On Leader Election in Multi-Agent Control Systems

Theodor Borsche1 and Sid Ahmed Attia2

1. Control Systems Group, Berlin Institute of Technology, Berlin, GermanyE-mail: tborsche@gmx.net

2. Control Systems Group, Berlin Institute of Technology, Berlin, GermanyE-mail: attia@ieee.org

Abstract: In this contribution, we discuss the election of an optimal leader out of a network of agents described by first integratordynamics and running a consensus algorithm. The network of agents may be a group of autonomous robots or more generallycommunicating vehicles, and the target is, for instance, to move the formation to a new location. A leader is said to be optimal ifit leads to a controllable network, and minimizes a quadratic cost of reaching a target for all the other agents of the network. Inthe first part, controllability conditions and a decentralized way of checking them are discussed. We then study the correlationbetween the value of a quadratic cost function measuring the leader performance and the network properties. Strong correlationis found between closeness and degree centrality indices of the agents and the cost of achieving the assigned tasks. This allows usto run the optimal leader election process without a central authority and without the nodes having full knowledge of the networktopology.Key Words: Consensus algorithms, controllability, multi-agent control systems, leader election, networked control systems,

centrality indices.

1 Introduction

The use of multi agents control systems is often suggested to

have many advantages over single-agent systems, for exam-

ple the spacecraft interferometry concept which consists in

distributing the instrumentation over small space crafts can

achieve measurements unachievable by other techniques.

Furthermore, using several low-cost agents introduce redun-

dancy and is therefore more fault-tolerant and resilient than

having one powerful and expensive agent. Multi-agent con-

trol systems consist of an interconnection of simple systems

communicating with each other, cooperating to achieve a

global task. In many of these problems, finding a consensus

on the distributed information held by each node is a central

part of the solution e.g., [1].

In this work, we are interested in an autonomy problem in

multi-agent systems. An agent is called autonomous, if it

decides how to relate its measurements to control inputs in

such a way that global goals are achieved. Two kinds of ap-

proaches for autonomous networks have recently been dis-

cussed, leader-less and leader-based approaches. Leaderless

approaches usually use a consensus algorithm. Both find-

ing an average consensus (e.g. [2, 3, 4]) and a maximum

consensus[5] have been studied recently. Under consensus,

the agents will eventually agree on a certain quantity. In

leader-based approaches one elects one or more agents as

leader(s). The leader is then seen as a role and is theoret-

ically amenable to change based on the task requirements

and its complexity, i.e., an agent at a certain point in time

may be more suitable than all the others. A leader election

process has then to be carried out at the beginning of each

important task and/or after a failure of the currently elected

leader e.g., a failure caused by an external disturbance.

The resulting system may or may not be controllable. In-

deed, the work in [6] on controlled nearest-neighbour inter-

actions is the first to discuss the controllability issue. Gen-

eral conditions for controllability are given there.Further an-

alytical work on this topic has been carried out recently in

[7]. More detailed conditions for controllability are derived

there focusing on the eigenvectors of the system. In our con-

tribution, we extend this work. More exactly, we require the

election process to yield a controllable system. That is, we

sort out nodes under whose leadership the system would not

be controllable. For that purpose we need a sufficient and

necessary condition which is decentralizable at an accept-

able computational complexity. For the description of the

problem we use a similar modelling framework as the one in

[6].

We investigate the relationship between the network topol-

ogy as measured by centrality indices and the optimal leader

(to be defined precisely later). The result here is a statisti-

cal study. We find with high confidence a strong correlation

between both the degree and the closeness centrality of the

network and the value of a quadratic cost index measuring

the leader performance. Optimal leaders are those who are

well connected within the network. The usefulness of this re-

sult is in minimizing the computational burden associated to

the leader election process. Indeed, using a naive approach,

it would have been necessary to solve many instances of an

algebraic Riccati equation in a decentralized way. On the

contrary, we present a tool with which it is feasible to elect

a near-optimal agent as a leader using an efficient, local, de-

centralized algorithm and with no Riccati equations solvers

needed.

We first discuss which nodes are potential leaders (i.e. lead

to a controllable system) (§2). Next, we define a cost index

depending on the choice of leader and introduce centrality

indices (§3). In (§4) we correlate cost and centrality, and

find that both degree and closeness correlate with cost. Con-

clusions and perspectives are given in (§5).

2 Controllability of consensus networks

In this section we review some results concerning control-

lability of consensus networks and the possibility to check

them in a decentralized way. In many applications it is of

interest to manipulate the states or at least the center of the

swarm using a control input. We require, that the leader we

102978-1-4244-5182-1/10/$26.00 c©2010 IEEE

elect leads to a controllable system. If controllability is sat-

isfied, a wide range of controllers can then be used.

Before we discuss controllability, we first define the con-

trolled agreement problem where an input is added to a

network running a consensus algorithm. The tools used in

the description of a controlled consensus network are then

briefly described.

2.1 Controlled Agreement

Consider a network G with bidirectional communication

links having a weight of one. The associated Laplacian L(G)has the degree of each node on its diagonal, is symmetric and

all rows and columns sum up to zero.

1 2

3

4 u

Figure 1: An example graph G

Example 1: Assume the unweighted, bidirectional graph in

Figure 1. The Laplacian is

L =

1 −1 0 0−1 3 −1 −10 −1 2 −10 −1 −1 2

(1)

Two ways of describing the controlled agreement problem

have recently been introduced. Both split the group into fol-

lowers and leader(s). The total number of nodes is n, the

number of followers nf and the number of leaders nl. Vec-

tors and Matrices are in bold face, whereas Laplacians are in

calligraphic font

x =

[

xf

xl

]

L(G) =

[

Lf lfl

lTfl Ll

]

(2)

Considering Example 1, the network of followers (nodes

1,2,3), leader (node 4) and their interaction would be

Lf =

1 −1 0−1 3 −10 −1 2

Ll =[

2]

lfl =

0−1−1

(3)

In [7], the network of followers Lf ∈ Rnf×nf is used as sys-

tem matrix A, whereas the interaction between the leader(s)

and followers lfl ∈ Rnf×nl is used as input vector B. The

system has therefore a dimension of n − nl, the follower

nodes are the states and the leader node(s) the input(s).

A = −Lf

x = xf

B = −lfl

u = xl

(4)

This modelling framework is used next.

In [8] the negative Laplacian is chosen to be the system ma-

trix A′, but with all rows corresponding to leaders being ze-

roed out. The input vector B′ has ones on these rows, and

zeros for the followers. All agents are states of the system,

the system has dimension n.

A′ =

[

−Lf −lfl

0 0

]

B′ =

[

01

]

(5)

This representation can be used if one has single-integrator

dynamics, that is if one can set the velocity of an agent rather

than the position.

In one representation the leader’s position is the control in-

put, while in the other the leaders velocity is controlled by

an additional input. It can be shown that both representations

are equivalent in terms of controllability, that is for control-

lability it is irrelevant which representation is chosen.

Proposition 2.1. A system with a leader, which is control-

lable when using the first representation (direct control of an

agent (4)) is also controllable when using the second rep-

resentation (control of the velocity of an agent (5)). Un-

controllable systems have the same rank deficiency in both

representations.

Proof. The system (A,B) is controllable, nl is the number

of leaders. The second system has the representation

A′ =

[

A B

0 0

]

B′ =

[

0

1

]

(6)

The identity matrix 1 has dimension Rnl×nl . The controlla-

bility matrix C′ is

C′ =

[

B′

A′B

′A

′A

′B

′ · · ·]

=

=

[

0 B AB · · ·

1 0 0 · · ·

]

=

[

0 C

1 0

]

(7)

Obviously, C′ has dimension dim(C) + nl, but also rank

rank(C) + nl. Therefore the rank-criterion gives identical

results for both representations.

2.2 Controllability test using eigenvectors

After having defined the controlled agreement, the next step

is to look for efficient and easy-to-use decentralizable cri-

teria to find all potential leaders of a network. We call a

node a potential leader, if the system is controllable with

only that node being connected to the control input. Even

though one could simply compute the controllability matrix

and check the rank criterion, doing so with local laws and

local information would take up an unacceptable amount of

time – especially since the system is dependent on the choice

of leader, which makes it necessary to carry out this compu-

tation for every single node. Considering the special set of

parameters in the consensus algorithm, it is also possible to

conclude on the controllability using the eigenvectors of the

system matrix. The following discussion concerning single-

leader networks is mainly based on the recent work in [7].

We slightly extend that work and show that the most promis-

ing proposition is not only necessary, but also sufficient.

The modelling framework defined in (4) is used. As the sys-

tem matrices differ depending on the choice of leader, an

index i is introduced to distinguish between systems under

different leadership.

Theorem 2.2. [7, Corollary 5.2] In the case of a single

leader, the network is controllable if and only if none of the

eigenvectors of Ai is orthogonal to 1.

To check this condition for every node, n eigenproblems

have to be solved. This condition is important and is needed

to prove Proposition 2.3.

2010 Chinese Control and Decision Conference 103

Next, consider the eigenvectors of the Laplacian ν{L}. It

was shown in [7, Proposition 5.4] that if the system is un-

controllable, there is a zero entry in (at least) one of the

eigenvectors of the Laplacian on the row corresponding to

the leader. On the inverse, a certain node chosen as leader

leads to a controllable system if there is no zero entry on the

corresponding row of any ν{L}.

It can easily be shown, that this condition is not only neces-

sary (zero entry ⇐ uncontrollable) but also sufficient (zero

entry ⇒ uncontrollable), giving on the inverse that a system

is controllable if and only if there is no zero entry on the

corresponding eigenvector row.

Proposition 2.3. Suppose there is an eigenvector of L(G)with a zero entry on the i-th row. Then the system (4) with

agent i as single leader is uncontrollable.

Proof. All eigenvectors of L(G) are orthogonal to 1, except

for the eigenvector corresponding to λ = 0 (this eigenvector

is exactly ν = α1 and has therefore no zero entry). Assume

an eigenvector with a zero component corresponding to the

leader i. If one removes the row containing the zero entry,

the remaining vector (ν⊥) will still sum up to zero.

[

1T 1]

[

ν⊥0

]

= 1T ν⊥ = 0 (8)

One can write

L(G)

[

ν⊥0

]

=

[

Ai lfl,i

lTfl,i di

] [

ν⊥0

]

=

=

[

Aiν⊥lTfl,iν⊥

]

= λ{L(G)},⊥

[

ν⊥0

] (9)

It is obvious, that lTfl,iν⊥ = 0 and also

Aiν⊥ = λ{L(G)},⊥ν⊥ (10)

which proofs that ν⊥ is also an eigenvector of Ai. Be re-

minded that the system (Ai,Bi) is uncontrollable exactly

if there exists an eigenvector ν{A} which is orthogonal to

1. Such an eigenvector exists if there exists an eigenvector

of L(G) with a zero component corresponding to the node

i.

We have obtained a necessary and sufficient condition which

returns all possible leaders by solving just one eigenprob-

lem. Other conditions so far required the solution of n eigen-

problems (e.g. 2.2), or have been only sufficient (returning

possible leaders only, but maybe not all of them).

A similar statement can be made about the eigenvalues of L.

If there is an eigenvalue with geometric multiplicity of more

than one, the eigenvectors span an eigenspace and are not

exactly defined. They can be chosen in such a way that there

is a zero entry on any one row of one of the eigenvectors.

Therefore no single leader can control the system. When

choosing potential leaders, this result has to be kept in mind.

Again, this condition concerns the complete graph instead

of the network of followers and has therefore to be checked

once only.

2.3 Decentralized eigenvector computation, Observ-

ability

While studying a different problem, an algorithm for solving

the eigenvector problem in a decentralized fashion was de-

vised by Kempe and McSherry[9]. That algorithm is based

on the QR-algorithm. It is also possible to decentralize the

Lanczos-algorithm. However, both approaches require mul-

tiple time-consuming consensus operations.

What is more, it can be proven that the system is observable

exactly if it is controllable if the output matrix and input ma-

trix are chosen identically (C = BT ).

3 Cost index and centrality indices

In §2 issues of controllability of multi-agent control net-

works were discussed. Potential leaders have been identi-

fied. Our aim is to choose the optimal leader from a list

of potential leaders. First, we have to give a criterion for

optimality. In §4 that criterion will be correlated to graph

properties, which will be introduced later in this section.

3.1 Cost index

To assess the energy needed by the system to transfer state

~x0 to ~xf the following cost index is used (for simplicity, in

most cases xf = 0 is assumed). This index also includes

the states in the cost functional. Assuming no constraint on

the velocity of the agents, especially on the velocity of the

leader, we can describe the system by linear dynamics. Fur-

ther assuming no constraint on the time allowed for achiev-

ing the control target, the infinite time cost functional can be

used.

J(x(t0),u(·)) =

∫ ∞

0

(uT Ru + xT Qx)dt (11)

The matrices R and Q are both symmetric. R is positive

definite, and Q is positive semi-definite. The control law

minimizing the cost functional is given by

u(t) = −R−1BT Px(t) (12)

whereas P is found by solving the matrix Riccati

equation[10]

AT P + PA − PBR−1BT P + Q = 0 (13)

The agent who minimizes the cost functional J shall be

leader.

It may be possible to implement the solution of the Riccati

equation in a decentralized manner. However, the compu-

tational burden would high, and running complex decentral-

ized algorithms (e.g. eigenvector algorithms) has shown to

be very time consuming. Within the network it is therefore

not realistic to solve the Riccati equation for every potential

leader to find the cost index. A correlation between easy to

check graph properties and the cost functional will therefore

be explored in § 4.

3.2 Centrality indices

We now introduce centrality indices, which are needed later

on for the correlation analysis to the cost functional just de-

fined. We also discuss how these may be computed in a de-

centralized way. Many different definitions of centrality are

104 2010 Chinese Control and Decision Conference

known. The most important of these for the work problem

under study are presented in the next paragraphs.

Different definitions for betweenness centrality have been

proposed over the past. The most straight forward is short-

est path betweenness[11]. It counts the number of shortest

paths between any two nodes going through a third node i.

Let nσst be the number of shortest paths between s and t, and

nσst(i) be the number of these paths going through node i.

The betweenness centrality cB is

cB(i) =∑

s 6=i

∑

t6=i,s

nσst(i)

nσst

(14)

It is a measure of the flow of information going through this

node. A node with a high betweenness centrality has a high

level of influence on the information flow in the network[12].

The degree centrality has a very simple definition, taking the

degree of each node as its centrality index. Usually, this

value is normalized to the maximum possible degree, which

is n − 1 (when self-loops do not count). It was decided to

take the inverse of the degree as centrality index to have a

positive correlation between cost and degree.

cD(i) =1

di

(15)

Eccentricity centrality is a simple measure giving the longest

shortest path between a certain node and any other node in

the network. If the node is providing information for the

network, minimal eccentricity guarantees minimal time until

every node receives a certain piece of information sent by the

center node. lσit is the length of the shortest path between i

and t, and V is the set of nodes. The eccentricity of agent i

is defined as[12]

cE(i) = maxt∈V

lσit (16)

The closeness centrality is related to the eccentricity, but its

definition is slightly more complex. For the closeness all

shortest paths starting at node i are calculated and the aver-

age of that is taken. A low closeness index means a node is

on the average very close to all other nodes in the network.

The closeness cC of agent i is defined as

cC(i) =

∑

t∈V lσit

n − 1(17)

As opposed to betweenness, degree, closeness and eccentric-

ity describe a high centrality if they assume a low value[12].

A few methods consider the dominating eigenvector for a

centrality measure, which can be seen as a measure tak-

ing into account the importance of the neighbour nodes as

well. Such centrality indices have been very successful in

many application, the most prominent algorithm using this

method is Google’s PageRank algorithm. We consider sym-

metric systems (bidirectional communication) only. It can be

proven, that in this case the feedback centrality is equivalent

to the degree centrality.

The reason to compare centrality indices with the cost index

is to find an easy way to assess optimality. Correlation will

be analysed in § 4. First, the centrality has to be computed in

a decentralized manner. Except for degree, which is trivial,

all centrality indices used are based on shortest paths.

Shortest paths are also needed in routing problems as they

arise in large networks. The Bellman-Ford algorithm[13]

can be used to compute shortest paths between any two

points. The routing information protocol uses this for com-

puting shortest paths between any two routers in a decentral-

ized manner [14]. This may easily be adapted to multi-agent

networks. The information needed to compute betweenness

may be gathered in a similar way, however it would be nec-

essary to keep information about the vertices which form the

shortest paths, and one would also have to keep track of all

shortest paths between two nodes.

Decentralized shortest path algorithms are considerably

more efficient than the solution of the eigenproblem. Even

though information has to propagate through the network, it

takes at most n−1 time steps to find the length of all shortest

paths between any two nodes. This is on the order of finding

an average consensus. Also, no complex computation has to

be performed at any node.

4 Statistical analysis of cost and centrality

We want to elect an optimal leader. In §3 we defined as op-

timal the one which minimizes a linear-quadratic cost func-

tional. However, we cannot compute the cost functional for

every node in a network. We therefore search for a correla-

tion between cost and graph properties. The graph properties

we discuss are centrality indices.

In this section we first introduce a few statistics tools needed

to assess the correlation. Mainly, we discuss how to compare

correlation coefficients from different networks with each

other. In §4.2 we then discuss the results for each of the

introduced centralities and decide which one is suitable to

serve as an indicator for optimal leadership.

4.1 Correlation coefficients from different experiments

When assessing the correlation between a centrality index

and the performance index for a single network, only the

potential leader nodes1 are regarded. In each network, we

compute the correlation between each centrality index and

the cost using Pearson’s correlation coefficient

r(x, y) =

∑n

i=1(xi − x)(yi − y)

√

∑n

i=1(xi − x)2

√

∑n

i=1(yi − y)2

(18)

A single network has only a few nodes and the data base is

therefore very small. It would be considerably more robust

to simulate multiple networks and to compute a common

correlation coefficient. To calculate a common correlation

coefficient rcom one may not simply take all data points from

all networks and calculate the correlation, nor is it legitimate

to take the average or mean of all correlation coefficients.

Rather, one has to calculate all correlation coefficients ri for

each simulation, check for a significant distribution and then

use equation (19) to compute rcom. One may either find a

χ2 distribution or a normal distribution of the ri. If a χ2-

distribution is found, the correlation coefficients are homo-

geneous. This means, that all experiments/ networks adhere

to the same ρ, that is that all networks have the exact same

correlation between the discussed centrality and cost. If on

1The simulations have also been performed taking into account all nodes

in the network. Preliminary results show similar results in both cases

2010 Chinese Control and Decision Conference 105

the other hand a normal distribution is found, one may com-

pute a mean correlation and standard deviation. Even though

every network may have a different correlation, it is still pos-

sible to predict with a certain probability the correlation for

an unknown network to be within the confidence interval.

Assume k simulations, and let ri be the correlation in exper-

iment i and ni the number of data points in that experiment.

The common correlation may then be computed as [15]

rcom =

∑k

i=1(ni − 1)ri∑k

i=1(ni − 1)(19)

Correlation coefficients are defined to be on an interval be-

tween -1 and 1. Distributions on the other hand are un-

bounded. To check for a χ2 or normal distribution, the cor-

relation coefficients have to be normalized [15, 16].

The null hypothesis to check for homogeneity of the ri is

H0 : ρ1 = · · · = ρk = ρ, with ρ being a hypothetical pa-

rameter. The χ2-test with bound χ2k;α in this case is defined

as[15]

χ̂2 =k∑

i=1

(ni − 3)(zi −△

z )2 ≤ χ2k;α (20)

with the zi being the mentioned normalized correlation co-

efficients and△

z being an estimation. Tables for the bound

χ2k;α can be found in any good statistics book (e.g. [15]). If

χ2 is smaller than the bound, the null hypothesis can not be

rejected, meaning with a probability of 1 − α that there is

a significant connection between the correlation coefficients

and a common correlation coefficient may be calculated.

There are many different approaches to check for a normal

distribution. The one used is the Lilliefors test, which can

for any kind of normal distribution[17]. The implementation

of Lilliefors test in the statistics toolbox of MatLab is used

to check for a normal distribution if the prior check using a

χ2 distribution is negative.

4.2 Common results for a large number of random net-

works

For the final analysis, 130 experiments have been made.

Later it is shown that this number is sufficient to achieve con-

fidence in the results. Random networks with a size between

6 and 162 nodes and an average degree between 2.2 and 5.3

have been used. As the cost index is dependent on the start-

ing position, an average cost for different random starting

positions has to be computed. 1000 different starting posi-

tions between -1 and +1 have been used for each network.

The states were then transferred to the origin by the leader

using a LQ-regulator. The costfunctional was computed for

the resulting state trajectories. An exhaustive discussion of

each single experiment would be time consuming and only

of limited use. In fact a trend supported by all simulations is

sought. The results of this analysis are presented in Table 1.

The first two columns present the total number of samples

and experiments, which is equivalent to the total number of

potential leaders and networks simulated. Eccentricity has

a lower network count, as NaN results of experiments have

been discarded. Such results are produced if a certain cen-

trality index is the same for each agent in the network, and

2As shown in [6] and also experienced during our simulations it is very

difficult to find larger networks controllable by a single leader

such is common for eccentricity. The value of pχ2 gives the

probability that the null hypotheses of a homogeneous dis-

tribution of the correlation coefficients can be refuted. If pχ2

is bigger than 1 − α, there is no homogeneous distribution.

A low probability is therefore desired. pL is derived from

the Lilliefors test and yields the probability that the null hy-

potheses of normal distribution must be refuted. High confi-

dence and low α can be assumed. The last four columns give

the actual findings, namely the 5% confidence interval rmin

to rmax and estimated rcom for each centrality index, and

the confidence pr that these results provide a correct trend.

A value of 1 gives 100% confidence up to the calculation ac-

curacy. This also justifies, why not more than 130 simulation

were carried out, a higher number would not have been able

to increase the confidence but would have made the analysis

harder due to rounding and sample size effects.

The reference index was created to test the method of anal-

ysis. No graph related properties are used for its calcula-

tion, instead the reference is only a random integer assigned

to each agent. ρ is known in this case to be exactly zero.

Therefore, a homogeneous χ2 distribution and no correla-

tion to the cost is expected. Both expectations are met. Let

it be further noted, that the confidence in a correlation is be-

low 88%, which is less than 1 − α. The null hypotheses

(H0 : ρ = 0) can therefore not be classified as negative, no

correlation can be assumed. The confidence interval reaches

from the negative to the positive and includes the hypotheti-

cal ρ = 0.

Centrality, betweenness, the adjusted betweenness and the

reference index all pass the χ2 test. For each of these indices

a common correlation coefficient can be computed. How-

ever, degree and eccentricity do not pass the χ2-test, that is

the null hypothesis of homogeneity has to be rejected. The

Lilliefors test on the other side clearly finds a normal distri-

bution in the z-transformed degree correlation coefficients.

That is, the correlation coefficients between degree and cost

have a normal distribution and a mean correlation coefficient

r can be computed. Eccentricity fails both distribution tests.

Therefore, neither rcom nor r may be calculated for eccen-

tricity, which is why no values for the mean correlation and

confidence interval are given in Table 1.

One can now make statements about unknown networks. For

centrality and betweenness one can predict the correlation

to cost (using Table 1, with a certain uncertainty), as it has

been shown that every network adheres to the same ρ. The

correlation of these two indices to cost is the same in every

network. Next, the correlation between degree and cost in

an unknown network can only be estimated. With a 95%

probability does it lie within the confidence interval given

in the table, but it will be different for each network! No

prediction at all can be made for correlation of eccentricity

and cost, as the correlation coefficients are not even normally

distributed.

For the leader election process, degree is the index that

should be used first. It has numerous advantages. Not only

has it the highest correlation to cost, it is easy to compute.

Any node knows at any time how many connections it has to

other nodes. One simply has to run a maximum consensus[5]

to decide what is the highest degree (lowest degree index) in

the network. However, there may be multiple nodes hav-

106 2010 Chinese Control and Decision Conference

samples homogeneity common correlation

n k pχ2 pL rmin rcom rmax pr

closeness 1072 130 0.3307 – 0.88492 0.90016 0.91347 1

eccentricity 772 90 >0.9999 >0.9999 – – – –

betweenness 1072 130 0.0492 – -0.61191 -0.65672 -0.69733 1

degree 1072 130 >0.9999 < 0.5 0.92289 0.93328* 0.94231 1

reference 1072 130 0.0321 0.2457 -0.037636 0.037381 0.11198 0.8743

Table 1: Combined correlation coefficients of all simulated networks.(*) As degree is not χ2 but normal distributed, this is the mean r.

ing maximum degree. One then can decide between those

nodes using closeness centrality. Closeness has also a very

strong correlation to cost and is easy to assess decentralized

and has low ambiguity. Should two nodes have same degree

and closeness, they have to be assumed to be equally good at

leading the network. Betweenness and eccentricity have to

be discarded. Betweenness because it performs worse than

closeness and is considerably more difficult to assess. Ec-

centricity because it is most ambiguous, with networks often

having same eccentricity at every node.

5 Conclusion

This paper discussed many aspects of leader election. Poten-

tial single leaders have been identified using an eigenvector

condition. A linear-quadratic cost index and centrality in-

dices were defined. Using statistical methods it was shown,

that the node with highest degree and closeness centrality is

the most cost-optimal leader. Average cost for reaching a

control target may be heavily reduced by choosing the opti-

mal leader over the worst case leader.

Further work could focus on decentralized controller design,

on larger networks which would require multiple leaders and

on systems with weighted, directed and/or dynamic links.

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