algorithm. The trajectory consists of a sequence of positions to be traversed by the agent and using
the genetic algorithm potential functions may be used to keep the agents from colliding into
obstacles, colliding which each other and simultaneously maintain a certain distance from an agent
to another. Results from test on 3 agent and 8 agent swarms showed that agents reached the
desired target while maintaining the swarm formation and also avoiding obstacles along the path.
Another algorithm for swarm formation and navigation was propos ed by Christopher Lum et
al. [10 ]. The algori thm was designed for autonomous vehicles to navigate from one point to another
in an obstructed environment having either obstacles or smaller areas referred to as choke -points in
the article, while avoiding collision between themselves. The approach is th at obstacles can be
avoided using a vector field generated using incomprehensible flow theory [11 ]. Smaller zones can
be traversed also using this approach, because agents will change the structure of the swarm to
permit each agent to pass through the narr ow zone and after passing through the swarm will take its
initial structure. A down side to this algorithm is that the vehicles may collide with one another when
avoiding obstacles by not maintaining the space between them. The collision avoidance algorit hm
can be compared to cars when entering a highway. The first agent maintains its speed while the
second agent lowers its velocity to allow the necessary distance between them so that the obstacle
can be avoided successfully. The algorithm was tested in th e Vehicle Swarm Technology Laboratory

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(VSTL) developed by the Boeing R esearch and Technology gr oup [12 ] where it successfully passed
several tests including one containing a narrow path where only one agent could pass.
2.2 Swarm intelligence
Swarm Intellig ence models are known as computational models inspired by natural swarm
systems. To the present day, many SI models based on different natural swarms have been
proposed. Some examples of swarm intelligence models are: Ant Colony Optimization [1 3], Particle
Swarm Optimization [ 14], Artificial Bee Colony [ 15], Bacterial Foraging [ 16], Cat Swarm Optimization
[17], Artificial Immune System [ 18], and Glowworm Swarm Optimization [ 19].
The first successful swarm intelligence model is Ant Colony Optimization (ACO), which w as
introduced by M.Dorigo, V. Maniezzo and A. Colorni [13]. This model was first used to solve discrete
optimization problems in the late 1980s. The inspiration for the model is the social behavior of ant
colonies. The model was applied on the well -known tr avelling salesman problem (TSP) to have a
comparison with the results obtained by other heuristic approaches. The article proposes three
algorithms: The Ant -density, Ant -quantity and Ant -cycle algorithm. Among these three the Ant -cycle
algorithm yields a b etter performance. The algorithm works as follows: Ants are placed in different
towns and an initial “trail intensity” is set on the edges connecting two towns. The “trail intensity” is
a number computed based on the path length between the neighbor towns. Each ant moves to a
neighbor town with a probability given as a function with two parameters. The two parameters are
the number of ants that have chosen the path in the past and the “visibility” parameters which says
that the closer town has a higher chan ge of being chosen. Each time an ant goes to another the trail
it leaves is summed to the previous trail left in the past. After every ant has moved new transition
probabilities are computed using the newly set trail values. After n -1 moves, where n is the number
of towns, the shortest path found by the ants is memorized. The process continued until the number
of iterations reaches a user -defined maximum number of iterations or every ant follows the same
path.
The Particle Swarm Optimization (PSO) algorithm proposed by J. Kennedy and R. C. Eberhart
[14] is a Swarm Intelligence model based on the behavior of bird flocks with the intent to simulate a
social milieu. Initially the agents were thought as a collision -proof birds with the purpose of
simulating a cho reography of a bird flock. As mentioned in their paper, the authors explain the
algorithm by its conceptual development. The first concept relied on two rules: the nearest -neighbor
velocity and craziness, the degree to which an individual may behave in a r andom manner.
Unfortunately the flock quickly settled to an unchanging direction. The second stage in the
development was the introduction of “roost”, a position that attracted birds until they finally landed
there, such as a food source. The third stage m eant the complete removal of the nearest neighbor
velocity vector and craziness because based on the newly found rules the optimization occurred
slightly faster. In the fourth stage the velocity are adjusted based on the distance from the best
locations. T he final algorithm even further reduced the velocities of the birds until a simple formula
was found. This algorithm was successfully used in training neural networks as effectively as the
usual error backpropagation method.
Sun-Chuan Chu and Pey -Wei Tsai [17] suggested a model of Swarm Intelligence based on the
behavior of cats, the Cat Swarm optimization (CSO) algorithm. By studying the behavior of feline
creatures they found two major behavioral traits which are modeled in the suggested Cat Swarm
Optimiza tion. These behavioral traits define the two sub modes of the algorithm, namely seeking
mode and tracing mode. Seeking mode is based on the fact that most felines spend their time
resting and in a state of alertness, looking around their environment for th eir next moves. Tracing
mode models the case when felines trace their targets. The two modes are combined using a
mixture ratio. The mixture ratio is allocated a very small value, because a cat spend most of its time

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in seeking mode, and too much time spen t in tracing mode can lead to an overuse of resources. The
algorithm can be summarized in six steps. In the first step N cats are created. The second step the
cats are randomly positioned in the M -dimensional solution space, randomly assigned velocities an d
randomly assigned a mode according to the mixture ratio. In the third step the fitness of the each
cat is computed with the fitness function, which represents the criteria for the chosen goal, and the
position of the best cat is memorized. In step four e ach cat is moved according to its current mode.
In step 5 new cats are created and assigned modes according to the mixture ratio. In the last step,
the sixth one, the termination conditions are tested and if not met repeat step 3 to 5.
Another Swarm Intell igence algorithm was designed by Xin -She Yang [ 20] based on the
echolocation of bats, namely the Bat Algorithm (BA). Bats use echolocation to detect prey, avoid
obstacles and locate crevices in the dark. Their pulses vary from species to species and are
correlated with their hunting strategies. Micro -bats use the time delay from the emission and
detection of the echo, the time difference between their two ears, and the loudness variation of the
echo to build up a three dimensional scenario of the surrounding . The Bat Algorithm is based on the
behavior of micro -bats. It has three basic idealized rules: all bats use echolocation to sense distance
and they also know the difference between food and obstacles, bats fly randomly with a fixed
frequency and varying w avelength and loudness and can adjust the wavelength or their emitted
pulses depending on the proximity of the target and the loudness varies between two know positive
constants. The algorithm starts with the initialization of the bat population and the de finitions of the
pulse frequency, pulse rates and the loudness. While a user -defined number of iterations haven’t
been reached new solutions are generated by adjusting frequency and updating velocities and
locations. If a random pulse rate is greater than the current rate a solution from the best solutions is
selected a local solution is generated around it. Other new solutions are generated by flying
randomly. If another solution has a lower value for loudness and it’s better than the previous one
the solu tion is kept, the pulse rate is increased, and the loudness reduced. In the last stage the bats
are ranked according to their solutions and the best solution is found.
D. Teodorovic and M. Dell’Orco [ 21] have also developed a model based on bee behavior.
The suggested Bee Colony Optimization (BCO) algorithm is an approach for solving combinatorial
problems characterized by uncertainty. Within the Bee Colony Metaheuristic, individuals, referred to
as artificial bees, collaborate in order to solve a difficult problem. Initially all bees are located in the
hive at the beginning of the search progress. During the search process each artificial bee constructs
a partial solution to the problem. Other bees add components to the current partial solution until
one fe asible solution is found. The search process iterates until a number of iterations, prescriber by
the decision -making process, is reached. The behavior of the bees is composed of two passes, the
“forward pass” and the “backward pass”. In the “forward pass” bees create several partial solutions
based on the collective experience from the past. In the “backward pass” the bees return to the hive
and all of them participate in the decision -making process. Based on the quality of the found
solutions each bee dec ides whether to abandon its current partial solution, to continue to expand its
solution without recruiting other bees or to continue to expand its solution by recruiting other bees
through dance. The algorithm was successfully applied to solve the Ride -Matching problem. This
problem consists of making a route and a schedule for vehicle and passengers for a whole week in
“the best possible way” respecting the following constraints: the total distance traveled by all
participants must be minimized, the total delay must be minimized and each vehicle must be used
approximately the same number of times as the other vehicles.
Another model of Swarm Intelligence based on the behavior of bees is the Honey -Bees
Mating Optimization (HBMO) algorithm proposed by O.B. H addad, A. Afshar, M. A. Marino and B.J.
Adams [ 22]. A honey -bee colony typically consists of a single egg laying queen, drones and worker
bees. The queen is the most important member of the colony because it produces new queen and
new worker bees. The sole purpose of the drones, male bees, is to mate with the queen to produce

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new bees. And the worker bees specialize in brood care. The most commonly used method for
founding a colony is called “swarming” in which the new colony is founded by a queen or more a long
with a group of workers from the original colony. The mating process takes place during mating –
flights far from the nest in which several drones mate with the queen forming the genetic pool of
the colony. Each time the queen lays fertilized eggs, she retrieves at random a mixture of the sperms
to fertilize the egg. Insemination ends with the death of the drone. The model proposed can be
compared with a set of transitions in a state -space where the queen moves between different states
and mates with the drone encountered at each state probabilistically. At the start of the mating
flight the queen is initializes with some energy and returns to her nest when that energy is depleted
within a certain threshold or her spermathecal is full. The algorithm can b e summarized in five main
stages. In stage one the mating -flight starts and the queen randomly selects drones to form the
spermatheca, and one drone is selected at random for the creation of broods. In stage two new
broods are create by crossover between t he drones’ genotypes within the queen’s. These broods are
the trial solutions. In stage three workers use heuristics to conduct local search on the broods. Stage
four adapts worker bees’ fitness based on the amount of improvement achieved on broods. And is
stage five the weaker queens are replaced by fitter broods.
3. Simple leader types
3.1 Collision avoidance leader
3.1.1 Description
Collision avoidance leaders are a different type of leaders that appear only when agents are
very close to obstacles with t he purpose of guiding the agents to change their direction and speed to
avoid that obstacle . Any agent can become a collision avoidance leader when it sense an incoming
obstacle, the constraint beeing that the agent should not already be under the influenc e of another
leader. This type of leader resembles to a degree the motivation type leader in the sense that it
adjust its speed and direction taking into accound the speeds and directions of its neighbors in such
a way as all of them succesfully avoid the obstacle. These leaders stay leaders just long enough to
avoid the obstacle. When reaching a certain distance from the obstacle the leader reverts to being a
simple agent.
The influence of this leader type on its neighbor resembles a wave of propagation, t hat start
from the agent closest to the obstacle and reaches all agents in the swarm up until a safe enough
distance.
Figure 1 shows a simulation with 6 obstacles and collision avoidance leaders. The leaders are
the onese having a white spot on the top and dark grey on the bottom. Followers of a leader have a
consistent color of dark grey. As described previously we can observe that the influence of the
leader propages similar to a wave on the neighboring agents. Also it can be observed that swarms
manage t o keep a certain distance from any obstacle. The number of agents than enter within the
region delimited by this distance is measured using the metrics area of influence described below.

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Figure 1 . Collision avoidance leaders
3.1.2 Obstacles
Obstacles are objects that agents must avoid and not collide with them. They are used to
simulate the real life object that can hinder the behavior of birds that try to form a stable flock.
Figure 2 shows a simulation with 6 obstacles. We can observe that the obstacle s are placed
randomly and influence the formation of a swarms with numerous agents. As the number of
obstacles increases it becomes increasingly hard for agent to form swarms because the swarm will
not have the necessary space to maintain it’s form and str ucture. Nontheless swarm are dynamic in
nature and they will inevitably split into multiple sub -swarms as they are about to collide with
different obstacles. The figure also reflects this. Several sub -swarms are formed that lack the
necessary space to join together to form a single swarm made of all the agents.
Obstacles can be either static or dynamic. The static obstacles are simple object placed at
random location that keep their location during the entire simulation time. Dynamic obstacles, in
contrast, also have a velocity and direction with which they move in the environment. Dynamic
obstacles are harder to avoid because agents must have a faster ability to sense the approaching
obstacle and act in consequence.
Two collision avoidance strategies have been studied of which only one stood out. The first
strategy was one in which only leaders could have the ability to avoid obstacles and the followers
would simply have followed the leader. This strategy did not work out as expected and a very big
number o f collisions occurred. The second strategy is one in which every agent has the capabilities of
avoiding an obstacle. When an agent senses an approaching obstacles it will adapt it’s speed and
direction in order to avoid that obstacle and for a limited time will neglect the imposed set of rules.
This behavior was chosen in order to reflect the real life behavior, that when faced with iminent
danger the survival instinct activates and the bird tries to escape alive regardless of its neighbors.

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Figure 2. Sc enario with 6 obstacles

3.1.3 Experimental results
The metric used to study the behavior of this type of leader is the area of influence metric.
This metric calculates the number of agents that enter a predefined circular area in the vecinity of an
obstac le. This area represents in fact a circle around the obstacle having as radius the distance at
which the avoidance collision leader detects the obstacle. It is used to show that the collision
avoidance type leader not only manages to ensure that itself and its neighbors avoid the obstacle
but also that they keep a certain distance between themselves and the obstacle, ability which simple
agents or the other leader types, motivation, independent and mixed do not have.
Results obtained in scenarios using the avoidance collision leader have been centralised in
Figure 3. Scenarios studied used a swarm population of 1000 agents, different number of obstacles
from both types, static and dynamic.

Figure 3. Number of collisions inside obstacle’s area of influenc e in scenarios with 1, 2,4, 8 and 10 obstacles

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As we can observe in scenarios with static obstacles the number of agents that entered an
obstacle’s area of influence was very low compared with the scenarios with dynamic obstacles. This
is primarily becaus e in scenarios with dynamic obstacles, as the number of obstacles increases the
distance between them decreases and when an agent is close to two or more obstacles it has a hard
time deciding which obstacle to avoid first, thus inevitably entering the area of influence of at least
one of the obstacles. In the case of static obstacles this number is approximately the same
regardless of the number of obstacles. This happens because a leader is not forced to enter the area
of influnce of multiple obstacles, an d by being initialized at random positions, the obstacles will be
located at significantly greater distances.
3.2 Motivation leader
3.2.1 Description
Another studied idea for the emergence of a leader is that the behavior of a leader can be
determined by the number of followers that the leader has. The agent that becomes a leader has a
specific speed and direction, but this speed can only be attained when all agents in the simulation
follow this particular leader. At each step the leader adopts new agents and adapts its speed based
on the percent of agents that follow him. Follower agents can also adopt new agents provided that a
new agent has at least two neighbors that follow the same leader. An agent that already follows a
leader cannot follow a new lead er except when it loses its influence. A leader can lose control over
an agent when that agent collides with an obstacle, moment in which that agent ignores all rules,
just to avoid that obstacle. If a leader has less than a certain number of followers for a period of
time, the leader loses its traits and becomes a simple agent again.
An example of a simulation with four such leaders whose behavior is based on the number
of followers is shown in Figure 4.

Figure 4. Simulation with 8 leaders whose behavior is based on the number of followers.

Emergence of a leader in a group
11
Such leaders tend to place themselves inside the sub -swarm of following agents, and adapt
their speed correspondingly. Alongside the behavior based on the number of followers these agents
also respect the basic rules for a simple swarm such as adapting the speed based on the mean
speed of its neighbors. So, the total speed of the agent at a certain time is computed with the
following formula:

speed = p * L + (100 – p) * N (1)

Where p is the percent of agents that follow this leader, L is the speed assigned to a leader
and N is the mean speed of the leader’s neighbors.
The figure also shows that sub -swarms that have a leader can collide into each other and
form a new swarm containing a ll the agents but with the same leaders. This happens because the
speed of an agent is influenced in equal proportion by the leader and by its neighbors. The speed of
an agent is computed as follows:

a = 0.5 * N + 0.5 * L
(2)
Where a is the agent spee d, N the mean speed of the neighbors and L the speed of the
leader. If a smaller sub -swarm collides with a bigger sub -swarm the agents from the smaller one will
eventually adopt the speed and direction of the greater one because the neighbor speed will cha nge
radically.
3.2.2 Experimental results
An initial study was performed in a scenario in which every agent can become a leader
regardless of the fact that he is part of a sub -swarm or has no neighbors. As such the likelihood of
leaders adopting only a few initial followers is stronger, and the number of leaders that may
disappear is increased.
Following we present results obtained using the metrics presented in Section 1.2.

Figure 5. Distribution of agents with leader in simulations with 2 (green), 4(bl ue) and 8(yellow) motivation
leaders

Figure 5 shows the distribution of agents that have a leader in simulations with 2, 4 and 8
leaders. The graph shows that, when comparing simulations with 2 and 4 leaders, as the number of
leaders increases, so does th e number of agents belonging to a leader. Also, as the number of
leaders increases, the variation between a state with no leader and a state belonging to a leader is

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lower. The same tendency can be found when comparing simulations with 4 and 8 leaders, the
number of agents belonging to a leader increases with the number of leaders and also the variation
between the two states mentioned earlier is smaller.
The major difference between the simulation involving 2 leaders and 8 leaders is that the
total number of agents belonging to all leaders is greater in the second case, and at certain times all
agents have a leader. In the simulation involving only 2 leaders the number of times all agents
belong to a leader is smaller than in the case of the simulation with 8 leaders, also the variation is
considerably higher.

Table 1
Average number of agents controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 146 81 45
2 145 86 46
3 81 47
4 87 44
5 46
6 42
7 42
8 40
Free agents 107 61 40

Table 1 shows the average number of agents per leader in simulations with 2, 4 and 8
leaders. For each simulation, the average number of followers per leader varies only slightly,
meaning that each leader has the same influence over the entire populatio n of agents. Also, when
comparing, simulations with 2 and 8 leaders we note that the number of free agents drops radically,
showing that almost all the population of agents tends to be adopted by a leader.

Figure 6. Distribution of zones with agents bel onging to a leader in simulations with 2(blue), 4(yellow) and
8(green) motivation leaders

Figure 6 highlights the distribution of zones that belong to a leader in simulations with 2, 4
and 8 leaders. The tendency observed is that the number of controlled zones decreases with the
number of leaders. This tendency was expected because with the increase in the number of leaders,
the number of followers also increases and a bigger stable swarm tends to form that occupies only a
small portion of the simulation s urface.

Emergence of a leader in a group
13
Table 2
Average number of zones controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 6 5 4
2 6 5 4
3 5 4
4 5 4
5 3
6 4
7 4
8 3
Free zones 88 84 82

Table 2 shows the average number of zones controlled by each leader in simulations with 2,
4 and 8 leaders. Because sub -swarms tend to be stable in simulations using this leader type, they
also tend to occupy as few zones as possible. Also sub -swarms interact with each -other, and such,
they may behave in the same ma nner and stay close to one another. This behavior is highlighted in
the table, in which each leader has almost the same number of zones occupied, as the others.
The other studied scenario is the one in which leaders can appear only in the middle of an
already formed swarm. The moment when the leaders appear varies depending on the moment that
the biggest formed swarm has a user -defined number of members.
Following we present results obtained using the defined metrics for simulations involving 2,
4 and 8 lea ders.

Figure 7. Distribution of followers in simulations with 2(green), 4(blue) and 8(yellow) motivation leaders

Figure 7 shows the number of agents with a leader in simulations with 2, 4 and 8 leaders.
The number of agents that have a leader is greate r in the simulations with 4 and 8 leaders because,
emerging when a relatively big sub -swarm is already formed, these leaders split that sub -swarm into
4, respectively 8 smaller sub -swarms that adopt new agents and the bigger the number of new sub –
swarms fo rmed the greater the coverage of the simulation area. Also the variation between a state
with no leader and a state with a leader is smaller as the number of leaders increase .
Comparing the results obtained in simulations involving 2 and 8 leaders, outline d in Figure 5 ,
we can observe a clear difference between the 2 simulations. In the second case, with 8 leaders, the
number of agents with leader reaches is equal to the total number of agents at several times during
the simulation whereas in the simulation with 2 leaders this number is significantly lower. Also the
variation between a state with a leader and a state without leader is much smaller in the 8 leader
simulation than in the 2 leader simulation.
The differences between simulation with 4 and 8 lead ers are not significant. The number of
agents belonging to a leader reaches the number of agents in the simulation in both cases, with a
greater tendency in the case of 8 leaders. The number of times this happens is relatively the same in

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both simulations, the only difference being in the stability of the sub -swarms. In the 4 leaders
simulation the number of changes between a state with a leader and a state without, for an agent, is
bigger than in the simulation involving 8 leaders.

Table 3
Average number of agents controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 141 84 42
2 149 81 46
3 82 44
4 84 42
5 41
6 46
7 40
8 46
Free agents 108 58 45

The average number of followers for simulations with agents placed near the center of an
already formed swarm do not vary considerably from the number in simulations with randomly
placed leaders, as outlined in Table 3. That is because, even though there isn’t any formed sub –
swarm in the simulations with randomly positioned leader s, they adopt quite fast new followers and
sub-swarms tend to form quite fast. And taking into consideration that a simulation last for 10000
iterations the already formed sub -swarm has a minor influence.

Figure 8. Distribution of zones with followers of a leader in simulations with 2(blue), 4(yellow) and 8(green)
motivation leaders

The same as in the simulations involving randomly positioned leaders the number of
controlled zones decreases when the number of leaders increases. In these simulations, part icularly,
involving an already formed sub -swarm, even fewer zones are controlled by the leaders, because of
their emergence inside an already formed swarm.

Table 4
Average number of zones controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 7 5 4
2 6 5 4
3 5 4
4 5 4
5 4
6 3
7 3
8 3
Free zones 88 84 83

Emergence of a leader in a group
15
Table 4 shows the average number of zones controlled by each leader in simulations with 2,
4 and 8 leaders. There are no observable differences when comparing with the scen ario in which
leaders appear randomly inside a swarm. Same as in the previous scenario, b ecause sub -swarms
tend to be stable using this leader type, they also tend to occupy as few zones as possible. Also sub –
swarms interact with each -other, and such, they may behave in the same manner and stay close to
one another. This behavior is highlighted in the table, in which each leader has almost the same
number of zones occupied, as the others.
3.3 Independent leader
3.3.1 Description
The second studied idea for the for the emergence of a leader is that a leader has a specific
behavior that he follows regardless of the number of followers, referred to as an independent
leader. The rules for adopting followers are the same as with the previous studied idea, new
followers must be neighbors of the leader, or have at least two neighbors that follow the same
leader. The speed and direction of the leader stays the same at all times, except when avoiding
obstacles when a special behavior for avoidance is performed after w hich the leader behavior is
resumed. The difference between the previous idea and this one regarding the speed of the
following agents is that the leader has a greater influence over its followers. The new formula for
computing a follower’s speed is:

a = p * L + ( 1 – p) * N (3)

Where p is set to 0.8, reflecting a greater influence from the leader. The influence factor is
set to this value because results have shown that leaders with this behavior tend to lose agents
faster when the influence factor is small.

Figure 9. Simulation with 8 independent leaders

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In this scenario a leader can only disappear when its allocated time expires regardless of the
number of followers. As opposed to the previous idea sub -swarm lack the tend ency to form bigger
sub-swarms containing different leaders because the leaders don’t depend on the behavior of other
agents. Loss of control over an agent happens as in the previous scenario when an agent avoids an
obstacle or the agent is adopted by anot her leader.
The agent that becomes a leader has a specific speed and direction, but this speed can only
be attained when all agents in the simulation follow this particular leader. At each step the leader
adopts new agents and adapts its speed based on the percent of agents that follow him. Follower
agents can also adopt new agents provided that a new agent has at least two neighbors that follow
the same leader. An agent that already follows a leader cannot follow a new leader except when it
loses its influ ence. A leader can lose control over an agent when that agent collides with an obstacle,
moment in which that agent ignores all rules, just to avoid that obstacle. If a leader has less than a
certain number of followers for a period of time, the leader los es its traits and becomes a simple
agent again.
An example scenario is shown in Figure 9.
3.3.2 Experimental results
Appearances of leaders in the first studied scenario are also random as those presented in
section 3.2.2. Every agent may become a leader r egardless of the fact that he is part of a sub -swarm
or has no neighbors. By contrast with the behavior of leaders in section 2.1, here leaders will
perform their defined behavior regardless of the number of followers and can only disappear when
their allo cated time expires.
Following we present results obtained using the metrics presented in Section 1.2.

Figure 10. Distribution of followers in simulations with 2(green), 4(blue) and 8(yellow) independent leaders

Figure 10 show s the distribution of agents which belong to a leader in simulations involving
2, 4 and 8 independent leaders. The variation between a state with a leader and a state without a
leader is big in all three simulations, showing that leaders cannot control a b ig number of followers
for a big amount of time. Also, in either simulations, the number of agents that follow a leader is
relatively small compared to the number of agents in the simulation. But as this number increases so
does the number of agents that a re being lost by a leader.
There is no significant change in behavior by increasing the number of leaders as shown
when comparing simulations with 2 and 8 leaders. Both simulation present big variations between
states with no leaders and states with a lead er for an agent. The number of adopted agents is bigger
in the simulation with 8 leaders, but these formed sub -swarms easily dissipate as the leader loses
control over its followers, fact shown by the almost vertical lines in the graph.

Emergence of a leader in a group
17
Comparing simulatio ns involving 2 and 8 leaders interesting results are shown. In the
simulation with 8 leaders the number of agents that follow a leader doesn’t reach the number of
agents in the simulation.
The variation is big in all simulations. This is because agents tha t follow a leader will
eventually adopt their leader’s behavior in terms of speed and direction. But, when reaching an
obstacle, the leader will lose control over that agent and the greater the speed of the leader, the
greater the change of meeting an obst acle.

Table 5
Average number of agents controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 126 70 34
2 121 66 35
3 66 33
4 68 34
5 36
6 36
7 35
8 35
Free agents 150 128 110

Table 5 shows the average number of follower s for each leader in simulations with 2, 4 and 8
randomly positioned leaders. The averages for each leader in a simulation have almost the same
value, showing that all leaders have the same influence over the entire population. The average
number of free a gents is big in all simulation. This happens because these leaders cannot maintain
control over a big swarm for a long period of time, and when meeting an obstacle, they tend to lose
all of their followers.

Figure 11. Distribu tion of zones with followers in simulations with 2(blue), 4(yellow) and 8(green) independent
leaders

Figure 11 shows that as we increase th e number of leaders the number of zones controlled
also increases. This happens because the simulation becomes more dynamic with the increase in
number of agents and even more agents will be influenced by a leader if that leader will not
maintain control over the agents for a long period of time.

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Table 6
Average number of zones controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 6 4 3
2 6 4 3
3 4 3
4 4 3
5 3
6 3
7 3
8 3
Free zones 89 84 80

The average number of zones controlled by leader in relatively small compared to the
dimensions of the swarm surface, as shown in Table 6. Thi s happens because a leader of this type
cannot control a big number of agents for a longer period of time and will lose control over them
and over the zones in which these agents were located. Also, due to the fact, that the leaders
behave the way they wan t multiple leaders may be located in the same zone at a particular time.
In this next studied scenario leaders can appear only in the middle of an already formed
swarm, the same as the second scenario in Section 3.2.2. The moment when the leaders appear
varies depending on the moment that the biggest formed swarm has a user -defined number of
members.
The results from simulations involving 2, 4, 6 and 8 leaders are as follows.

Figure 1 2. Distribution of followers in scenarios wit h 2(green), 4(blue) and 8(yellow) independent leaders

There is not noticeable difference between simulations with 2 and 4 leaders, as shown in
Figure 1 2. Each simulation has a high degree of variation between a state with no leader and a state
with leader for an individual and the number of agents that have a leader is relatively the same for
each simulation. At a small number of moments the number of agents is bigger in the simulation
with 4 leaders than in the simulation with 2 leaders.
In the case when comparing simulations with 2 and 8 leaders we denote the behaviors vary
only slightly. The number of agents adopted by a leader tends to be a little bigger for the scenario
with 8 leaders but the variation is as big for both simulations. The noticeable dif ference between the
2 simulations is the number of agents that are lost by a leader when colliding with an obstacle,
number represented by the almost vertical lines for the 2 leaders simulation.
For 8 leaders, the number of agents that have a leader is sli ghtly bigger than in previous
simulations but also the variation between a state without a leader and a state with a leader for an
agent is bigger. As we can observe, in the simulation with 8 leaders the number of agents without a
leader doesn’t drop below 100 agents, except isolated cases, because when a leader loses control

Emergence of a leader in a group
19
over that particular individual another leader will adopt him. A bigger number of leaders, thus,
provides a better coverage of the surface in which agents behave.

Table 7
Average numb er of agents controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 130 65 37
2 126 66 38
3 71 35
4 65 40
5 40
6 34
7 37
8 30
Free agents 141 128 101

There are no significant differences between independent leaders positi oned near the
center of an already formed sub -swarm and randomly positioned leaders. In both cases the average
number of free agents is quite big. Due to the fact that these simulations are more dynamic than
those with motivation leaders, the formed sub -swarm will be split quite rapidly by the emerging
leaders, and the simulation will grow similar to one with randomly positioned independent leaders.

Figure 1 3. Distribution of followers in simulations with 2(blue), 4(yellow) and 8(green) independent leaders

The number of zones controlled by all leaders in simulations with this type of leaders is similar to
the number of zones controlled by randomly placed leaders. The formed swarm is rapidly torn apart
by the different behaviors of the emerging leaders and soon agents will behave similar to their new
leader which have a greater influence.

Table 8
Average number of zones controlled by each leader
Leader no. 2 leaders 4 leaders 8 leaders
1 6 4 3
2 6 4 3
3 4 3
4 5 3
5 3
6 3
7 3
8 3
Free zones 90 85 80

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The average number of zones controlled by each leader remains almost the same as in
simulations with randomly placed leaders. Provided that, these leaders will behave only the way
they want, the formed sub -swarm wil l split quite rapidly and the environment will revert to one
similar to that in which leaders appear randomly.
3.4 Mixed leader
3.4.1 Description
The mixed leader is a leader type that enherits the behavioral properties of both the
motivation type leader a nd of the independent type leader. The change in behavior occurs when a
certain threshold is surpassed, the threshold being a defined number of agents. At the start of the
simulation each mixed leader acts as an independent type leader having a fixed speed and
wandering around the environment and adopting new agents. When an experimentally observer
number of followers have been adopted the mixed leader switches its behavior to resemble the
motivation type leader in order to keep these followers for a greate r amount of time and
successfully adopt new agents to the same sub -swarm.
This type of leader succesfully combines the most important behavioral patterns of the
previous leader types. For the motivation part is remains capable of maintaining for a bigger p eriod
of time a bigger number of followers than the independent type leader and for the independent
type, when it remains without followers is easily wanders around the environment at full speed to
adopt new followers.
The same formulas that guide the beha vior of the motivation and independent type leaders
are kept. The motivation type leader has a defined speed and direction and moves around taking
into consideration the speed and direction of its followers. The bigger the number of followers, the
less inf luence followers have on leader. When all agents have been adopted the follower’s influence
dissapears and the leader will behave according to its definec leader speed and direction. The
independent type leader also has a defined speed and direction, but i ts movement is not influenced
by the number of followers. It will always move with the predefined speed in the predefinec
direction.

Figure 14. Simulation with 8 mixed leaders

Emergence of a leader in a group
21
Figure 6 highlights the different behavior of mixed type leaders. We can obs erve that different
sub-swarms have been formed and they tend to join together as in the simulations with the
motivation leader type. The green leader, in contrast, does not have the neccesary number of
followers to become a motivational type leader and be haves similar to the inependent leader until it
gathers that number of followers.
3.4.2 Experimental results
The first metric used to study the behvior of the mixed leader is the number of followers that a
leader has. The following figures show comparison s between the three types of leaders, motivation,
independent and mixed, in simulations with 1, 2, 4 and 8 static and dynamic obstacles and varying
number of leaders.

Figure 15. Number of agents with leader for 1 leader in scenarios with 1,2,4,8 obstacl es

Figure 15. shows the differencs between the three leader type in regard to the number of
agents with leader in simulations with 1 leader. The main observation is that the independent leader
performes the worst of the three types of leaders when it come s to the number of agents that follow
a leader and that for a small number of obstacles the motivation leader performes best. In contrast,
in simulations with a bigger number of obstacles the mixed leader has the upper hand having the
biggest number of fol lowers for both static and dynamic obstacles.

Figure 16. Number of agents with leader for 2 leaders in scenarios with 1,2,4,8 obstacles

Figure 16. reflects the same tendencies as the previous figure, the mixed leader performing
best in scenarios with b igger number of both static and dynamic obstacles, the difference being that
the total number of agents with leader is bigger.

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Figure 17. Number of agents with leader for 4 leaders in scenarios with 1,2,4,8 obstacles

Figure 1 8. Number of agents with leader for 8 leaders in scenarios with 1,2,4,8 obstacles

Figures 17 and 1 8 reflect the same behavior but with a greater number of total agents that
have a leaders, suggesting a direct proportional relation between the number of leaders in a
simulation and the total number of agents that have a leader. Also, as in previous cases, the mixed
leader outstands the independent and motivation leaders in simulations with bigger number of
obstacles.
In conclusion, we can say that the total number of agent s is propo rtional with the number of
leaders in the simulation, regardless of the leader type. Mixed leader provide the best results in
simulation with bigger numbers of obstacles, both static and dynamic. Motivation leaders provide
better results than the mixed lea der but only in simulations with smaller number of obstacles. And
simulations with the independent type leader result in the smallest number of total agents with
leader.
The second metric used is the average number of agents per leader that studies an ave rage
of how many followers does a leader type have in different simulations. The average is calculated
taking into consideration the number of leaders within a given simulation. In consequence the
average will be greater in simulations with 1 leader than i n the simulations with 8 leaders. What we
want to study is the differences in behavior of the three leader types in regards to this metric.

Emergence of a leader in a group
23
Figure 1 9. Average number of agents per leader, simulation with 1 leader, motivation (Mo), independent (I),
mixed (Mi)

Figure 1 1 shows the average number of agents in simulations with 1 leader of each type. In
simulations with smaller number of obstacles motivational and mixed leader types have
approximately the same average number of followers, whereas the independe nt leader
performance is worst with the smaller number of average followers. In contrast, in simulations with
bigger number of obstacles the mixed leader clearly surpasses the motivational type leader with a
bigger average.

Figure 20. Average number of agents per leader, simulation with 2 leaders, motivation (Mo), independent (I),
mixed(Mi)

Figure 20 reflects the same behavior as in the simulation with 1 leader but here, the average
number of agents is split between the 2 leaders, and such the averages are smaller. The independent
leaders maintains its behavior providing the smallest avarages of the three leader types and the
mixed leader surpassing the motivational leader and the indpendent leader in simulations with a
bigger number of both static and d ynamic obstacles.

Figure 21. Average number of agents per leader, simulation with 4 leaders, motivation (Mo), independent (I),
mixed(Mi)

Figures 21 and 22 study the same metric in simulations with 4 and 8 leaders. The same
patters is observed, with th e mixed leader performing better in simulations with bigger number of
obstacles, the independent leader providing the lower averages and the motivation leader providing
the best results in environments wi th smaller number of obstacles. Also the avarages ar e smaller
when dealing with dynamic obstacles versus static obstacles.

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Figure 22. Average number of agents per leader, simulation with 8 leaders, motivation (Mo), independent (I),
mixed(Mi)

In conclusion, the mixed leader has better results in simulatio ns with bigger number of
obstacles, motivation leaders perform best only in simulations with smaller number of obstacles and
the independent leader has the worst results regardless of the number of obstacles or obstacle type.
Energy consumption is a metric used to study how different types of leader will behave in a
constricted enviroment. Energy is defined as an external quantity that is used by leaders for
movement. Each leader begins with the same quantity and consumes it based on the leader type
and bas ed on the number of followers that the leader has. This metric is used to highlight which type
of leader perfomes better when an environmental constraint is imposed. A higher energy
consumption coefficient suggest a loss in performance for a leader and als o that the leader lacks a
certain ability to adapt itself to the environment.
Energy is consumed differently by the motivation and the independent leaders. The
motivation leader consumes energy inversely proportional to the number of followers that it has .
When more than half of the agents from the entire population are adopted, they start to compense
the energy consumption, and the energy consumption becomes negative, because the leader is
pushed forward by the followers instead of its own forces. The ind ependent leader, in exchange,
does not take into account the number of followers with concerns to the energy consumption.
Having its own speed and direction it will consume a energy amount relative to its speed at any
given time.
Following are the results obtained when studying the energy consumption in simulations
with 1, 2, 4 and 8 obstacles and varying number of mixed leaders.

Figure 23. Energy consumption, 1 Mixed leader

Figure 23 show the energy consumption for a simulation with 1 mixed leader. We can
observe that the average energy consumption for the leader when it is of the motivational type is
equal to 0. That is because in this simulation the leader manages to adopt a sufficient number of
agents that compensate for the consumption of energy. On the other hand the independent type

Emergence of a leader in a group
25
has a significantly high amount of energy consumption which is not influen ced by the number of
obstacles.

Figure 24. Energy consumption, 2 Mixed leaders

In the case of two leaders, as shown in Figure 24 the energy c onsumption for the motivation
type leader increases. In contrast with the previous scenario with only one leader, in this case
leaders cannot adopt all of the agents and thus will have to consume their own energy. Still the
energy consumption is substantia lly lower than the independent leader consumption.

Figure 25. Energy consumption 4 Mixed leaders

Figure 26. Energy consumption, 8 Mixed leaders

Figures 25 and 26 show the same behavior when it comes to energy consumption. When
leaders are of the inde pendent type their energy consumption is not influenced by number of
obstacles or by the number of leaders within the simulation. When leaders are of the motivation
type energy consumption coeficients vary. In the case of 4 leaders we observe a lower consu mption
than in the case of 8 leaders. This is due to the average number of followers than a leader has,

ALEXANDRU -CATALIN CIOBANU
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average which is smaller in simulations with 8 leaders. Also we can observe that the number of
obstacles does not influence energy consumption on neithe r of the two simulations .
4. Shape forming leader
For the last semester, I began implementing a new type of leader that re -organizes its
followers to change their position to form a predefined shape, the shape forming leader. This type of
leader is built on top of the mixed leader and switches to the shape forming behavior when the
number of followers reached the required number to form the shape. The leader is based on the
behavior of the shepherding agents presented by Christopher Vo[25 ] is his dissertat ion. The
difference between the leaders implemented and those presented by Christopher is that his leaders
resemble obstacles and agents will change their behavior when approaching these type of agents
while the leaders presented here influence the agents and the agents adopt with small steps the
behavior of the leaders.
4.1 Subleaders
A subleader is an agent that has only a subset of the capabilities of a full leader. It emerges
with the scope of reorganizing its immediate neighbors to change position such that the entire
swarm can reorganize into the desired geometric shape. The main leader chooses which of its
followers can become a subleader based on their heartbeat and on their position. Each geometric
shape has different key positions that need to be t aken into account when reorganizing the swarm.
For example the square has four corners that need to be occupied with agents while the triangle has
only three corners. A subleader has a predefined lifetime and becomes a simple agent after it “lives”
for a n umber of frames or after it has finished its behavior.
There are two main types of behaviors used for the subleaders. The main one and the most
used is the INWARD behavior, in which an agent close to the edge of the swarm and having the
smallest heartbeat is selected to move toward the position of the leader. By doing this it influences
the immediate neighbors to move in the same direction and such the total surface of the swarm
decreases. This behavior is useful when forming a circle or a triangle.

Figure 27. Subleader with INWARD behavior

Emergence of a leader in a group
27
Figure 6 highlights the INWARD behavior of a subleader. As we can see the subleader was
chosen from the agents close to the edge of the swarm and moves toward the position of the main
leader. The main leader moves onl y in small steps to always be in the center of the swarm. In this
case the desired shape is that of a circle and by choosing agent from different parts of the swarm to
become subleaders the swarm will eventually resemble a Circle.
The second behavior is th e OUTWARD behavior which is to some degree the opposite of the
previous one. In this case agent to become subleaders are selected close to the initial leader and
move toward key positions of the swarm in order to expand the swarm only in certain directions . For
example in the case of the square we want subleaders to move towards the corners.

Figure 28. Subleaders with OUTWARD behavior

Figure 28 highlights the OUTWARD behavior of a subleader. The subleaders were chosen
from agents close the leader positi oned in the center of the swarm. The subleaders only affect a
small number of agents close to them and influence them into moving toward the desired key
locations. In this case the desired geometric shape is a square and subleaders will move toward the
four corners to gather agents towards that particular locations.
The OUTWARD behavior is mostly used after the INWARD behavior when the swarm shape
resembles to a certain degree that of a Circle. In such a case the height and the width of the swarm
are appro ximately the same and there are only the corners that need to be filled.
4.2 Algorithms, methods, approach
The following two states are the ones that are dealing with the shape forming algorithm. In
the Create_Subleaders state the leader creates subleader s that will rearrange the simple agents.
These subleaders are chosen from the agents that are the furthest from the leader and will move
toward the position of the leader. The major problem here was how to choose these subleaders
without performing redunda nt operations each turn to find the location of each agent. I solved this
problem by using a variant of the heartbeat algorithm , which is also used to check if an agent is still
under the influence of the leader. The algorithm works as follows. At each ste p the leaders sends a γ
value, “the heartbeat”, increased by one. Each neighbor receives that value, compares with its own

ALEXANDRU -CATALIN CIOBANU
28
and sends it forward to their respective neighbors excepting the neighbors from which it has
received that value. If the value is low er that its own it will ignore it since it is an older heartbeat. If
the value is greater, the agent’s value is updated to the new value. If for δ number of frames the
agent doesn’t receive a new timestamp, then it means that the agent has lost contact wit h the
swarm and is no longer under the influence of the leader.
As such, agents near the margins of the swarm have the lowest heartbeats. The leader only
has to see which followers have the lowest values for the heartbeat and create a new sub -leader.
Thes e sub -leaders only live for a predetermined period and only influence the agent that are
neighbors to them.
The behavior of the sub -leaders is built upon the collision avoidance leader. The difference is
that these sub -leaders try to avoid a virtual obstac le and will move towards the position of the main
leader to avoid that obstacle. Tests performed showed that the influence of the sub -leader
propagates the same as the influence of the collision avoidance leader. As we can see in Figure 5 the
yellow sub -leader has adopted only the followers close to him and are guiding them towards the
main leader of the swarm (the agent that is surrounded by a circle in the figure).
In the following state, Wait_For_Sub leaders , the leader waits for the current sub -leaders t o
finish their behavior and checks the degree of similarity between the goal shape and the shape of
the swarm. If the expected similarity is obtained the leader stops creating new sub -leaders and will
switch its behavior to Switch_Shape or Finish_Movement .
The Switch_Shape state is chosen when the current shape formed is not the desired shape.
To form the square or the triangle shape the swarm must resemble a circle and afterwards the
necessary adjustments to form the desired final shape will be performed. After switching from the
Circle shape to the desired shape the leader will start to create new subleaders for the target shape
and will switch to the Create_Subleaders .
The Finish_Movement state is chosen when the desired shape is reached and the main
leader will keep the swarm in place such that none of its followers move to ruin the swarm’s shape.
The method to check if the shape of the swarm resembles that of a circle resembles the
Circle Fit Least Squares algorithm [ 28][29]. Given a set of point, in or case the positions of the agents,
the algorithm tries to compute the best circle that has the center in the center of the set of points
and the smallest radius that best surrounds the points. Here the circle radius must be equal to the
value provided by a formula that calculates the best radius based on the number of followers and
the model size of the agents.
In the case of forming a Triangle bary centric coordinates are used [ 30]. These coordinates
are used with specific formulas that check if a 2D Point is inside a triangle represented by its 3
corners, also 2D Points. In our case the points are the agent locations and the corners of the triangle
are computed also with a formula based on the number of agents and the size of the agents.
The rules used when forming a square a more complex than those in the case of a circle or a
triangle. After forming a circle subleaders will adopt the OUTWARDS behavior and will move towards
the corners of the square in order to change the swarm shape. In consequence the coo rdinates of
the corners must be calculated to be able to point the subleaders into the correct directions. For
that the swarm is split into a number of different symmetrical groups having the same width and
height that will be numbered.
These groups are t hen used to compute their centers and the subleaders will be sent
towards those centers to rearrange the agents such that they occupy the entire surface of the
subgroup.
Figure 29 shows how a swarm that resembles a circle is split into 16 different groups. As we
can observe the corner groups are not fully defined and the agents need to be pushed in order to
define them. Especially the bottom left corner must be reshaped having only a few agents.

Emergence of a leader in a group
29

Figure 29 . Swarm split into 16 groups

Figure 30 highligh ts only the 4 groups that define the corners of a swarm. In this figure the
square is almost formed and the corners are better defined than in the previous scenario. By the
looks of it only another set of subleaders will rearrange the agents to create a go od enough square
figure.

Figure 30 . Square formation with corners highlighted

ALEXANDRU -CATALIN CIOBANU
30

Figure 31 . Triangle formation with base corners highlighted

Figure 31 shows a swarm forming a triangle shape. Subleaders have an INWARD behavior
such that move agents move toward the corners of the base to better define them. The bottom
corner is defined quite well and being far enough from the center of the swarm will not be forced to
change their positions.
4.3. Experimental results
The experimental results are shown as pi ctures, as a means of comparing the initial shape
and the final shape of the swarm. The highest similarity between the goal shape and the swarm
shape was obtained when re -arranging the swarm as a circ le

Figure 32. Initial swarm

Figure 32 shows the init ial state when a leader become a shape forming leader after it has
reached the required number of followers. Agent have been adopted on all sides of the swarm and
thus the initial shape is chaotic. At this point the leader chooses sub -leaders from the fart hest agents
to push the agents towards its position. Sub -leaders must also be chosen from all sides of the swarm

Emergence of a leader in a group
31
such that the swarm maintains its center. If the swarm center moves the main leader will move to
the new center and the other agents will follo w.

Figure 33. Circle shape

Figure 33 shows the final state of the simulation in which a shape forming leader tries to re –
arrange its followers to take the shape of a circle. As we can observe, the shape of the swarm
resembles that of a circle but still needs refinement. Also, we can observe the yellow sub -swarm
formed around a sub -leader that emerges at the frontier of the swarm and pushes agents towards
the position of the leader agent.

Figure 34. Square shape

Figure 34 shows the final state of the simulation in which a shape forming leader tries to re –
arrange its followers to take the shape of a square. The degree of similarity is smaller than in the
previous simulation and requires more fine -tuning. One improvement could be to use a bigger
number of sub -leader so that more agents change their position at once such that there is the same
number of agents on the left and the right side of the leader.

ALEXANDRU -CATALIN CIOBANU
32

Figure 35. Triangle shape
The experimental results when trying to form a triangle shape is shown i n Figure 35. The
degree of similarity is even smaller than in the simulation with the square shape but there is room
for improvement.
One such improvement could be to split the swarm into several sub -swarms that form
smaller parts of the initial shape and after reaching a predefined degree of similarity, unite the
swarms into the initial one. Also, more tests need to be performed to test the influence of the sub –
leader on its followers, considering that the agent are also under the influence of the main sh ape
forming leader.

Figure 35. Circle, Square and Triangle shapes.

Figure 35 shows a scenario with three shape forming leaders each trying to form a different
geometrical shape. As we can observe the circle and triangle shapes are quite accurate while the
square needs more refinement, especially in the top -right corner.
5. Conclusions
I have implement ed four different types of leaders, the motivation leader, which has its
behavior based on the number of followers, the independent leader that behaves in a predefined
manner regard less of the number of followers, the collision avoidance leader that emerges to
prevent agents from colliding with obstacles and the shape forming leader that organizes its
followers so that the shape of the swarm imitates a simpl e geometric shape such as a circle, square
or triangle.

Emergence of a leader in a group
33
Results have shown that the leader based on the number of followers has a greater
influence over the total population of the swarm and can more easily adopt new agents and
maintain a stable swarm than the independent leader. Also the average number of agents that this
leader can control over the course of a simulation is considerably greater than in the case of the
independent leader.
The independent leader, in exchange, is better at controlling a bigg er percent of the swarm
surface. Simulations involving this type of leader are more dynamic, and even though , this leader
cannot maintain control over a big number of followers for a longer period of time, agents
influenced by him will change their behavio r more rapidly to match that of the leader, and such
traverse a bigger portion of the swarm surface.
Also, in simulations involving independent leaders instead of a tendency to form a bigger
stable swarm, the tendency is for a leader to split the sub -swarm of another leader and “steal” some
of the followers that are in its view range. By doing this the simulation maintains its dynamic
character because it rarely converges to a single stable swarm. The study of this behavior is intended
for further research.
In contrast with the previous mentioned idea, in simulations involving motivation -based
leaders the general tendency is to form a single big swarm containing multiple sub -swarms
controlled by different leaders. Because agent are not only influenced by the ir leader but also by
their neighbors, on a larger scale smaller sub -swarms are influenced by bigger sub -swarms. For
example when a small sub -swarm collides with a greater one, the first one will eventually adopt the
behavior of the greater one and both wi ll form a new swarm with two different leaders. The study of
this behavior is also intended as further research.
The mixed leader has a greater influence over the total population of the swarm and can
more easily adopt new agents and maintain a stable swar m than both the independent leader and
motivation leader, especially when big number of obstacles are present. In simulations with a small
number of the motivation leader performed better but when the number of obstacles was increased
the mixed leader clea rly separated from the previous types of leaders.
The collision avoidance leaders are best suited when the purpose is to avoid as many
obstacles as possible while also keeping a certain distance from them. This leader type can easily
guide itself and the s urrounding neighbor in order to avoid both types of obstacles, static and
dynamic.
The shape forming leader cannot influence it’s followers on its own to form a complex shape
and must rely on the help of sub -leaders. Results obtained have shown that one su bleader is able to
re-arange the swarm into simple shapes such as circles. Using more subleaders I managed to obtain
swarms that take the shape of squares and triangles.

ALEXANDRU -CATALIN CIOBANU
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Appendix A

C++ implementation of the algorithm by which a shape forming leader switches states.

__device__ bool changeLeaderState(AgentProperties* cap, AgentProperties* nap, Agent3DProperties * d_a3pv,
SimulationDetails* d_sd)
{
switch (cap ->state) {
case GatherFollowersState: nap ->state = SlowDownState; return true;
case MoveToSwarmCenterState:
if (isLeaderAtCenter(cap, nap, d_sd, SwarmCenter)) {
nap->state = SlowDownState;
return true;
}
break;
case MoveToEnvCenterState:
if (isLeaderAtCenter(cap, nap, d_sd, EnvCenter)) {
nap->state = SlowDownState;
return true;
}
break;
case ShapeMovementState:
if (isShapeMovementPerformed(cap, d_sd)) {
nap->state = SlowDownState;
return true;
}
break;
case SlowDownState:
if (!needsToSlowDownSwarm(cap, d_ sd)) {
nap->state = changeNewStateFromPrevState(cap, d_sd);
return true;
}
break;
case ChooseShapeState:
chooseNewLeaderShape(cap, nap, d_sd);
if (!isLeaderAtCenter(cap, nap, d_sd, EnvCenter)) {
nap->state = MoveToEnvCen terState;
}
else {
nap->state = ShapeMovementState;
}
break;
case CreateSubLeaders:
if (areThereAnySubLeaders(cap, d_sd)) {
nap->state = WaitForSubLeaders;
return true;
}
break;
case WaitForSubLeaders:
if (haveSubLeadersFinished(cap, nap, d_a3pv, d_sd) &&
swarmNotMoving(cap, nap, d_a3pv, d_sd)) {
if (isShapeFormed(cap, nap, d_a3pv, d_sd)) {
if (d_sd ->lp[cap ->leaderIndex].initialShape) {
nap->state = SwitchShape;
}
else {
nap->state = FinishMovement;
}
}

Emergence of a leader in a group
35
else {
nap->state = CreateSubLeaders;
}
return true;
}
break;
case SwitchShape:
nap->state = CreateSubLeaders;
return true;
break;
case FinishMovement:
nap->state = FinishMovement;
}
return false;
}

__device__ void performStateBehavior(AgentProperties* cap, AgentProperties* nap, Agent3DProperties* d_a3pv,
SimulationDetails* d_sd)
{
switch (cap ->state) {
case GatherFollowersSta te: setShapeFormingFollowers(cap, nap, d_a3pv, d_sd, false); break;
case SlowDownState: slowDownSwarm(cap, nap, d_sd); break;
case MoveToSwarmCenterState: moveLeaderToCenter(cap, nap, d_sd, SwarmCenter); break;
case MoveToEnvCenterState: moveLeaderTo Center(cap, nap, d_sd, EnvCenter); break;
case ShapeMovementState: moveInShape(cap, nap, d_sd); break;
case CreateSubLeaders:
case CreateSquareSubLeaders:
createSubleaders(cap, nap, d_a3pv, d_sd);
moveLeaderToCenter(cap, nap, d_sd, EnvCente r, true); break;
case WaitForSubLeaders:
case SwitchShape:
case FinishMovement:
moveLeaderToCenter(cap , nap, d_sd, EnvCenter, true); break;
}
}

__device__ void changeBehavior(AgentProperties* cap, AgentProperties* nap, Agent3DProperties* d_a 3pv,
SimulationDetails * d_sd)
{
if (cap ->isLeader && cap ->leaderType != ShapeSubLeader) {
if (checkNumberOfFollowers(cap, nap, d_sd, d_sd ->lp[cap ->leaderIndex].shapeAgents * 0.9, d_sd ->lp[cap –
>leaderIndex].shapeAgents)) {
if (!changeLeaderStat e(cap, nap, d_a3pv, d_sd)) {
performStateBehavior(cap, nap, d_a3pv, d_sd);
}
else {
setStateProperties(cap, nap, d_sd);
nap->previousState = cap ->state;
}

d_sd ->lp[cap ->leaderIndex].state = nap ->state;
d_sd ->lp[cap ->leaderIndex].previousState = cap ->state;
}
else {
nap->state = GatherFollowersState;
setShapeFormingFollowers(cap, nap, d_a3pv, d_sd, true);
}
}
}

ALEXANDRU -CATALIN CIOBANU
36
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