Controllability of homogeneous single-leader networks

Abstract

This paper addresses an aspect of controllability in a single-leader network when the agents are homogeneous. In such a network, indices are not assigned to the individual agents and controllability, which is typically a point to point property, now becomes a point to set property, where the set consists of all permutations of the target point. Agent homogeneity allows for choice of the optimal target point permutation that minimizes the distance to the system's reachable subspace, which we show is equivalent to finding a minimum sum-of-squares clustering with constraints on the cluster sizes. However, finding the optimal permutation is NP-hard. Methods are presented to find suboptimal permutations in the general case and the optimal permutation when the agent positions are 1-D.

Full text

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Controllability of Homogeneous Single-Leader Networks
Philip Twu, Magnus Egerstedt, and Simone Martini
Abstract—This paper addresses an aspect of controllability
in a single-leader network when the agents are homogeneous. In
such a network, indices are not assigned to the individual agents
and controllability, which is typically a point to point property,
now becomes a point to set property, where the set consists of all
permutations of the target point. Agent homogeneity allows for
choice of the optimal target point permutation that minimizes
the distance to the system’s reachable subspace, which we show
is equivalent to finding a minimum sum-of-squares clustering
with constraints on the cluster sizes. However, finding the
optimal permutation is NP-hard. Methods are presented to find
suboptimal permutations in the general case and the optimal
permutation when the agent positions are 1-D.
I. INTRODUCTION
Research in multi-agent systems has mainly focused on
designing decentralized controllers that allow for agents
to autonomously achieve global goals, such as reaching
consensus (e.g. [1], [2], [3], [4], [5]) or achieving formations
(e.g. [6], [7]). However, in many of the intended applications
for multi-agent systems, such as search and rescue, it is more
likely that agents will be working closely with humans as
opposed to acting completely autonomously. The research
problem that arises involves understanding how a human
controller can affect and interact with an entire network of
agents, without directly communicating with each of them.
In this paper, we focus on when a human takes control
of a single agent within the network, while all other agents
are executing a nearest neighbor averaging rule. The agents
thus form a single-leader network where the positions of
all follower agents within the network can be affected by
controlling the leader agent (e.g. [8], [9], [10], [11]). We
investigate the controllability problem when the network
consists of homogeneous agents. Since agents are inter-
changeable, it does not matter which agent goes where, as
long as there is an agent at each of the target locations.
Controllability is now no longer a point to point property
of the system, but instead becomes a point to set property,
where the set consists of all permutations of the target point.
Homogeneity in the system lets us choose a permutation
of the target point that is closest to the system’s reachable
subspace. However, we show that finding such an optimal
permutation is NP-hard.
The outline of this paper is as follows: Section 2 presents
system dynamics for a single-leader network. Section 3
Philip
Electrical and Computer Engineering, Georgia Institute of Technology,
Atlanta, GA30312, USA,
magnus@ece.gatech.edu
SimoneMartiniiswith
Center,“E.Piaggio”,University
s.martini@ingegneria.pisa.it
TwuandMagnusEgerstedtarewith theSchool of
Email:
ptwu@gatech.edu,
TheInterdepartmental
ofPisa,
Research
Email:Italy,
reviews previous work on the controllability of a single-
leader network and discusses the problem of controllability
in a homogeneous single-leader network, as well as the
computational complexity of finding the optimal permutation
of a target point that is closest to the system’s reachable
subspace. Finally, Section 4 explores methods to find the
optimal permutation of a target point both in the general
case and the special case of 1-D agents.
II. SYSTEM DYNAMICS
Consider a team of N +1 agents, numbered 1,...,N +1,
with positions xi∈ Rn, for i = 1,...,N + 1, respectively.
Let the information flow amongst agents in the network
be represented by an undirected static graph G = (V,E),
where V = {v1,...,vN+1} and (vi,vj) ∈ E if and only if
information flows between agents i and j. The neighbor set
Ni = {j|(vi,vj) ∈ E} represents the set of indices of all
agents that share an edge with agent i in E.
Suppose the agents form a single-leader network where all
followers execute a nearest neighbor averaging rule, while
the leader’s position is the external input u. Without loss
of generality, assume the N + 1th agent is the leader while
agents 1,...,N are followers. The agents’ dynamics are:



The adjacency matrix of G is the (N + 1) × (N + 1)
symmetric matrix A where ai,j, the element in the ith row
and jth column, is given by
˙ xi
=
−
?
j∈Ni
(xi− xj) ,∀i = 1,...,N
xN+1
=u.
(1)
ai,j=
?
1
0
(vi,vj) ∈ E
otherwise.
(2)
The degree matrix of the graph G is a (N + 1) × (N + 1)
diagonal matrix ∆, where
∆i,j=



|Ni|
i = j
0 otherwise.
(3)
Finally, the graph Laplacian matrix L is given by
L = ∆ − A,
(4)
which can be decomposed into blocks
L =
?Lf
ℓT
ℓ
ξ
?
,
(5)
where the dimension of Lfis N ×N, ℓ is N ×1, and ξ ∈ R.
Let x =?xT
1,...,xT
N
?T∈ RNnbe the concatenated posi-
tions of all follower agents, where xj= [xj,1,...,xj,n]T∈

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Rn, for j = 1,...,N. Define di : RNn→ RN, for
i = 1,...,n, as a function that returns the positions
of the N follower agents along the ith dimension, i.e.,
di(x) = [x1,i,...,xN,i]T. The dynamics of the follower
agents’ positions along the ith dimension are given by
di(˙ x) = −Lfdi(x) − ℓui,
(6)
where uiis the ith element of u. Since the dynamics along
each dimension are decoupled, the dynamics of x can be
written using ⊗, the Kronecker product, as the linear system
˙ x = −(Lf⊗ In)x − (ℓ ⊗ In)u,
(7)
where Inis the n × n identity matrix.
III. CONTROLLABILITY IN HOMOGENEOUS
SINGLE-LEADER NETWORKS
Many applications of multi-agent systems, such as the de-
ployment of mobile sensor networks, require a large number
of homogeneous agents that are initially in close proximity
to spread out and reach a desired target point (positions of
the follower agents). Analyzing the controllability of single-
leader networks allows us to understand, for a given network
topology, the range of target points that can be achieved from
a human user steering the network through the leader.
A. Controllability of Single-Leader Networks
In a single-leader network, the dynamics of the follower
agents along each dimension are decoupled and given by
the linear system (6). Treating each dimension separately,
the reachable subspace is given by the range space of the
controllability Grammian Γ, where
Γ =
?
−ℓ Lfℓ ... (−Lf)N−1(−ℓ)
?
.
(8)
The reachable subspace of a single-leader network was found
in [10] to have an interesting interpretation involving the
graph topology. Before stating this result, we must first
review some definitions from [10], [12], and [13].
Definition 3.1: Givena vertex
{C1,...,CM} be a partition of V , where Ci ⊂ V for
i = 1,...,M, C1∪ ... ∪ CM = V , and Ci∩ Cj = ∅
when i ?= j. We will call each Cia cell.
Definition 3.2: Given a vertex v and a cell C, the node
to cell degree gives the number of vertices in cell C
that share an edge with v, and is given by deg (v,C) =
card({v′∈ C|(v,v′) ∈ E}).
For example, in Figure 1(b), C1,C2,C3are cells that parti-
tion the vertices in the network and deg (v2,C3) = 3.
Definition 3.3: An external equitable partition (EEP) is
a partition Π such that ∀C ∈ Π,v ∈ C and v′∈ C ⇒
deg (v,C′) = deg (v′,C′) ∀C′∈ Π − {C}.
Definition 3.4: An EEP is leader-invariant if the vertex
corresponding to the leader agent belongs to its own cell.
Definition 3.5: A leader-invariant EEP is maximal if it has
the fewest number of cells in any leader-invariant EEP.
For example, Figure 1(a) is a leader-invariant EEP, while
Figure 1(b) is a maximal leader-invariant EEP.
set
V ,let
Π=
V4
V5
V6
V2
V3
V1
(a) The trivial leader-invariant EEP.
V4
V5
V6
V2
V3
V1
C1
C2
C3
(b) The maximal leader-invariant EEP.
Fig. 1.
where V1 is the vertex for the leader agent. (a) shows the trivial leader-
invariant EEP, while (b) gives the maximal leader-invariant EEP. Since the
two partitions are different, the network is not completely controllable.
Two examples of leader-invariant EEPs of a single-leader network,
With these definitions, we now state a result relating con-
trollability of a network to its maximal leader-invariant EEP.
In [10] it was shown that follower agents within the same cell
of the maximal leader-invariant EEP asymptotically approach
the centroid of agents within that cell. That result is stated
again below for easy reference.
Theorem 3.1: Assume a single-leader network has an in-
formation flow graph with a maximal leader-invariant EEP of
k cells, numbered 1,...,k, that do not contain leader agents.
In [10] it was shown that the range space of the controlla-
bility Grammian for (6), the follower agent dynamics along
any dimension, is given by
R(Γ) = span{w1,...,wk}
(9)
where wi∈ RNand wi,j, the jth element of wi, is defined
by
?
From the above theorem, it is seen that a network is com-
pletely controllable and can reach any target point only when
the maximal leader-invariant EEP is trivial, i.e., each follower
agent is contained within its own cell. When multiple agents
are within the same cell, they asymptotically approach each
other. Referring back to Figure 1, we see that the trivial
leader-invariant EEP is not the same as the maximal leader-
invariant EEP so the network is not completely controllable.
wi,j=
1
0
vj∈ cell i
otherwise.
(10)
B. Optimally Reachable Target Points
The set of reachable target points in a network is restricted
by the choice of leader agent and the network topology. In

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many situations, a tolerable margin of error may be allowed
between where the agents are located and where the user
desires them to be. Thus, we will investigate how close a
network can get to a given target point.
We will model agents initially being at close proximity
to one another by assuming zero initial conditions on the
positions of the follower agents in the network. Such an
assumption greatly simplifies the analysis of controllability,
since agents within the same cell start and stay together.
Assumption 3.1: The agent positions, x, are initially 0.
With zero initial conditions on x, a target point of follower
agents xT∈ RNnis reachable if and only if di(xT) ∈ R(Γ),
for i = 1,...,n. Depending on the network topology,
the system of follower agents is not always completely
controllable and so may not be able to reach a target point
perfectly. Therefore, for a given xT, it is useful to find the
optimal reachable target point x∗(xT), which minimizes
J (xT,x) = ||xT− x||2=
n
?
i=1
||di(xT) − di(x)||2, (11)
such that di(x) ∈ R(Γ), for i = 1,...,n.
Theorem 3.2: For a given xT, the optimal reachable
x∗(xT), where di(x∗(xT)) ∈ R(Γ), for i = 1,...,n, that
minimizes (11) is
x∗(xT) =?WWT⊗ In
W =
||w1||...
?xT,
?
(12)
where
?
w1
wk
||wk||
,
(13)
and w1,...,wkare as given in (9).
Proof: Minimizing ||xT− x||2is equivalent to min-
imizing ||di(xT) − di(x)|| individually for each i because
the dynamics along each dimension are decoupled. The
Hilbert Projection Theorem says that the optimal reachable
di(x∗(xT)) ∈ R(Γ) that minimizes ||di(xT)−di(x)|| is the
projection of di(xT) onto the subspace R(Γ). The reachable
subspace R(Γ) is spanned by vectors w1,...,wk as given
in (9). Therefore, the optimal choice of di(x) is given by
di(x∗(xT)) =
k
?
j=1
wT
jdi(xT)
||wj||2
wj.
(14)
Letting W be as defined in (13), (14) can be rewritten as
di(x∗(xT)) = WWTdi(xT).
Since this holds for i = 1,...,n, x∗(xT) is written as (12).
The expression determined for x∗(xT) has an interesting
and intuitive interpretation that will be useful later. Define
gi : RNn→ Rn, for i = 1,...,N, as a function that
returns the n dimensional coordinates of the ith agent, i.e.,
gi(x) = xi. Further, define m : {1,...,N} → {1,...,k}
as a function that takes in an index of a follower agent and
returns the index of the cell it belongs to in the maximal
leader-invariant EEP. Let m−1be the inverse image function
that takes in a cell number and returns a set containing the
indices of the follower agents that belong to that cell.
Corollary 3.1: For a given xTand corresponding x∗(xT)
that minimizes (11),
gi(x∗(xT)) =
1
|m−1(m(i))|
?
j∈m−1(m(i))
gj(xT).
(15)
In other words, agent positions in cell j of x∗(xT) are all
located at the centroid of the agent positions in cell j of xT.
Proof: From the definition of vectors wjgiven in (10),
the expression for di(x∗(xT)) in (14) can be interpreted.
The numerator term of each summand wT
along the ith dimension of the positions of all agents in cell
j in the target point. That quantity is then divided by ||wj||2,
which is the number of agents in cell j, so the result is the
centroid along the ith dimension of all agent positions in
cell j of xT. Finally, that value is multiplied to wj, thereby
assigning it to the ith dimensional component of the positions
of all agents in cell j of x∗(xT). Since this is true for along
all dimensions i = 1,...,n, the result is that agent positions
in cell j of x∗(xT) are all located at the centroid of the
agent positions in cell j of xT.
With an expression for x∗(xT), it is now possible to
compute the minimum cost associated with any given xT.
Corollary 3.2: For a given xTand corresponding x∗(xT),
the minimum cost J∗(xT) = J (xT,x∗(xT)) is
jdi(xT) is the sum
J∗(xT) = xT
Proof: Plugging in the expression (12) for x∗into the
cost (11), expanding the norm-squared, and noticing that the
term INn− WWT⊗ Inis symmetric results in
T
?INn− WWT⊗ In
?xT.
(16)
J∗(xT) = xT
T
?INn− WWT⊗ In
?2xT.
Expanding the squared term and making use of the fact that
the columns of W are orthonormal results in (16).
C. Homogeneous Networks
Equation (16) represents the cost associated with the
closest that a particular single-leader network can reach a
target point xT. Notice that xT represents the specification
to have each agent i be located at gi(xT), for i = 1,...,N.
However, in a network of homogeneous agents, the roles of
agents are interchangeable and so it makes no difference if
instead we ask agent i to go to gj(xT) and agent j to go
to gi(xT). In fact, any permutation of the agent indices in
xT to some (P ⊗ In)xT, where P is a permutation matrix,
ends up specifying the same target point if all we care about
is the presence of an agent at each of the target positions.
However, the new target point may be “more reachable” in
the sense that J∗((P ⊗ In)xT) < J∗(xT).
Example 3.1: Consider a 1-D single-leader network with
N = 3 follower agents as illustrated in Figure 2(a), where
agents 1 and 2 are in cell 1 and agent 3 is in cell 2 of
the maximal leader-invariant EEP of the network. The range
space of the controllability Grammian is thus
R(Γ) = {w1,w2} =





1
1
0

,


0
0
1




.

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V1
V2
V3
V0
C2
C1
C0
(a) The single-leader network used in Example
3.1, where the leader agent’s vertex is V0.
X3
X1,X2
XT,1
XT,2
XT,3
(b) The closest the agents in the network (circles) can
reach target point xT (X’s).
X3
X1,X2
XT,3
XT,1
XT,2
(c) The closest the agents in the network (circles) can
reach the permuted target points PxT (X’s).
Fig. 2.
given in (a). (b) shows the closest the agents in the network can reach
xT= [1 9 10]T, while (c) shows the closest the agents can reach PxT=
[9 10 1]T. Notice that the PxT results in an error less than xT and so
PxT is the better specification of the target point.
The topology of the single-leader network in Example 3.1 is
Let xT=?
1910
?Tand


P =
0
0
1
1
0
0
0
1
0

,
then J∗(xT) = 32, while J∗(PxT) = 0.5. Therefore,
as illustrated in Figures 2(b) and 2(c), PxT is a better
specification of the target point than xT.
To exploit the advantage of homogeneity towards a net-
work’s controllability, it is necessary to solve the following
problem that will be the focus of the rest of this paper:
Problem 3.1: Given a single-leader network and target
point of follower agents xT. Let P be the set of all N×N
permutation matrices. Find P∗such that
P∗= argmin
P∈P
J∗((P ⊗ In)xT).
(17)
Calculating P∗can be viewed as finding the optimal
specification of a target point. A more intuitive interpretation
of finding P∗is to treat it as a constrained clustering problem
on the N target positions g1(xT),...,gN(xT).
Definition 3.6: A multiset is a collection of objects in
which order is ignored, but where multiplicity is significant.
For example, M1 = {1,3,4}, M2 = {1,3,4,4}, and
M3= {1,4,3,4} are all multisets. M2= M3, but M1?= M2
and M1?= M3. Also, |M1| = 3, while |M2| = |M3| = 4.
Definition 3.7: Given a multiset S, a clustering of S is a
partitioning of the elements of S into multisets c1,...,ck.
Now, let S be a multiset of agent positions. Within each
cluster ci, define the distortion measure of that cluster as
?
D(ci) =
z∈ci
||z − θ(ci)||2,
(18)
where θ(ci) is the centroid of all positions in ci. Define the
cost of a clustering as the total distortion measure, given by
H (c1,...,ck) =
k
?
i=1
D(ci).
(19)
Problem 3.2: The Euclidean minimum sum-of-squares
clustering problem is to find a clustering c∗
a multiset of positions S, so as to minimize (19).
Theorem 3.3: Suppose a single-leader network has a max-
imal leader-invariant EEP of exactly k cells containing
follower agents, numbered 1,...,k. Finding the optimal
permutation P∗for a target xT in Problem 3.1 is equivalent
to solving Problem 3.2 for the multiset of target positions,
S = {g1(xT),...,gN(xT)}, under the constraint that |ci| =
|m−1(i)|, the number of agents in cell i, for i = 1,...,k.
Proof:
Given a permutation matrix P, let p
{1,...,N}→{1,...,N} take in an agent index
and returns the permuted index such that, for j
1,...,N, gj(xT)=gp(j)((P ⊗ In)xT). Let ci
{gj((P ⊗ In)xT)|m(j) = i}, for i = 1,...,k, be a cluster-
ing of S = {g1(xT),...,gN(xT)}, where target positions
in (P ⊗ In)xT with indices in cell i are assigned to ci.
Notice that |ci| = |m−1(i)|, the number of agents in each
cell i, for i = 1,...,k. Considering different permutations
of agent indices for the target point xT is equivalent to
considering different cell assignments of the target positions,
which is equivalent to considering clusterings c1,...,ckof
S. The cost (16) associated with a chosen permutation P of
target positions can be rewritten using (11) and (15) as
1,...,c∗
k, given
:
=
=
J∗((P ⊗ In)xT)
n
?
N
?
N
?
k
?
k
?
which shows that the cost is equivalent to (19).
Given the P∗that solves Problem 3.1, an optimal
clustering
c∗
k
that
the constraint that |ci| = |m−1(i)|, for i = 1,...,k,
can be computed by the polynomial-time algorithm:
Let c∗
for i = 1,...,N do
Add gi((P∗⊗ In)xT) to c∗
end
Alternatively, given an optimal clustering c∗
trix P∗can be computed by the polynomial time algorithm:
=
i=1
||di((P ⊗ In)xT) − di(x∗((P ⊗ In)xT))||2
=
i=1
||gi((P ⊗ In)xT) − gi(x∗((P ⊗ In)xT))||2
=
i=1
||gi((P ⊗ In)xT) − θ?cm(i)
?
?||2
=
i=1
j|m(j)=i
||gj((P ⊗ In)xT) − θ(ci)||2
=
i=1
?
z∈ci
||z − θ(ci)||2= H (c1,...,ck),
1,...,c∗
solves Problem 3.2under
1,...,c∗
kbe empty multisets;
m(i);
1,...,c∗
k, the ma-

Page 5

Let Q = R = {1,...,N}, and P∗= 0 (N ×N matrix);
for i = 1...,k do
for each z ∈ c∗
Find any j ∈ Q such that gj(xT) = z;
Remove j from Q;
Find a b ∈ m−1(i) such that b ∈ R;
Remove b from R;
Set the element P∗
end
end
ido
b,jto 1;
Thus, finding P∗
findingan
{g1(xT),...,gN(xT)}, that minimizes (19) subject to |ci|
equaling the number of agents in cell i, for i = 1,...,k.
Viewing the problem of finding the optimal permutation
for a target specification as a size-constrained version of
the Euclidean minimum sum-of-squares clustering problem
given in Problem 3.2 is very useful because it allows us find
the computational complexity associated with the task.
Theorem 3.4: The problem of finding the optimal permu-
tation matrix P∗in Problem 3.1 is NP-hard.
Proof: It was shown in [14] that the Euclidean mini-
mum sum-of-squares clustering problem described in Prob-
lem 3.2 is NP-hard by using a reduction from the DENSEST
CUT problem for the case of k = 2 clusters. Using almost the
same procedure, we will show that the optimization version
of the MAX BISECTION problem, which was shown in
[15] to be NP-hard, reduces to the size-constrained Euclidean
minimum sum-of-squares problem for k = 2 clusters.
Let G = (V,E) be an undirected graph. Define B1,B2
as a partition of V such that |B1| = |B2| =
is assumed to be even. The MAX BISECTION problem is
to find B∗
E (B1,B2) = {(vi,vj) ∈ E|vi∈ B1and vj∈ B2}.
Arbitrarily number and orient the edges in E as
e1,...,e|E|so that each ei is an ordered pair of vertices.
Define the incidence matrix I as a N ×|E| matrix such that
for each ek= (vi,vj) ∈ E, Ii,k= −1 and Ij,k= 1. Have
x1,...,xN ∈ R|E|be such that xT
I. Define the multiset S = {x1,...,xN}. Have c1,c2 be
two clusters that partition S subject to the size constraint
|c1| = |c2| =N
Bi= {vj|xj∈ ci}, for i = 1,2.
Let the function φj : R|E|→ R take in a vector and
return the jth element of its argument. Computing the total
distortion of the cluster as in (19), we have
in Problem 3.1 is equivalent to
clustering
c∗
optimal
1,...,c∗
k
for
S=
N
2, where N
1and B∗
2so as to maximize |E (B1,B2)|, where
iequals the ith row of
2. Let B1and B2be a partition of V , where
H (c1,c2)=
2
?
|E|
?
i=1
?
2
?
z∈ci
||z − θ(ci)||2
=
j=1i=1
?
z∈ci
(φj(z) − φj(θ(ci)))2
If ej ∈ E (B1,B2), then either φj(z) equals 1 for
exactly one z ∈ c1 and equals −1 for exactly one z ∈ c2
with all others equaling 0 and thus φj(θ(c1)) =
φj(θ(c2)) = −2
2
Nand
N, or the same statements above but with
c1 and c2 switched. Furthermore, if ej / ∈ E (B1,B2), then
φj(θ(c1)) = φj(θ(c2)) = 0. Using these properties:
H (c1,c2)=
?
?
?
+ |E (B2,B2)|)
2|E| −4
N|E (B1,B2)|.
e∈E(B1,B2)
2
?
?2?
?
i=1
??N
2− 1
??2
N
?2
+1 −2
N
+
?
e/ ∈E(B1,B2)
2
=21 −2
N
|E (B1,B2| + 2(|E (B1,B1)|
=
Choice of B∗
2, that minimizes H (c1,c2) also maximizes |E (B1,B2)|,
since |E| and N are constant. Therefore, the NP-hard MAX
BISECTION problem reduces to size-constrained Euclidean
minimum sum-of-squares, which is equivalent to finding P∗
in Problem 3.1, and so finding P∗is also NP-hard.
1and B∗
2, or equivalently the choice of c∗
1and
c∗
IV. FINDING THE OPTIMAL SPECIFICATION OF A TARGET
NETWORK
In order to exploit homogeneity in the controllability of a
single-leader network, it is necessary to compute the optimal
permutation matrix P∗in Problem 3.1 or equivalently, the
optimal clustering c∗
in Theorem 3.4 it was shown that the complexity of the
problem in the general case is NP-hard. This section explores
heuristic-based and approximation algorithms for solving the
general problem, as well as the special case of the problem
where agents positions are 1-D.
1,...,c∗
kfrom Theorem 3.3. However,
A. Heuristic and Approximation Algorithms
A popular method for finding locally optimal solutions
to the Euclidean sum of squares problem is the k-means
algorithm (e.g. [16]). However, Theorem 3.3 adds equality
constraints on the size of individual clusters. In [17], a
constrained k-means clustering algorithm is proposed that
finds locally optimal clusterings which minimize (19), where
the minimum size of individual cluster can be specified.
Equality constraints on the cluster sizes are imposed when
minimum cluster sizes are chosen to sum to N.
B. Special Case: 1-D Networks
Recall that we assumed a network of N agents with a
maximal leader-invariant EEP of k cells containing follower
agents. In the general multi-dimensional case, finding P∗
in Problem 3.1 involves considering at most N! possible
permutation matrices, or equivalently N! clusterings by
Theorem 3.3. However, in the special case of 1-D networks,
only k! clusterings need to be considered by exploiting a
special property of constrained clustering in 1-D networks.
Definition 4.1: Given a clustering c1,...,ckof a multiset
S of 1-D points, a cluster ci is compact if ∄xi1,xi2 ∈ ci
and ∄xj∈ cjsuch that xi1< xj< xi2, ∀j ?= i.

Page 6

Lemma 4.1: The optimal clustering c∗
tiset S of 1-D points, which minimizes (19) with |ci|
predefined for i = 1,...,k, involves only compact clusters.
Proof: We start by showing that elements in every non-
compact clustering can always be reassigned to decrease (19)
without changing the cluster sizes. Assume c1,...,ck are
not all compact, then ∃xa1,xa2 ∈ ca and xb ∈ cb such
that xa1 < xb < xa2, for some ca and cb where a ?= b.
Furthermore, define the function
1,...,c∗
kof a mul-
Ho(c1,...,ck,m1,...,mk) =
k
?
i=1
?
z∈ci
(z − mi)2,
where H (c1,...,ck) ≤ Ho(c1,...,ck,m1,...,mk) with
equality if and only if mi = θ(ci), for i = 1,...,k. The
total distortion cost of the clustering can be rewritten as
H (c1,...,ck)=Ho(c1,...,ck,θ(c1),...,θ(ck))
Q − 2R(ca,cb,θ(ca),θ(cb)),=
where
Q =
N
?
i=1
x2
i+
k
?
i=1
|ci|θ(ci)2− 2
k
?
i=1,i?=a,b
?
θ(ci)
?
z∈ci
z
?
and
R(ca,cb,ma,mb) = ma
?
z∈ca
z + mb
?
z∈cb
z.
If θ(ca) ≥ θ(cb), assign ˆ ca the |ca| largest elements of
ca∪ cb, while giving ˆ cbthe remaining elements. Otherwise,
if θ(ca) < θ(cb), then let ˆ cahave the |ca| smallest elements
of ca∪ cb, while ˆ cbgets the rest. Furthermore, define ˆ ci=
ci∀i ?= a,b. Notice that |ˆ ci| = |ci|, for i=1, ..., k. Then since
after the reassignment, θ(ˆ ca) ?= θ(ca) and θ(ˆ cb) ?= θ(cb),
H (c1,...,ck)=
≥
=
Q − 2R(ca,cb,θ(ca),θ(cb))
Q − 2R(ˆ ca,ˆ cb,θ(ca),θ(cb))
Ho(ˆ c1,..., ˆ ck,θ(c1),...,θ(ck))
H (ˆ c1,...,ˆ ck).>
Thus, whenever a clustering c1,...,ckis not all compact, it
is always possible to obtain a new clustering ˆ c1,...,ˆ ckwith
a lower total distortion. Since there are only a finite number
of ways to cluster points in S, an optimal cluster must exist
and it must involve only compact clusters.
Knowing the optimal clustering is compact reduces the
number of clusterings that need to be searched.
Theorem 4.1: Finding c∗
1-D network requires considering at most k! clusterings.
Proof: Since points are 1-D, only the ordering of the
k compact clusterings matter in finding c∗
most only k! clusterings need to be considered.
1,...,c∗
kto minimize (19) for a
1,...,c∗
k. Thus, at
V. CONCLUSIONS
This paper extended the notion of controllability in a
single-leader network to the case of homogeneous agents.
By taking advantage of the fact that agents are interchange-
able, controllability in the homogeneous setting was phrased
as a point to set property of the system, where the set
corresponds to all permutations of a target point. It was
shown that different permutations of a target point may be
at different distances from the system’s reachable subspace,
and so is more reachable than others. Finding the optimal
permutation of a target point that is closest to the network’s
reachable subspace was shown to be equivalent to solving an
Euclidean minimum sum-of-squares clustering problem with
constraints on the cluster sizes. However, the task was shown
to be NP-hard. As a consequence, methods were presented
for finding suboptimal permutations in the general case, as
well as finding the optimal permutation in the special case
when the network consists of 1-D agents.
Acknowledgments
This work is partially supported by the Office of Naval
Research through the MURI Heterogeneous Unmanned Net-
worked Teams (HUNT).
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