Pinning control of networks dimensionality reduction through simultaneous block-diagonalization of matrices Shirin Panahi1Matteo Lodi2Marco Storace2and Francesco Sorrentino1a

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Pinning control of networks: dimensionality reduction through simultaneous

block-diagonalization of matrices

Shirin Panahi,1Matteo Lodi,2Marco Storace,2and Francesco Sorrentino1, a)

1)University of New Mexico, Albuquerque, NM, US 80131

2)University of Genoa, Via Opera Pia 11A, 16154, Genova, Italy

In this paper, we study the network pinning control problem in the presence of two diﬀerent types of coupling:

(i) node-to-node coupling among the network nodes and (ii) input-to-node coupling from the source node to

the ‘pinned nodes’. Previous work has mainly focused on the case that (i) and (ii) are of the same type. We

decouple the stability analysis of the target synchronous solution into subproblems of the lowest dimension

by using the techniques of simultaneous block diagonalization (SBD) of matrices. Interestingly, we obtain

two diﬀerent types of blocks, driven and undriven. The overall dimension of the driven blocks is equal to the

dimension of an appropriately deﬁned controllable subspace, while all the remaining undriven blocks are scalar.

Our main result is a decomposition of the stability problem into four independent sets of equations, which

we call quotient controllable, quotient uncontrollable, redundant controllable, and redundant uncontrollable.

Our analysis shows that the number and location of the pinned nodes aﬀect the number and the dimension

of each set of equations. We also observe that in a large variety of complex networks, stability of the target

synchronous solution is de facto only determined by a single quotient controllable block.

In this paper we consider dynamical networks

formed of coupled identical oscillators and use

pinning control to synchronize all of the network

nodes on a given target time evolution. The

problem of stability of the entire network about

the target solution is then studied by lineariz-

ing the network dynamics about the target so-

lution. Diﬀerent from previous work, we focus

on the general case that the node-to-node cou-

pling among the network nodes is diﬀerent from

the input-to-node coupling from the source node

to the ‘pinned nodes’. The stability problem is

then decoupled into the stability of independent

subsystems of the lowest dimension. Our analy-

sis is relevant to the analysis of several complex

networks for which we see that stability of the

target synchronous solution is often determined

by the maximum Lyapunov exponent associated

with only one subsystem that we call ‘quotient

controllable’.

I. INTRODUCTION

Gaining control over collective network dynamics is a

hot topic in both graph and control theory. In particu-

lar, pinning control is a feedback control strategy largely

used for imposing synchronization or consensus in com-

plex dynamical networks1–16. Speciﬁcally, one or more

virtual leaders (the so-called sources) are added to the

network and deﬁne its desired trajectory. Each source di-

rectly controls only a small fraction of the network nodes

(the pinned nodes), by exerting a control action that is a

a)Electronic mail: fsorrent@unm.edu

function of the pinning error vector, whose i-th compo-

nent is given by the diﬀerence between the output of the

considered source and the output of the i-th node. Pin-

ning control of time varying networks has been studied

in17–20. Our ability to impose a desired synchronous so-

lution to a dynamical network can have important appli-

cations in diﬀerent ﬁelds of science, such as in physical21,

social22, multi-agent23, and biological networks24 and

pinning control is a widely adopted solution25.

Here we consider the problem of pinning control of

undirected networks of dynamical systems, for the case

in which the node-to-node connectivity is diﬀerent from

the connectivity exerted on the pinned nodes, which is

relevant to a variety of realistic scenarios and engineer-

ing applications. As an example, imagine a biological

network that one wants to synchronize on a speciﬁc time

evolution, by pinning some of the network nodes. How-

ever, the type of forcing one may be able to exert on the

network is likely going to be diﬀerent from the biologi-

cal node-to-node interactions between the network nodes.

This motivates the study of a problem in which the node-

to-node coupling among the network nodes is diﬀerent

from the coupling exerted on the pinned nodes. Simi-

lar versions of this problem have been previously inves-

tigated in26–29 using a Lyapunov function (V-stability)

which provides a suﬃcient stability condition. In this

paper, we investigate stability using linearization, which

provides both necessary and suﬃcient conditions.

We study the stability of the target synchronous solu-

tion in an undirected network of coupled dynamical sys-

tems, subject to the control action of one source node.

Our goal is to reduce (through proper transformations)

the stability analysis into simpler problems, which can

be analyzed independently of one another. This prob-

lem presents mathematical challenges that require the

introduction of a speciﬁc formalism; in particular, we

use the techniques for simultaneous block diagonaliza-

tion of matrices (SBD)30–32 which allow the reduction

arXiv:2210.06410v1 [eess.SY] 12 Oct 2022

of the stability problem in problems of lower dimension.

Previous work has successfully applied this approach to

both the stability of complete synchronization33 and clus-

ter synchronization34. However, no characterization was

provided about the dimensions of the independent sets of

equations in which the stability problem is reduced. The

dimension of these sets of equations (which are related

to the diagonal blocks of some matrices) is the lowest

and we show that it may vary based on the structure of

the network and the choice of the pinned nodes. Once

these sets/blocks are found, a maximum Lyapunov expo-

nent (MLE) can be associated to each of them in terms

of a Master Stability Function. The target synchronous

solution is stable only when the MLEs corresponding to

all the sets/blocks are negative, which provides a simple

criterion for assessing stability.

In summary, by combining MSF and SBD we study the

stability of the target solution in the considered networks

subject to pinning control; this leads to a decomposition

of the stability problem into four diﬀerent types of inde-

pendent equations, which we call quotient controllable,

quotient uncontrollable, redundant controllable, and re-

dundant uncontrollable. The insight we provide on both

the sizes and ‘roles’ of the blocks is the main contribution

of this paper. As stated before, though there is previous

work on the SBD reduction, to the best of our knowl-

edge, no paper has explained the ‘reason’ for the blocks

resulting from application of the SBD decomposition.

To carry out the stability analysis, we resort to two

alternative transformations: one (T) based on the SBD

decomposition and one ( ˆ

T) based on the concepts of con-

trollable subspace and equitable clusters. We are the ﬁrst

ones to apply the SBD approach to the pinning control

problem and in so doing we establish a connection be-

tween the blocks in which the stability problem is reduced

and the particular choice of the network connectivity and

of the pinned nodes. In particular, we show that the sizes

of the blocks provided by a ﬁnest SBD are the same as

for the blocks generated by the transformation ˆ

T, which

has a clear interpretation in terms of controllability and

quotient graphs. We prove that the transformation ˆ

also provides a ﬁnest SBD. This is another important

contribution of this paper.

The rest of the paper is organized as follows. In Sec. II

we introduce the problem of pinning control of networks.

In Sec. III a stability analysis is derived for a network

with pinned nodes. The solution to the stability prob-

lem consists of two steps presented in Secs. IV and V.

First, we use the SBD method to reduce the dimension of

the stability problem. Then, by using two appropriately

deﬁned transformation matrices, we obtain the transfor-

mation ˆ

Twhich decouples the stability problem into four

independent blocks. We then compare the application of

the SBD transformation (T) with the transformation ˆ

The eﬀects of the selection of diﬀerent pinned nodes on

multiple driven blocks is studied in Sec. VI. Application

of the theory to larger complex networks is considered in

Sec. VII. Finally, the conclusions are drawn in Sec. VIII.

II. PROBLEM DEFINITION: PINNING CONTROL OF

NETWORKS

We consider an undirected network of Ncoupled dy-

namical systems, which evolve in time according to the

following equation:

xi(t) = F

F(x

xi(t))+

j=1

Aij [G

G(x

xj(t))−G

G(x

xi(t))] i= 1,··· , N

(1)

where x

xiis the m-dimensional state of node i,F

F:Rm→

Rmis the function that governs the dynamics of each

node when isolated, and G

G:Rm→Rmis the node-

to-node coupling function. The network topology is de-

scribed by the adjacency matrix A. If there is a con-

nection between the nodes iand jthen Aij =Aji = 1

otherwise Aij =Aji = 0. Eq. (1) can be rewritten as:

xi(t) = F

F(x

xi(t)) +

j=1

LijG

G(x

xj(t)) i= 1,··· , N, (2)

where the Laplacian matrix L=A−Dand Dis a diago-

nal matrix, such that Dii =PjAij . This network allows

a completely synchronized solution x

x1(t) = x

x2(t) = ··· =

xn(t) = x

xs(t), which obeys,

xs(t) = F

F(x

xs(t)).(3)

A relevant question that links control theory to graph

theory is: how can control inputs be introduced to force

the network state to the target synchronous state?

Equation (3) allows for a number of diﬀerent syn-

chronous solutions determined by its initial condition.

The problem studied in pinning control is how control

inputs generated by source nodes can be designed to en-

force stability of a particular ‘target’ synchronous solu-

tion, xt(t) produced by the initial condition x

xt(t) = F

F(x

xt(t)), x

xt(0) = x

t.(4)

In order to address this problem, we introduce control

inputs ui(t) as pinning control signals,

xi(t) = F

F(x

xi(t)) +

j=1

LijG

G(x

xj(t)) + ui(t),(5a)

ui(t) = γri[H

H(x

xt(t)) −H

H(x

xi(t))],(5b)

i= 1,··· , N

where the binary scalar ri= 1 (ri= 0) if node iis

pinned (not pinned), and the scalar γ > 0 measures the

strength of the control coupling. Here, H

H:Rm→Rmis

the source-to-pinned-node coupling function. Note that

the control action is only directly active on the pinned

nodes. For instance, consider the network shown in Fig.

1. The network consists of N= 11 nodes coupled via

blue edges. A source node that provides the target tra-

jectory is shown in red. The control input is directly

applied (red directed edges) from the source node to the

pinned nodes 1, 2, and 3, which are shown in black.

Deﬁnition 1. Network with inputs. A network with

inputs is represented by a pair of N×Nmatrices Land

R, where the Laplacian matrix Ldescribes the network

connectivity and the diagonal matrix Ris such that Rii =

ri, i = 1, .., N. We call Vthe set of the Nnetwork nodes,

VPthe set of the spinned nodes, and VNP the set of τ=

N−snon pinned nodes, VP∪VN P =Vand VP∩VNP =∅.

Deﬁnition 2. Extended network. Given a network

with inputs, its extended network is represented by the

(N+ 1) ×(N+ 1) directed Laplacian matrix ˜

L,35 deﬁned

below,

L=(L−R)N×Nr

01×N0(6)

where the column vector r

r= [r1, r2, ..., rN]Tand 01×Nis

the zero row-vector of dimension N.

Figure 1. A network of N= 11 nodes subject to pinning

control. The source node is shown in red. The pinned nodes

(i.e., the nodes that receive the source signal) are shown in

black. The remaining network nodes are in blue. The node-

to-node network connections are in blue. Red connections

carry the target solution x

xt(t) (Eq. (4)) from the source node

to the pinned nodes.

Previous work (see e.g.,35,36) has analyzed stability of

the target synchronous solution by considering pinned

nodes with the same coupling function as the node-to-

node coupling function, i.e., with H

H=G

G. Here instead

we consider a generalization where the eﬀect of the source

node’s time evolution on the pinned nodes is given by a

diﬀerent coupling function.

III. STABILITY ANALYSIS

When all the x

xi(t) converge on the target solution

xt(t), the control inputs ui(t) converge to zero. To study

the stability of the synchronous solution, a small per-

turbation δx

xi= (x

xi−x

xt) is considered37. Lineariza-

tion of Eq. (5) about the target synchronous solution

x1(t) = x

x2(t) = ··· =x

xn(t) = x

xt(t) yields, see also35,

δ˙

xi(t) = DF

F(x

xt(t))δx

xi(t) + PN

j=1 Lij DG

G(x

xt(t))δx

xj(t)

−γriDH

H(x

xt(t))δx

xi(t),

(7)

i= 1,··· , N, where Dis the Jacobian operator.

By stacking all perturbation vectors together in one

vector δX

X= [δx

1, δx

2, . . . , δx

N]T, Eq. (7) can be rewrit-

ten as

δ˙

X(t)=[IN⊗DF

F(x

xt(t)) + L⊗DG

G(x

xt(t))

−γR ⊗DH

H(x

xt(t))]δX

X(t),(8)

where ⊗is the Kronecker product or direct product. Our

goal is to break the (N×m)-dimensional Eq. (8) into a

set of independent lower-dimensional equations, thus per-

forming a dimensional reduction. An inherent diﬃculty

is due to the fact that, with the exception of a few speciﬁc

cases38, it is not possible to simultaneously diagonalize

both matrices Rand L. Instead, in what follows we seek

a transformation that decouples the set of Eqs. (8), by

simultaneously block diagonalizing Rand L33.

IV. DIMENSIONALITY REDUCTION OF THE

STABILITY PROBLEM THROUGH SBD

Deﬁnition 3. Simultaneous Block Diagonalization

of Matrices (SBD)30–32.Given a set of qsquare N-

dimensional matrices M1, M2,··· , Mq, an SBD transfor-

mation is an orthogonal square matrix T(transformation

matrix) with dimension Nsuch that

T−1MkT=⊕l

j=1Bk

j, k = 1,2,··· , q, (9)

where the symbol ⊕is the direct sum of matrices, lde-

notes the number of blocks, each block Bk

jis a square

matrix with dimension bjand Pl

j=1 bj=N.

Deﬁnition 4. Finest SBD. A ﬁnest SBD is an SBD

for which the resulting blocks can not be further reﬁned30.

In particular this means that the size of the blocks can

not be made smaller and that the number of blocks, l, is

largest among all SBDs.

Remark 1. A ﬁnest SBD is unique only in the number

and sizes of the irreducible blocks30.

Remark 2. The blocks resulting from a ﬁnest SBD

are also matrices that belong to irreducible matrix ∗-

algebras according to the structure theorem for matrix

∗-algebras.31,32

Application of the SBD technique to complete synchro-

nization of networks was ﬁrst proposed in Ref.33 and only

recently applied to cluster synchronization in Ref.34.

Once we obtain T=SBD(R, L), the matrices Rand

Lare transformed as follows,

T−1LT =LT=⊕l

j=1 ˆ

Lj,(10a)

T−1RT =RT=⊕l

j=1 ˆ

Rj,(10b)

where LTis the transformed matrix L,RTis the trans-

formed matrix R, and the irreducible block pairs ( ˆ

Lj,ˆ

Rj)

have the same dimensions bj,j= 1, ..., l. One of the

main contributions of this paper is to investigate the di-

mensions of the block pairs (ˆ

Lj,ˆ

Rj), as a result of the

original network topology (through L) and the choice of

the pinned nodes (through R).

A. Finding the matrix P

The procedure described in Ref.39 to ﬁnd the ﬁnest

SBD transformation, T, requires two steps. First, ﬁnding

a matrix Pwhich commutes with each member of the

set of matrices (here, Rand L), that is P R =RP and

P L =LP , is selected. Second, the transformation matrix

Tis constructed to have columns corresponding to the

eigenvectors of the commuting matrix P.

Here, we describe the approach to compute the matrix

Pthat commutes with both Rand L. Without loss of

generality, we apply the same permutation to both ma-

trices Rand Lsuch that the matrix Rcan be rewritten,

R=Is0

00N−s,(11)

where Isis the identity matrix of size s,0N−sis the

square zero matrix of dimension N−s, and sis the num-

ber of pinned nodes.

Lemma 1. Any matrix Pthat commutes with the matrix

Rin (11) has the following block-diagonal structure,

P=P10

0P2(12)

where the block P1has dimension sand the block P2has

dimension N−s.

Proof. First, we consider the matrix Rof Eq. (11) and a

generic matrix P=P1P12

P21 P2with sub-blocks having the

same dimensions as those of the matrix R. Then, from

the commutation equation P R =RP we obtain

P1P12

P21 P2Is0

0 0(N−s)=Is0

00(N−s)P1P12

P21 P2

P10

P21 0(N−s)=P1P12

00(N−s).

(13)

Therefore, to fulﬁll the commutation equation P R =RP ,

the sub-blocks P12 and P21 must be zeros and the matrix

Pis block diagonal with diagonal-blocks P1and P2in

which each block has the same dimension as the diagonal-

blocks of matrix R.

Corollary 1. The transformation matrix Talso has the

same block-diagonal structure,

T=T10

0T2,(14)

where T1is the matrix of eigenvectors of P1and T2is the

matrix of eigenvectors of P2.

Lemma 2. Application of the block diagonal transfor-

mation Tto the matrix R, transforms Rback to itself,

i.e., T−1RT =RT=R.

Proof. By considering the matrix Rof Eq. (11) and the

matrix Tof Eq. (14) and by using the fact that a block-

diagonal matrix can be inverted block by block, we have

T−1RT =T−1

0T−1

2Is0

00(N−s)T10

0T2=

Is0

00(N−s).

(15)

The above lemma has important consequences. In fact,

as we will see in Sec. IV B, this will lead to a decomposi-

tion of the stability problem into equations that can be

of either one of two types: driven or undriven.

Remark 3. Lemma (2) shows that RT=R. This is

true independent of any permutation of the rows and

columns of the matrix R. In the examples that follow

we will sometimes show the matrix RTin a form that

corresponds to permutations of rows and columns of the

matrix Rin Eq. (11).

B. Driven and Undriven blocks

As stated before, the purpose of using the SBD trans-

formation is to break the stability problem of Eq. (8) into

a set of independent equations of lowest dimension.

Due to the block-diagonal structure of both matrices

LTand RTand to Eq. (11), they can be decoupled into

the pairs (Ld, Rd) and (Lud,0N−c),

LT=Ld0

0LudRT=Rd0

0 0(N−c),(16)

where the square matrices Ld, Rdhave size c≥sand the

square matrix Lud has dimension N−c. We remark that

in general the blocks Ldand Rdare composed of smaller

diagonal blocks. In particular, if c=s, then Rd=Is,

otherwise Rdis equal to a diagonal matrix with sentries

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Pinningcontrolofnetworks:dimensionalityreductionthroughsimultaneousblock-diagonalizationofmatricesShirinPanahi,1MatteoLodi,2MarcoStorace,2andFrancescoSorrentino1,a)1)UniversityofNewMexico,Albuquerque,NM,US801312)UniversityofGenoa,ViaOperaPia11A,16154,Genova,ItalyInthispaper,westudythenetworkpinningc...

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Pinning control of networks dimensionality reduction through simultaneous block-diagonalization of matrices Shirin Panahi1Matteo Lodi2Marco Storace2and Francesco Sorrentino1a

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