Enhancing Resilience of Strong Structural Controllability in

Leader-Follower Networks

Vincent Schmidtke

and Olaf Stursberg

Control and System Theory, EECS Dept., University of Kassel, Germany

Keywords:

Strong Structural Controllability, Leader-Follower Networks, Edge Augmentation, Resilient Control.

Abstract:

This paper explores measures of edge augmentation to enhance resilience of strong structural controllability

for control systems modeled as leader-follower networks. Unlike existing methods which typically increase the

number of leaders, the proposed approach achieves resilience by strategically adding edges, thus maintaining

leader sets with a small cardinality. Using the zero forcing method, conditions are derived to enhance resil ience

either for speciﬁc agents or for the entire network. Numeric simulations validate the approach and show its

effectiveness in large and complex networks.

1 INTRODUCTION

Networked Control Systems (NCS) are integral to

modern engineering, enabling distributed coordina-

tion and scalability in complex systems such as smart

grids, trafﬁc systems, swarms and biological net-

works. In these systems, it is frequently desired to

control the network by injecting control actions only

for a small subset of nodes. This leads to the notion

of leader-follower frameworks, where leaders have a

control input, whereas followers do not h ave one (Por-

ﬁri and di Bernardo, 2008; Egerstedt et al., 2012).

While this can simplify the con trol architectur e, the

system may g et more fr a gile with respect to malfunc -

tions or attacks, if the loss of an agent and/or its co n-

nections signiﬁcantly impairs the system’s functio n-

ality (Pasqualetti et al., 2020).

Controllability and resilience are thus important

properties when investigating in how far such systems

can maintain their function if subject to internal or

externally triggere d changes. Controllability ensure s

the ability to let the system transition from any ini-

tial state to a desired state, while resilience pertains to

the system’s capacity to withstand (or recover from)

disruptions. These properties are c rucial, especially

in safety-critical applications where failures can have

catastrophic co nsequences.

This paper focuses on the notion of strong struc-

tural controllability (SSC) which requires controlla-

bility for varying inte rconnections of states (and thus

structures) of the system to be con trolled. This is par-

https://orcid.org/0009-0006-6322-103X

https://orcid.org/0000-0002-9600-457X

ticularly advantageous in NCS, since controllability

depends on the network topology and the leader set,

rather than only particular weights assigned to con-

stantly existing interconnections. This makes SSC

suitable for real-world applications, in which infor-

mation on ed ge weights may be uncertain, time-

varying, noisy, or even vanishing (Ch apman and Mes-

bahi, 2013; Monshizad e h et al., 2014).

In order to assess SSC of leader-follower networks

as well as its resilience against changing network

topologies, this paper utilizes the m ethod of zero forc -

ing sets, a graph -theoretic appro ach that is frequently

used in the context of SSC for NCSs (Monshizadeh

et al., 2014; Abbas et al., 2020b; Schmidtke et al.,

2024). In the context of SSC, the following exist-

ing papers have proposed edge augmentation, i.e. the

addition of edges to a graph, as a means to modify

the system structure for enhanced contr ollability. In

(Chen et al., 2019), minimal edge sets are computed

to render non-SSC system s structura lly controllable,

whereas (Abbas et al., 2020a) and (Mousavi et al.,

2021) identify a set of edges that can be added without

violating SSC. These approaches leverage e dge aug-

mentation as a design tool to either restore or preserve

controllability.

A certain body of literature in the given con-

text has examined the relationship between graph-

theoretic robustness measur e s and the minimum num-

ber of required leaders: The m ost often used mea-

sure is the Kir choff index, which formulates the sys-

tem’s resilience against disconnection. (Pasqualetti

et al., 2020) and (Abbas et al., 2020b) investigate the

trade-off between an increasing connectivity and the

minimum number of required leader agents, which is

136

Schmidtke, V. and Stursberg, O.

Enhancing Resilience of Strong Structural Controllability in Leader-Follower Networks.

DOI: 10.5220/0013826200003982

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 22nd International Conference on Informatics in Control, Automation and Robotics (ICINCO 2025) - Volume 1, pages 136-144

ISBN: 978-989-758-770-2; ISSN: 2184-2809

also discussed as a central controllability measure in

(Egerstedt et al., 2012). T he work in (Abbas e t al.,

2024) identiﬁes edge additions leading to a reduction

of th e required number of leader agents, while in-

creasing the robustness measure. A design pr ocedure

which builds a graph which is SSC with the highest

possible Kirc hoff ind ex for speciﬁed graph parame-

ters is presen te d in (Patel et al., 2024).

The notion of robustness addressed in the afore-

mentioned work differs from the approach to r e -

silience taken in this pap er. Rather than focusing

on graph connectivity, resilience of controllability is

here speciﬁcally considered with re spect to preserv-

ing SSC in case of structur a l d isruptions such as node

or edge failures. Resilient SSC is addressed only in

very few contributions: In (Schmidtke et al., 2024),

conditions have been speciﬁed which allow to check

which agents can be removed from the system, while

maintaining SSC with the cur rent leader set. T he pa-

pers (Abbas, 2023) and (Alameda et al., 2024) exploit

methods to specify a set of lea der agents which guar-

antees that a certain number of nodes or edges c an be

removed, while keeping the system SSC. However,

these methods require the introduction of far more

leader ag ents – this, on the one hand, leads to leader

selection p roblems which are known to be NP-hard in

general (Aazami, 2008). On the other hand, this ap-

proach contradicts the goal to have as few leaders as

possible. Thus overall, a signiﬁcant gap in literature is

the enha ncement of resilience without incre a sing the

number of leader ag ents. This paper addresses this

gap by proposing schemes to augment the graph by

additional edges in order to achieve strong structural

controllability, including the case that agents leave

and/or join the network.

The paper is structured as follows: Section 2 intro-

duces preliminaries on th e system class, the property

of stron g structural controllability, and the zero fo rc-

ing method. Section 3 f ormally states the problem

of en hancing resilien t strong struc tural controllability

through modifying the edges of the graph. Section 4

presents the main results, while Section 5 provides a

numeric example to illustrate the approach. Finally,

Section 6 concludes the p aper, summ arizing th e ﬁnd-

ings and their implications.

2 PRELIMINARIES

Networks are in this paper represented by a graph

G = (V ,E,W ), in which V = {1,...,N} is the set of

agents, E the set of edge s, and W the set of weights

assigned to each e dge according to w

i j

: E → R \ 0.

For simp liﬁcation, it is assumed that (i, j) ∈ E ⇔

( j, i) ∈ E, which simpliﬁes the notation. However,

i j

6= w

is allowed, making th e graph direc te d. It

is im portant to note, that the upcoming results can

be straightforwardly extended to cases where the as-

sumption of bi-directional edges is dropped. Edges

(i,i) for represen ting self-loops are permitted. Let all

neighbors of a node i be collected in a set N

= { j ∈

V |( j,i) ∈ E, j 6= i}.

The set of agents is divided into a set of leader

agents V

= {l

,. .. ,l

} ⊆ V and the set of follower

agents V \ V

. The m

leaders have a control input

(t) and the dynamics:

˙x

(t) = w

· x

(t) +

∑

j∈N

i j

· x

(t) + u

(t), (1)

while the dynamics of the followers:

˙x

(t) = w

· x

(t) +

∑

j∈N

i j

· x

(t) (2)

lacks an input. Note that with this deﬁnition the lead-

ers ar e allowed to interact dy namically with follower

and leader agents. The in itial state of any agent is de-

noted by x

(0) = x

i,0

The glob a l dynamics of all ag ents can be

derived b y introducing the state vector x(t) :=

(t),x

(t),... ,x

(t)]

∈ R

, the input vector

u(t) := [u

(t),u

(t),... ,u

]

∈ R

, the weight ma-

trix W with elements w

i j

, and the leader selection ma-

trix B ∈ [0 , 1]

N×m

accordin g to:

B =

(

i j

= 1 if i = l

0 otherwise.

(3)

The g lobal dynamics of the network ca n then by writ-

ten as:

˙x(t) = W x(t) + Bu(t), (4)

in which the leader agents can be controlled, whereas

the follower agents only respond to the beh avior of

the leader agen ts.

2.1 Strong Structural Controllability

As is well known, a system (4) is controllable if the

following matrix has full rank (Antsaklis and Michel,

1997):

R(W,B) =



B W B .. . W

n−1



, (5)

In networked systems, however, edge weights are of-

ten either unavailable or time-varying, rendering the

direct use of (5) impractical. T hen, strong structural

controllability (SSC) becomes relevant, as it assesses

controllability sole ly based on the system’s structu re,

indepen dent of the speciﬁc edge weights. To deﬁne

Enhancing Resilience of Strong Structural Controllability in Leader-Follower Networks

137

SSC, the following family of matrices associated with

a given graph

G is conside red:

W (

G) = {W ∈ R

N×N

: for i 6= j, w

i j

6= 0 ⇔ ( j, i) ∈ E}.

(6)

This matrix family includes all matric e s for which

the nonzero off-diagonal entries corre spond exactly

to the edges of

G. Note that the d ia gonal elements of

every W ∈ W (

G) can be arbitrary. With this notation,

SSC can be deﬁned as follows:

Def. 1 (Strong Structural Contro llability). A linear

system according to (4) is SSC if and only if the pair

(W,B) is controllable for all W ∈ W (

G), where B rep-

resents the agents through which inputs are applied.

Aiming at SSC according to this deﬁnition re-

quires that any possible matrix W of the given dimen-

sion needs to be considered, while the zero e ntries W

are encod ed also by the fact that a corre sponding edge

is not de ﬁned in E. Thus, it is sufﬁcient (without loss

of generality) from here on to refer the considerations

on SSC to a reduced version G = (V ,E) of the graph

G without weights.

2.2 SSC and Zero Forcing

The use of zero forcing as a method to examine SSC

is motivated by its direct co rrelation to network con-

trollability on graphs, as explained in (Monshizadeh

et al., 2014), see also Th e orem 1. Zero forcing is

deﬁned as follows (AIM Minimum Rank - Special

Graphs Work Group, 2008):

Def. 2 (Zero Forcin g). For a graph G = (V ,E), let

⊂ V , |V

| > 0 be chosen arbitrarily as an initia l

set of colored nodes. Let then V be partitioned into

and a set V

of initially uncolored nodes accord-

ing to V

∩ V

= ∅ and V

∪ V

= V . If a colored

node i then only has one uncolored neigh bor j, the n j

is forced to be colored by i (denoted by i ⇀ j), i.e. j

is inserted into V

and removed from V

. This color-

ing p rocess continues until n o further coloring can be

executed, leading to ﬁnal colored and uncolored sets

and V

1 2

= {1,4}

(G,V

) = {4 ⇀ 2,1 ⇀ 3,3 ⇀ 5}

(G,V

) = {4 ⇀ 2,2 ⇀ 5,5 ⇀ 3}

D(G,V

) = V

(G,V

) = {2, 3}

Figure 1: Example for zero forcing with V

as ZFS with

minimal cardinality.

Def. 3 (Derived Set). If zero fo rcing has started from

and terminated with V

, the latter set is also

called th e derived set and is denoted by D(G,V

) :=

Def. 4 (Zero Forcing Set). If a selected set of initially

colored n odes V

leads to D(G, V

) = V (i.e. all

nodes are colored at the end of zero forcing), V

called zero forcing set (ZFS) .

In ad dition the following terms are intr oduced:

Def. 5 (Forcing Chain). A forcing chain (FC)

C (G,V

) contains the list of forces i ⇀ j obta ined

during zero forcing for the graph G, where this list is

sorted according to th e sequence of forces carried out

during the forc ing procedure when starting from V

Def. 6 (Forcing Agen ts). For an agent j ∈ V , the

set F

(G,V

) contains all forcing agents (FA) i ∈ V ,

which can color j throu gh a force i ⇀ j con tained in

any fo rcing chain C (G,V

An example that illustrates these quantities can

be found in Fig. 1. As shown there, forcing chains

generally lack uniqueness, while the derived set

D(G,V

) remains unique ( A IM Minimum Rank -

Special Graphs Work Group, 2008). For any set of

FA, the inequality |F

(G,V

)| ≤ |N

| is always sat-

isﬁed beca use only neighbors are capable of color-

ing an agent. When V

constitutes a ZFS, it en-

sures that |F

(G,V

)| ≥ 1 for all j ∈ V

, guarantee-

ing that at least one FC for every u ncolored agent ex-

ists. Accordingly, |F

(G,V

, j)| = 0 for all j ∈ V

since these agen ts are colored initially.

The following theorem from (Monsh iz adeh et al.,

2014) establishes that zero forcing can determ ine if

a leader set V

guaran tees strong structu ral controlla-

bility fo r the system deﬁned by the graph G:

Theorem 1 ((M onshizadeh et al., 20 14)). System (4)

with W ∈ W and B deﬁned by the leade r set V

is SSC

if and only if V

is a ZFS for G.

2.3 Resilient SSC

In network controllability, resilience refers to the sys-

tem’s ability to remain controllab le despite agent mal-

functions or edge failures (Abbas, 2023). In terms of

structural controllability, the concept of a ZFS is ex-

tended to l-le aky forcin g sets, deﬁned as follows:

Def. 7 (l-Leaky Forcing Set). The leader set V

is a l−leaky forcing set (l−LFS), if ∀i ∈ V \ V

(G,V

) ≥ l + 1.

Building on the previous deﬁnition, (Abbas, 2023)

demonstra te s that an l-LFS provides resilience ag a inst

l malfunctions, where a malfunction m ay involve a

node departure or edge removal. ( N ote that the case

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

138

(G,V

) = {1}

′

) = {1; 4}

Figure 2: In graph G, node 1 has only node 5 as its only one

forcing agent, i.e., the system is no longer SSC if node 1 is

removed from the graph. In G

′

, which is obtained by adding

the edge indicated by the red dashed line to G, node 5 has

an additional forcing agent, and therefore the graph remains

SSC, if node 1 i s removed.

of entering agents is not considered here, but results

on this situation can be found in (Schmidtke et al.,

2024).

3 PROBLEM STATEMENT

While in (Abbas, 2023) the l-LFS guarantees re-

silient SSC by introducing additional leader agents,

the present work aims at improving resilience through

edge augmentation, i.e., by adding new edges to the

graph.

Formally, given a gra ph G = (V ,E ) with a ZFS

, the addressed problem is to enhance resilience by

adding an edge (i, j) (where i, j ∈ V and (i, j) 6∈ E)

such that:

(G,V

)| < |F

′

)|, (7)

for G

′

= (V ,E ∪ (i, j )).

This approach, which can be applied iteratively,

follows the principle of the l-LFS methodology by in-

creasing the n umber of FA for critical nodes. How-

ever, this is ach ieved through the addition of edges

rather th an by expanding the leader set. The method

is par ticularly well-su ited for scena rios in whic h bot-

tlenecks or vulnerable regions of the network ar e

known in advance. In such cases, edge augmenta-

tion can be applied in a targeted manner to strengthen

the resilience of speciﬁc nodes, thereby improving

resilience without the need to intr oduce additional

leader agents.

For instance, in scenarios like the one in Fig. 2,

resilience is ensured by leveraging the result from

(Schmidtke et al., 2024): SSC is preserved, when

an agent j leaves G, provided that for all uncolored

neighbors i ∈ N

the following holds:

(G,V

) \ j| > 0. (8)

To date, no publishe d meth od systematically explores

possibilities of modifyin g the set of FA through leader

selection or edge augmentation. Thus, the following

section proposes a framework for strategically mod-

ifying the network topology to enhance resilience of

SSC.

4 IMPROVING RESILIENCE BY

EDGE AUGMENTATION

The primary objective o f this section is to establish

conditions that guarantee (at least loca lly) enhanced

resilience of SSC by augmentin g the set E of the

graph G by an additional edge (i, j). To ac hieve this,

the no tion of a forcing graph is introduced. The latter

enables the analysis o f the forcing agents of G by

examining how the addition of the edge affects the

network’s resilience. Based on this analysis, a main

theorem is formulated stating conditions under which

the augmentation leads to improved resilience. Fur-

thermor e, this section shows that, under the derived

conditions, a ZFS in G retains the SSC properties,

ensuring that resilience is either glo bally or locally

enhanced.

A forcing graph is introduced to determine whic h

nodes are c olored before a speciﬁc uncolored n ode,

under the conditio n that a designated co lored node

does not color any o ther node. The formal deﬁnition

is as follows:

Def. 8. Give n a graph G = (V ,E), let i ∈ V

be an

uncolored node and j ∈ V

a colored node. A forc-

ing graph

i j

= (

V ,

E) is constructed by in troducin g

two dummy nodes, α and β, thus

V = {V ∪ {α; β}} ,

and a modiﬁed set of edges is obtained from

E =

{E ∪ {(α,l) | l ∈ N

} ∪ {(α, j),(β, j)}}. In here, α

is connected to all neighbors of i and to j, while β is

only connected to j.

An example of a forcing gra ph is shown on the

left h and side of Fig. 3, where the graph from Fig. 2

is expanded by the nodes i = 5 and j = 4.

Note that the forcing graph is a direct extension of

the extended graph, introduced in (Schmidtke et al.,

2024), w hich is denoted in this paper by

i{}

. In

this extende d graph, which can be used to determine

the FA of node i over all FC (see (Schmidtke et al. ,

Enhancing Resilience of Strong Structural Controllability in Leader-Follower Networks

139

2024)), the node α is only connected to the neighbors

of i, and β has no neighbors.

The following result can be obtained for the de-

rived set of the forcing graph:

Lemma 1. Using a ZFS V

l,1

for G, the following con-

dition holds true for the forcing process on the forcing

graph

i j

for every p air (i, j) with i ∈ V

, j ∈ V

{α,β,i} 6∈ D(

i j

l,1

). (9)

Proof. Based on the deﬁnition o f

i j

, it follows

that N

⊂ N

. Consequently, node i can only b e

colored after node α has been colored. Given that

\ N

= { j}, node j is the only node being able to

color α. Furthermore, due to the construction of

i j

the condition N

= { j} always holds. Neither α nor

β are initially colored, and node j c annot color α and

β. Thus (9) follows. 

The forcing graph can be used to draw conclusions

about the forcing process for G based on the derived

set:

Lemma 2. Consider V

l,1

as a ZFS for G. For all

v ∈ D(

i j

l,1

) with i ∈ V

and j ∈ V

, an FC

C (G,V

l,1

) = { c

,. .. ,c

|V |−|V

l,1

} exists with c

= u

′

⇀

′

such that:

• ∃ c

∈ C (G,V

l,1

) with:

= u

′

⇀ v, c

= ˜v ⇀ i, i

< i

(10)

• ∄ c

∈ C (G,V

l,1

) with:

= j ⇀ ˜u and i

< i

. (11)

Proof. Lemma 1 states that a forcing graph

i j

for a ZFS of G prevents the coloring of agent i.

Consequently, v ∈ V

∩ D(

i j

l,1

) implies the

presence of a FC C (G,V

l,1

) in G with the same

leader set V

l,1

, where all nodes v are co lored prior to

the coloring of node i, according to (10). Moreover in

i j

, node j consistently has two uncolored neighbors

that remain uncolored (as established by Lemma 1),

ensuring that all n odes v can be colored before node

i without node j coloring any of its neighbors. This

directly validates cond ition (11). 

Corollary 1. If v 6∈ D(

i j

l,1

) with V

l,1

as a ZFS

for G, then ∄ C (G, V

l,1

) such that conditions (10) and

(11) are satisﬁed.

It is shown next how the for cing graph can be used

in order to ﬁnd an edge (i, j) to speciﬁcally add node

j to the set of FA of another node i:

Theorem 2. Consider i ∈ V

, j ∈ V

, i 6∈ N

and

l,1

as a ZF S for G. Adding the undirected ed ge (i, j)

to G, resulting in G

′

= (V , E

′

) with E

′

= {E ∪(i, j)}

leads to j ∈ F

′

l,1

) if and only if the following

condition holds for all nod es l ∈ N

l ∈ D(

i j

l,1

). (12)

Proof. Based on Lemma 2, condition (12) satisﬁed

∀l ∈ N

implies that a FC C (G,V

l,1

) exists for which

(10) and (11) holds. This implies th at all neig hbors of

j can be colored before i is colored, without requiring

j to force any of its neighbors. Thus, the force j ⇀ i

is possible in G

′

, leading to j ∈ F

′

l,1

To demo nstrate the necessity of condition (12), con-

sider the case in which the edge (i, j) is added to G,

while

∃ l

′

∈ N

: l

′

6∈ D(

i j

l,1

) (13)

applies. According to Corollary 1, ∄ C (G, V

l,1

) such

that l

′

is co lored while (10) and (11) are true, i.e.

either j ⇀ l

′

or i

> i

with c

: ˜v ⇀ l

′

, c

= u

′

⇀ i.

In the ﬁrst case, edge (i, j) prevents j ⇀ l

′

because

i is an additional uncolored neighbor, making it

impossible f or j to color i or l

′

, consequently

j 6∈ F

(G,V

l,1

). In the second ca se, j 6∈ F

(G,V

l,1

)

follows directly from the fact that i is colore d before

′

, implying that up to the point of the coloring of

i, node j has more than one uncolored neighbors,

preventing j ⇀ i. Thus (12) has to apply ∀l ∈ N

such that j ∈ F

′

l,1

) after adding (i, j) to graph

G, thereby establishing the nece ssity of (12). 

Fig. 3 shows an example in which condition (1 2)

holds ∀l ∈ N

, such that adding edge (5, 4) to the

graph resu lts in 4 ∈ F

′

) (as already illustrated

in Fig. 2). Note tha t the computational complexity of

verifying condition (1 2) for a speciﬁc pair of nodes

is equivalent to that of computing D(

i j

l,1

), for

which the effort is of order O(N + |E|) (see (Brimkov

et al., 2019)).

In order to show that adding the edge (i, j) by The-

orem 2 increases the cardinality of the set of FA of i,

as required in (7), the following result is derived ﬁrst:

Lemma 3. For l ∈ F

(G,V

l,1

), let an edge (i, j) b e

added to G u nder the conditions of Theorem 2, lead-

ing to G

′

. Then l ∈ F

′

l,1

) holds.

11 22

αα

ββ

)

Figure 3: Left: The forcing graph corresponding to graph G

from Fig 2 with i = 5 and j = 4. Right: The derived set of

the forcing graph wi th V

being a ZFS for G.

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

140

Proof. Sinc e l is a forcing agent of i in graph G, l

and all its neighbors except of i are colored bef ore i is

colored. This can be expressed by the forcing graph

(cf. (Schmidtke et al., 2024) Theorem 3):

l ∈ D(

i{}

l,1

) ∧ N

\ i ⊂ D(

i{}

l,1

). (1 4)

Theorem 2 ensures that all neigh bors of j can be col-

ored by other nodes than j, before i is color ed. There-

fore, no c oloring in the derived set of the forcing

graph depends on j, implying:

i j

l,1

) = D(

i{}

l,1

). (15)

From th is equality, it directly follows that:

l ∈ D(

i j

l,1

) ∧ N

\ i ⊂ D(

i j

l,1

). (16)

This implies that adding the edge (i, j) does not

change the coloring of all FA l ∈ F

(G,V

l,1

) in G

′

since the coloring of each of these no des does not de-

pend on the co loring of j, as shown by (15). T hus,

each FA can still be colored before i is colored. This

eventually means th at:

l ∈ F

(G,V

l,1

) =⇒ l ∈ F

′

l,1

), (17)

holds if G

′

results from adding (i, j) to G under th e

condition of Theorem 2 holds. 

Lemma 3 shows that all FA in G remain FA in

′

, if the edge is added under the conditions of Theo-

rem 2. T hus, as a conseque nce of Lemma 3 together

with Theorem 2, the following result can be stated

which directly validates (7):

Corollary 2. If the graph G is augmented by adding

the edge (i, j) and if Theorem 2 holds, then

(G,V

l,1

)| < |F

′

l,1

)|, (18)

holds true.

While augmenting G with an edge (i, j) under the

conditions of Th eorem 2 gu arantees an increase in the

number of FA for no de i, Corollary 2 does not pro-

vide insights into the set of FA of other nodes. To

address this shortcom ing, the following proposition

demonstra te s that a ZFS of G remains valid for the

augmen te d gra ph G

′

under the same conditions:

Proposition 1. If V

l,1

is a ZFS of G a nd if an edge

(i, j) is added to G accrding to Theorem 2, V

l,1

re-

mains a ZFS for the resulting graph.

Proof. The graph G

′

follows from G, by connecting

an uncolored node i with a co lored node j, while l ∈

D(G

i j

l,1

) holds ∀l ∈ N

with V

l,1

being a ZFS for

G. This condition directly implies:

l ∈ D(G

{} j

l,1

) ∀ l ∈ N

, (19)

where G

{} j

is the forcing graph with α a nd β only

connected to node j. Th is means tha t all neighbo rs

of j can be colored by other nodes in G

{} j

, ensuring

that the addition of the edge (i, j) does not disrupt the

zero f orcing process initiated by V

l,1

. Therefore, V

l,1

remains a ZFS of G

′

. 

Since V

l,1

remains a ZFS in G

′

, the fo llowing con-

dition holds true ∀i

′

∈ V

\ i:

′

l,1

) ≥ 1. (20)

Comparing the set F

′

l,1

) of each ag ent i

′

∈ V

i to F

′

l,1

), the following two cases can occur:

′

l,1

)| ≥ |F

′

(G,V

l,1

)| ∀i

′

∈ V

\ i or (21)

∃i

′

∈ V

\ i : |F

′

l,1

)| < |F

′

(G,V

l,1

)|. (22)

In case (21), the number of FA of all uncolored node s

stays the sam e or has increased. This is the desired

case, as the resilience of the whole networked system

has increased. In the secon d case (22), the number

of FA has decreased for at least one unco lored node

in the graph. This implies that, while the resilience

has increased for at least node i, it has decreased fo r

at least one oth e r node. Thus, in this case, only a

local increase in resilience is achieved. This can be

advantageous, for instance, when a f orcing agent is

added to a node with a small FA set, even if anoth er

node with a large FA set loses one of its FAs in the

process.

Fig. 2 illustrates the case (21). I nterestingly, in

this case, the aug mentation by the edge (5,4) makes

the leader set V

= {1; 2; 3; 4; 8;9} a 1−LFS of graph

′

, meaning that the system stays SSC despite any

removal of an edge or a node.

Investigating how to ensure the desired case (21)

will be a focus of f uture work. The following section

investigates how frequently the conditions introduced

hold in random graphs, allowing for the addition of

edges while enhancing resilient strong structural co n-

trollability.

5 NUMERICAL EXAMPLE

This section investigates the effectivene ss of the pro-

posed approach through numerical experiments for

random gra phs. The study proceeds in two parts:

ﬁrstly, the proposed approach is evaluated w ith re-

spect to graph size, network density, and leader set

size, and secondly, the approach is compared to an

existing method from literature.

Enhancing Resilience of Strong Structural Controllability in Leader-Follower Networks

141

5.1 Analysis Across Network

Parameters

The following three questions are addressed through

numerical experiments:

• How frequently ca n an edge be found that en-

hances resilience , either locally or globally, de-

pending on the size and density of the network?

• Which share of edges contributes to resilience of

the whole network?

• How does the size o f the leader set affect the oc-

currenc e of edges which enh ance resilience?

To investigate these questions, Erd¨os–R´enyi graphs

G(N, p) are used, in which N represents the set of

nodes and p denotes the probability of an edge ex-

isting between two nodes (Erd¨os an d R´enyi, 1959).

A higher value of p results in a den ser network, as it

increases the expected number of edges in the graph .

The ana lysis focuses on the addition of edges to crit-

ical nodes, deﬁned as node s w ith a forcing agent set

of cardinality one, i.e., K : {i ∈ K | F

(G,V

) = 1}.

These node s represent structural bottlenecks in the

sense that SSC depends on a single neighbor to ensure

controllability. The removal of this neighbor r esults in

a loss of SSC. To mitigate this v ulnerability, the pos-

sibility of adding FA by adding edges is examined,

which would enhance resilience without introducin g

further leader agents.

The set

E consists of edge s satisfying Theorem 2,

with i ∈ K and j ∈ V

. The sub set of edges that

enhance resilience for the whole network is denoted

by E

⋆

, mea ning that the condition (21) holds wh e n

any edge in E

⋆

is added to G. Furthermore, the ratio

⋆

|/|

E| reﬂects the share of added edges for which

condition (21) h olds, relative to all edges that can be

added according to Theorem 2. For all of the upcom-

ing tests, 10 instances of graphs are randomly gen-

erated and the average is shown below (conside ring

only connected graphs G(N, p)). In all tests V

l,min

denotes the leader set with min imal cardinality deter-

mined by exhaustive search.

First, the dependency of the sizes of K ,

E, and

⋆

on the size |V

| o f the leader set is examined for a

ﬁxed number of agents N = 20 and an edge probabil-

ity of p = 0.2. To increase the size of each leader set,

an additional leader is randomly chosen from among

the unc olored nodes. The results are presented in Ta-

ble 1.

For the leader set V

l,min

, only very few edges in

E c a n be identiﬁed, and none of them belongs to E

⋆

This occurs because, in the smallest possible leader

set conﬁguration, there is typic ally not enough redun-

dancy to color the neighbors of a leader if the leade r

Table 1: Evaluation of the effect of the size of the leader set

size on the number edges which can be added (according

to Theorem 2) to critical nodes K in random graphs with

N = 20 and p = 0.2.

|K | |

E| |E

⋆

| |E

⋆

| · |

−1

| = |V

l,min

| 7.6 0.5 0 0

| = |V

l,min

| + 1 5.8 5.8 2.6 0.45

| = |V

l,min

| + 2 3.6 6.6 3.1 0.47

| = |V

l,min

| + 3 2.1 5.2 3 0.58

itself is prevented f rom coloring, as required by The-

orem 2. As a result, edges in

E are rarely identiﬁed

in this setting. Increasing the cardinality of the leader

set leads to a reduction in |K |, while the size of

E in-

creases. Notably, the portion of edges in

E that also

belongs to E

⋆

appears to grow with the size of the

leader set. Addin g just one agent to V

l,min

already re-

sults in a signiﬁcant increase in both |

E| and |E

⋆

|. For

this reason, this conﬁguration is used in the f ollowing

simulations.

Next, the inﬂuence o f the network density on th e

applicability of Theorem 2 is investigated. To this

end, the edge probability p is var ie d, where smaller

values result in sparser networks and larger values

correspo nd to denser top ologies. The results for a

ﬁxed number of N = 20 nodes are summarized in Ta-

ble 2.

Table 2: Investi gation of the effect of the network density

on the number of edges which can be added according to

Theorem 2, considering criti cal nodes K in random graphs

with N = 20 and |V

| = |V

l,min

| + 1.

p |V

| |K | |

E| |E

⋆

| |E

⋆

| · |

−1

0.15 6.4 8.6 14.8 7 .5 0.51

0.2 7.1 5.8 5.8 2.6 0.45

0.25 8.6 5.8 5.5 4.1 0.75

0.3 9.5 5.1 4.3 3.3 0.77

0.35 10. 5 3.4 4.4 1.8 0.41

With increasing network density, a larger leader

set is required to ensure SSC, while the number of

critical nodes K decreases – an effect already ob-

served in Table 1. I n con trast, sparser networks allow

for a greater number of edges in

E, as redundancy

in forcing is easier to establish. The share of edges

in E

⋆

also belonging to

E ranges between 41% and

77%, with the maximum observed at p = 0.3 and the

minimum at p = 0 .35. Investigating the cause o f this

variation is subject of future work.

Lastly, the inﬂuence o f the number of n odes N o n

ICINCO 2025 - 22nd International Conference on Informatics in Control, Automation and Robotics

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the applicability of Theorem 2 is examined. To this

end, N is va ried while keeping the edg e probability

p and |V

| constant. The results are summarized in

Table 3.

Table 3: Effect of the graph size on the number of edges

which can be added according to Theorem 2 for cri ti-

cal nodes K in random graphs w ith p = 0.2 and |V

| =

l,min

| + 1.

N |V

| |K | |

E| |E

⋆

| |E

⋆

| · |

−1

10 4.3 3 3.7 2.7 0.73

15 5.5 5.8 8.4 4.6 0.55

20 5.8 5.8 5.8 2.6 0.45

25 10 .4 6.2 5.8 2.7 0.47

30 13 .3 6.5 7.4 4.2 0.57

As the g raph size increases, more leader agents

are required to ensure SSC, and the number of critical

nodes K also grows. However, the size of

E a ppears

to be largely inde pendent of N. The same holds for

the ratio |E

⋆

|/|

E|, with the highest value observed

for N = 10, followed by N = 30, an d the lowest for

N = 20 .

In summary, increasing the cardinality of the

leader set slightly beyond V

l,min

has the most signiﬁ-

cant impact. The n umber o f edges in

E increases with

sparser and larger graphs. In contrast, the portion of

edges in E

⋆

relative to

E (i.e., those improving re-

silience at the network level) appears largely indepen -

dent of the graph size and density, ranging be twe en

41% and 77%.

5.2 Comparison of Edge Augmentation

and an LFS Approach

The proposed method is now compared to the use

of a 1-LFS (cf. Section 2.3), i.e., a set of leaders

that guarantees SSC is maintained in the case a sin-

gle agent leaves the network. The evaluation is per-

formed on the Erd¨os–R´enyi graph G with N = 30

and p = 0.12 shown in Fig. 4. A critical node is as-

sumed to be k nown: with the ze ro-for c ing set V

{1,2,3,4,6,10,20,23, 29, 30} (of size |V

l,min

| + 1),

node 11 is critical since F

(G,V

) = {12}. Thus, if

node 12 leaves the graph, node 11 cannot be colored

and th e network loses the SSC property.

To evaluate condition (1 2), the derived set on the

forcing graph

i j

is computed for i = 12 an d j ∈ V

For j = 1 , Theo rem 2 holds, implying th at adding the

edge (1,11) to the graph expands the set of forcing

agents of node 11 to F

′

) = {1,12}. Conse-

quently, even if node 1 2 leaves, th e system remains

SSC since the coloring of no other agent depends

solely on node 12. The evaluation of Theorem 2 re-

quires 60 ms in MATLAB R2021b on a machine with

32 GB RAM and an Intel Core i7-14700 processor.

In contrast, the 1-LFS V

l,1-LFS

{1,2,4,5,7,10,11,13, 14, 20,23,30}, obtained

by using the heuristic from (Abbas, 2023), is com-

puted in 4.12 s and introduces two additional leaders

in comparison to V

. Since the procedure is heuristic,

minimality of the set size is not guaranteed. Usin g the

integer-programming-based method f rom ( A la meda

et al. , 2024) to compute a 1-LFS of minimal cardi-

nality does not produce a solution within one hour

of computation, which illustrates the hardness of the

problem.

Figure 4: Erd¨os–R´enyi graph G(30,0.12) used for evaluat-

ing resilience. Node 11 is a critical node when considering

the zero-forcing set V

= {1,2,3, 4,6,10,20,23,29,30} (in-

dicated by the black nodes). Adding the dashed edge (1,11)

expands t he set of forcing agents of node 11 by including

node 1 to enhance the resilience of the network. T he red

nodes indicate a 1−LFS for the network.

6 CONCLUSIONS

Conditions were spe ciﬁed for identifying edges that

can be added to a network in order to enhance resilient

strong structural controllability, eith er locally or f or

the entire network. The proposed method is based on

the notion of forcing graphs, which extend the origi-

nal graph by connecting dummy agents in a speciﬁc

manner. Utilizing the concept of zero-forcing, a con -

dition was estab lished that must hold fo r the neigh-

bors of a lead er agent in the forcing graph. This con-

dition ensures that adding an edge from the leader

agent to a fo llower agent in c reases the follower’s re-

silience.

Simulation studies demonstrate that such edges

can frequently be identiﬁed when the size of the

leader set exceeds the minimally required size to

Enhancing Resilience of Strong Structural Controllability in Leader-Follower Networks

143

achieve strong structur al controllability. Furthermore,

the sparser and larger the graph is, the more edges can

be found that enhance network resilience. Notably,

the portion of added edges that improve resilience

globally, rather than just locally, remains independent

of the graph’s sparsity and size.

Future research should explore how the proposed

method can be extended to directly ide ntify edges

which enhance resilient SSC for the entire network.

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