Graph-theoretic connectivity control of mobile robot networks

doi:10.1109/JPROC.2011.2157884

INVITED

PAPER

Graph-Theoretic Connectivity

Control of Mobile

Robot Networks

This paper develops an analysis for groups of vehicles connected by a

communication network; control laws are formulated to accomplish tasks

requiring rendezvous, and swarm in group formations.

By Michael M. Zavlanos, Member IEEE, Magnus B. Egerstedt, Senior Member IEEE,and

George J. Pappas,

Fellow IEEE

ABSTRACT

|

In this paper, we provide a theoretical framework

for controlling graph connectivity in mobile robot networks.

We discuss proximity-based communication models composed

of disk-based or uniformly-fading-signal-strength communica-

tion links. A graph-theoretic definition of connectivity is pro-

vided, as well as an equivalent definition based on algebraic

graph theory, which employs the adjacency and Laplacian

matrices of the graph and their spectral properties. Based on

these results, we discuss centralized and distributed algorithms

to maintain, increase, and control connectivity in mobile robot

networks. The various approaches discussed in this paper

range from convex optimization and subgradient-descent algo-

rithms, for the maximization of the algebraic connectivity of

the network, to potential fields and hybrid systems that main-

tain communication links or control the network topology in a

least restrictive manner. Common to these approaches is the

use of mobility to control the topology of the underlying com-

munication network. We discuss applications of connectivity

control to multirobot rendezvous, flocking and formation con-

trol, where so far, network connectivity has been considered an

assumption.

KEYWORDS

|

Algebraic graph theory; convex and subgradient

optimization; graph connectivity; hybrid systems

I. INTRODUCTION

Mobile robot networks have recently emerged as an inex-

pensive and robust way of addressing a wide variety of

tasks ranging from exploration, surveillance, and recon-

naissance, to cooperative construction and manipulation.

The success of these stories relies on efficient information

exchange and coordination between the members of the

team. In fact, recent work on distributed consensus and

state agreement has strongly depended on m ultihop com-

munication for convergence and performance guarantees

[1]–[14].

Multihop communication in multirobot systems has

typically relied on constructs from graph theory, with

weighted proximity and disc-based graph s g aining the most

popularity. Besides their simplicity, these models owe

their popularity to their resemblance to radio signal

strength models, where the signals attenuate with the dis-

tance [15]–[17]. I n this context, multi hop communication

becomes equivalen t to network connecti vity, d efined as

the property of a graph to transmit information between

any pair of its nodes.

Network connectivity has been widely studied in the

area of wireless and ad hoc networks. Of great im portance

in this field is the power management of the nodes for

optimal routing and lifetime of the network, while ensur-

ing connectivity [18]–[23]. This rese arch has given rise to

connectivity or topology control algorithms that regulate

the transmission power of the nodes and, therefore, their

communication range. Approaches range from cone-based

Manuscript received May 13, 2010; revised October 14, 2010 and March 15, 2011;

accepted May 8, 2011. Date of publication July 12, 2011; date of current version

August 19, 2011. The work of M. M. Zavlanos and G. J. Pappas was supported by the

ONR HUNT MURI and ARO SWARMS MURI projects. The work of M. B. Egerstedt was

supported by the ONR HUNT MURI project.

M. M. Zavlanos is with the Department of Mechanical Engineering, Stevens Institute of

Technology, Hoboken, NJ 07030 USA (e-mail: michael.zavlanos@stevens.edu).

M. B. Egerstedt is with the Department of Electrical and Computer Engineering,

Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:

magnus@ece.gatech.edu).

G. J. Pappas is with the Department of Electrical and Systems Engineering, University

of Pennsylvania, Philadelphia, PA 19104 USA (e-mail: pappasg@seas.upenn.edu).

Digital Object Identifier: 10.1109/JPROC.2011.2157884

Vol. 99, No. 9, September 2011 | Proceedings of the IEEE 15250018-9219/$26. 00



2011 IEEE

[24]–[26] to distributed algorithms that do not involve any

position information of the nodes [27], [28]. Related is also

work on asymptotic bounds on the number of neighbors

required to ensure connectivity in randomly deployed

networks [29], as well as on the critical interference above

which connectivity is lost [30]. However, this type of work

focuses more on the power consumption and routing

problem than t he actuation and control.

Although networks have long served as models of local

interactions in the field of mobile robotics (Fig. 1), until

recently their structural properties have been assumed and

decoupled from the control objectives, as in the case of

connectivity in distributed consensus [1]–[14]. A first at-

tempt to control the network structure was with the design

of networks with maximal connectivity, where eigenvector

structure-based approaches for tree networks [31], [32] were

followed by optimization-based approaches applied to more

general networks [33], [34]. These results were derived for

static, state-independent, networks. Recently, controllabil-

ity frameworks for state-dependent graphs were also pro-

posed [35]. Nevertheless, the first work to treat connectivity

as a control objective was [36], in the context of multirobot

rendezvous. Since then, a large amount of research has been

targeted in this direction, and a wide range of applications

and solution techniques have been proposed.

A metric that is typically employed to capture connec-

tivity of robotic networks is the second sm allest eigenvalue



2

ðLÞ of the Laplacian matrix L of the graph, also known

as the algebraic connectivity or Fiedler value of the graph.

It is well known that 

2

ðLÞ is a concave function of the

Laplacian matrix, and when positive definite, it implies

network connectivity [37]–[40]. This has given rise to

optimization-based connectivity controllers that rely on

maximization of the Fiedler value [41], [42]. Since 

2

ðLÞ is

a function of the network’s structure via the Laplacian

matrix, connectivity algorithms th at relied on it were ini -

tially centralized [41]. Only recently have there been sub-

gradient algorithms for its distributed optimization [42].

Furthermore, the Fiedler value is a nondifferentiable

function of the L aplacian matrix, which presents difficul-

ties in designing feedback controllers to maintain it

positive de finite. Ways to overcome this problem involve

either positive definiteness constraints on the determinant

of the Laplacian matrix that is a differentiable function of

the Laplacian [43], or distributed consensus on either

Laplacian eigenvectors [44], [45] or on the network struc-

ture itself [46] for local estimation of the Fiedler value of

the overall network.

Alternatively, connectivity can be captured by the sum

of powers

P

K

k¼0

A

k

of the adjacency matrix A of the net-

work, which represents the number of paths up to length K

between every pair of nodes in the graph [40]. By defini-

tion of graph connectivity, if this number is positive

definite for K ¼ n  1 and all pairs of nodes, then the

network is connected (n denotes the number of nodes).

For originally connected networks, maintaining positive

definiteness of all positive entries of

P

K

k¼0

A

k

for any

K  n  1, maintains paths of maximum length K between

agents and, as shown in [47], is sufficient to maintain

connectivity of the network. This typically gives rise to

optimization-based connectivity controllers [47], [48] that

are often centralized due to the multihop agent depen-

dencies that are introduced by the powers of the adjacency

matrix. Since smaller powers correspond to shorter de-

pendencies (paths), distribution is possible as K decreases.

If K ¼ 1, conn ecti vity maintenance reduces to preserving

the links of a connected spanning subgraph of the network

and due to differentiability of the adjacency matrix, often

results in feedback solution techniques. Discrete-time

approaches are discussed in [36], [49], and [50], while

[51]–[56] rely on local gradients that may also incorporate

switching in the case of link additions. Switching between

arbitrary spanni ng topologies has also been studied wi th

the spanning subgraphs being updated by local auctions

[46], distributed spanning tree algorithms [57], combina-

tion of information dissemination algorithms and graph

picking games [58], or intermediate rendezvous [ 59], [60].

This class of approaches are typically hybrid, combining

continuous link maintenance and discr ete topology

control. The algebraic connectivity 

2

ðLÞ and number of

paths

P

K

k¼0

A

k

metrics can also be combined to give

controllers that maintain connectivity, while enforcing

desired multihop neighborhoods for all agents [61].

The results discussed above have been successfully

applied to multiple scenarios that require network con-

nectivity to achieve a global coordinated objective. Indi-

cative of this work is recent literature on connectivity

preserving ren dezvous [36], [52], [56], [ 62], [63], flock-

ing [55], [64], and formatio n control [56], [59], where so

far connectivity had been an assumption. Further exten-

sions and contributions involve connectivity control for

double integrator agents [49], agents with bounded inputs

[65]–[67] and indoor navigation [61], as well as for

communication based on radio signal strength [68]–[71]

and visibility constraints [36], [62], [72]–[74]. Periodic

connectivi ty for robot teams that need to occasionally split

in order to achieve individual objectives [75] and sufficient

conditions for connectivity in leader–fo llower networks

Fig. 1. Networks have long served as models of local interactions in

the field mobile robotics. Robots are typically associated with the

nodes of a graph and communication links with the edges.

Zavlanos et al.: Graph-Theoretic Connectivity Control of Mobile Robot Networks

1526 Proceedings of the IEEE | Vol. 99, No. 9, September 2011

[76], also add to the list. Early experimental results have

demonstrated efficiency of these algorithms also in practice

[75],[77],[78].

In this paper, we focus on the works of [41]–[43], [46],

[56], and [64], since they are the first to have formally

addressed connectivity control of mobile networks for a

wide range of applications and solution techniques. Our

contribution is to present a cohesive overview of the key

results in these papers in a unified framework. This includes

basic notions of network connectivity and control-theoretic

methods for connectivity guarantees and convergence. The

results discussed in this work incorporate a variet y of

mathematical tools, ranging from spect ral graph theory and

semidefinite programming, to gradient-descent algorithms

and hybrid systems. A byproduct of this work is to classify

the available literature with respect to the connectivity

metrics and solution techniques and provide a basis upon

which future resear ch can be built.

The rest of this paper is organized as follows. In

Section II, we develop graph-theoretic models of comm u-

nication and discuss network conn ectivity. In Section III,

we present ce ntralized [41] and distribut ed [42] optimi-

zation-based approaches to maximizing the algebraic

connectivity of a network, while i n Section IV, we discuss

gradient-based f eedback controllers that rely on the

spectral properties of the network [43]. In Section V, we

introduce distributed hybrid solu tions to the problem [46],

[56], while in Section IV, we discuss application of

connectivity control to connectivity preserving rendezvous

[56], flocking [64], and formation control [56].

II. CONNECTIVITY IN MOBILE

ROBOT NETWORKS

Consider n points robots in R

d

and let x

i

ðtÞ2R

d

denote

the position of robot i at time t  0. The robots can be

described by either single integrator models

_x

i

ðtÞ¼u

i

ðtÞ (1)

where u

i

ðtÞ2R

d

denotes the control input to robot i at

time t,ordoubleintegratormodels

_x

i

ðtÞ¼v

i

ðtÞ (2a)

_v

i

ðtÞ¼u

i

ðtÞ (2b)

where v

i

ðtÞ2R

d

denotes the veloc ity of robot i at time t.

Assume further that the robots have integrated wireless

communication capabilities and denote by ði; j Þ acommu-

nication link between robots i and j.Withevery

communication link ði; jÞ, we associate a weight function

w : R

d

 R

d

! R

þ

such that

w

ij

ðtÞ¼wx

i

ðtÞ; x

j

ðtÞ



¼ fx

ij

ðtÞ



2



(3)

for some f : R

þ

! R

þ

,wherex

ij

ðtÞ¼x

i

ðtÞx

j

ðtÞ.

1

We

choose the function f to be a decreasing function of the

inter-robot distance kx

ij

ðtÞk

2

such that

1   G fx

ij

ðtÞ



2



 1; if x

ij

ðtÞ



2

G 

1

and

0  fx

ij

ðtÞ



2



G ; if x

ij

ðtÞ



2

>

2

for 0 G 

1

G 

2

and small enough 0 G  G 1(Fig.2).This

definition captures the fact that signal strength between

wireless robots is strong up to a distance 

1

and then

decreases rapidly until it practically vanishes beyond a

distance 

2

.

The system described above gives rise to a weighted

state-dependent graph

G ¼ðV; WÞ

where V ¼f1; ...; ng denotes the set of nodes indexed by

the set of robots and W : V  V  R

þ

! R

þ

denotes

the set of edge weights, such that

Wði; j; tÞ¼w

ij

ðtÞ

for i; j 2 V and with w

ij

ðtÞ as in (3). The set

~

EðtÞ¼

fði; jÞjw

ij

ðtÞ > 0g is called the set of directed edges of G,

while the unordered pair fi; jg is an edge of G if w

ij

ðtÞ > 0

or w

ji

ðtÞ > 0. If w

ij

ðtÞ¼0impliesw

ji

ðtÞ¼0forall

i; j 2 V, then the weights are called weakly symmetric

and the graph is called undirected. On the other hand, if

w

ij

ðtÞ¼w

ji

ðtÞ for all i; j 2 V, then the weights are call ed

symmetric. Clearly, if a graph has symmetric weights, then

it is also undirected. Throughout this paper, we assume

graphs G with symmetric weights that additionally have no

loops, i.e., w

ii

ðtÞ¼0foralli 2 V.Wealsodefinethesetof

neighbors of node i 2 V by N

i

ðtÞ¼fj 2 Vjði; jÞ2

~

EðtÞg,

which in the case of undirected graphs results in a mutual

adjacency relationship between nodes, i.e., if i 2 N

j

ðtÞ

then j 2 N

i

ðtÞ. Similarly, we define a directed path of

1

We denote by R

þ

the set ½0; 1Þ and by R

þþ

the set ð0; 1Þ.

Zavlanos et al.: Graph-Theoretic Connectivity Control of Mobile Robot Networks

Vol. 99, No. 9, September 2011 | Proceedings of the IEEE 1527

length k by a sequence of k þ 1distinctnodesi

0

; i

1

; ...;

i

k

2 V such that ði

p1

; i

p

Þ2

~

EðtÞ for all 1  p  k.Ifthe

graph G is undirected, then so are its paths. An important

topological invariant of graphs is graph connectivity, which

for the case of undirected graphs is defined as follows.

Definition 2.1 (Graph Connectivity): We say t hat an undi-

rected graph G isconnectedifforeverypairofnodesthere

exists a path starting at one node and e nding at the o ther.

Network connectivity is an important property of

robotic networks designed to achieve global coordinated

objectives, since it ensures information sharing via multi-

hop c ommunicatio n paths between members of the team.

This property can be efficiently captured using an equiv-

alent algebraic representation of graphs by the adjacency

and Laplacian matrices.

A. Algebraic Definitions of Connectivity

We define the adjacency matrix AðtÞ2R

nn

þ

of the

weighted graph G with entries

AðtÞ½

ij

¼ w

ij

ðtÞ: (4)

Clearly, if the network has symmetric weights, then the

adjacency matrix is a symmetric matrix. Furthermore, if

the weights satisfy w

ij

ðtÞ2f0; 1g [Fig. 2(a)], then the

powers of the adjacency matrix of a graph are closely

related to network connectivity. In particular, we have the

following result [40].

Theorem 2.2 (Graph Connectivity): The entry ½A

k

ðtÞ

ij

of

the matrix A

k

ðtÞ is the number of paths of length k from

node i to node j in G. Therefore, the graph G is connected

if and only if there exists an integer K such that all the

entries of the matrix C

K

ðtÞ¼

P

K

k¼0

A

k

ðtÞ are nonzero.

Note that the integer K in Theorem 2.2 is upper

bounded by n  1,sincethisisthelengthofthelongest

possible path in a network of n nodes. Note also that for

any K  n  1 the ine quality

C

K

ðtÞ½

ij

> 0

enforces paths of maximum length K between nodes i and j

in V. It is sh own in [47] that, for initially co nnected

Fig. 2. Different choices for the function f. In particular, (a) fðyÞ¼1 if y  

2

;(b)fðyÞ¼1=1 þ e

ðyÞ

with  ¼ð2=ð

2

 

1

ÞÞ logðð1  Þ=Þ,

and  ¼ð

1

þ 

2

Þ=2;(c)fðyÞ¼ð1=ð

1

 

2

ÞÞy ð

2

=ð

1

 

2

ÞÞ if 

1

 y G 

2

;and(d)fðyÞ¼e

ðy

1

Þ

if y >

1

,with ¼ð1=ð

2

 

1

ÞÞ logð1=Þ.

The above plots are for 

1

¼ 0:6, 

2

¼ 1:4,and ¼ 0:01.

Zavlanos et al.: Graph-Theoretic Connectivity Control of Mobile Robot Networks

1528 Proceedings of the IEEE | Vol. 99, No. 9, September 2011

networks, requiring that ½C

K

ðtÞ

ij

> 0, fo r any K  n  1,

whenever ½C

K

ð0Þ

ij

> 0 is sufficient for network connecti-

vity for all time t  0. This result can be easily understood

if applied for K ¼ 1, where it states that maintaining all

one-hop links o f an originally connected network is suffi-

cient for connectivity for all time. In what follows, when

relying on the m atrix C

K

ðtÞ to ensure connectivity, we only

consider the case K ¼ 1. The general case is discussed in

[47] and [48].

Alternatively, graph connectivity can be captured using

the Laplacian matrix LðtÞ2R

nn

of the network G,

which is defined by

LðtÞ½

ij

¼

w

ij

ðtÞ; if i 6¼ j

P

s6¼i

w

is

ðtÞ; if i ¼ j.



(5)

If DðtÞ¼diagð

P

n

j¼1

w

ij

ðtÞÞ denotes the diagonal matrix of

degrees of the network, also called the Valency matrix of

G, then the Laplacian matrix can be written as

LðtÞ¼DðtÞAðtÞ:

The L aplacian matrix of a network G with symmetric

weights is always a symmetric positive-se midefinite

matrix with spectral properties closely related to network

connectivity, as it can be seen from the following

theorem [40].

Theorem 2.3: Let

0  

1

LðtÞðÞ

2

LðtÞðÞ

n

LðtÞðÞ

be the ordered eigenvalues of the Laplacian matrix LðtÞ.

Then, 

1

ðLðtÞÞ ¼ 0 with corresponding eigenvector 1,i.e.,

the n  1 vector of all entries equal to 1. Moreover,



2

ðLðtÞÞ > 0 if and only if G is connected.

Besides an indicator of connectivity, the second

smallest eigenvalue 

2

ðLðtÞÞ of the Laplacian matrix of

G, also called the algebraic connectivity or Fiedler value

of the network, is also a measure of the robustness of

the network to link failures, captured by the notion of

k-connectivity [40].

Definition 2.4 (k-Connectivity): Let ðGÞ be the mini-

mum number of edges that if removed from G increase

its number of connected components. Then, for any

k  ðGÞ the undirected graph G is called k-connected.

TheedgeconnectivityðGÞ and algebraic connectivity



2

ðLðtÞÞ are related by the inequality [40]



2

LðtÞðÞðGÞ:

Therefore, if 

2

ðLðtÞÞ > k  1, then the network G is

k-connected. Note that if k ¼ 1, then k-connectivity

reduces to the usual definition of connectivity

(Definition 2.1). The results discussed above give rise

to the following statement of the connectivity control

problem.

Problem 1 (Network Connectivity Control): Given an

initially connected state-dependent network G,design

distributed controllers fu

i

ðtÞg

n

i¼1

for the robots so that

the closed-loop system (1) or (2) guarantees that G is

k-connected for all time.

In what follows, we discuss optimization [41], [42] and

feedback-based [43], [46], [56] solutions to Problem 1 that

employ both connectivity metrics de velo ped above, i.e.,

the adja cency matrix AðtÞ and its powers as well as the

algebraic c onnectivity 

2

ðLðtÞÞ.Weunifytheseap-

proaches under a common control framework and charac-

terize them with respect to the amount of distribution they

possess.

III. OPTIMIZATION-BASED

CONNECTIVITY CONTROL

Observe that 

2

ðLðtÞÞ is a concave function of LðtÞ in the

space 1

?

givenbytheinfimumofasetoflinearfunctions

in LðtÞ,i.e.,



2

LðtÞðÞz

T

z  z

T

LðtÞz

for all z 2 1

?

,orequivalently



2

LðtÞðÞ¼inf

z21

?

z

T

LðtÞz

z

T

z

: (6)

Therefore, maximization of 

2

ðLðtÞÞ gives rise to

optimization-based approaches to the connectivity control

problem. In other words, a sufficient solution to Problem 1

can be obtained by solving the optimization problem

max

x2R

dn



2

LðxÞðÞ (7)

where x ¼½x

1

x

2

... x

n



T

2 R

dn

denotes the vector of all

robot positions. The two approaches to this problem that

we discuss rely on concavity of the state-independent

problem

max

L2S

n



2

ðLÞ (8)

Zavlanos et al.: Graph-Theoretic Connectivity Control of Mobile Robot Networks

Vol. 99, No. 9, September 2011 | Proceedings of the IEEE 1529

Graph-theoretic connectivity control of mobile robot networks

Figures

Citations

Heterophilious Dynamics Enhances Consensus

Graph-Theoretic Connectivity Control of Mobile Robot Networks This paper develops an analysis for groups of vehicles connected by a communication network; control laws are formulated to accomplish tasks requiring rendezvous, and swarm in group formations.

Self-Triggered Consensus for Multi-Agent Systems With Zeno-Free Triggers

Interconnected dynamic systems: An overview on distributed control

Distributed Coordination Control for Multi-Robot Networks Using Lyapunov-Like Barrier Functions

References

Convex Optimization

Consensus problems in networks of agents with switching topology and time-delays

Consensus and Cooperation in Networked Multi-Agent Systems

Coordination of groups of mobile autonomous agents using nearest neighbor rules

Flocks, herds and schools: A distributed behavioral model

Related Papers (5)

Distributed Coordination Control of Multiagent Systems While Preserving Connectedness

Consensus and Cooperation in Networked Multi-Agent Systems

On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian

Flocking for multi-agent dynamic systems: algorithms and theory

Coordination of groups of mobile autonomous agents using nearest neighbor rules