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An Efﬁcient Approach to Enhance the Robustness

of Scale-Free Networks

Syed Minhal Abbas1, Nadeem Javaid1,∗, Muhammad Usman2, Shakira Musa Baig1,

Arsalan Malik1, Anees Ur Rehman1

1Department of Computer Science, COMSATS University Islamabad, Islamabad 44000, Pakistan

2Department of Electrical and Computer Engineering, COMSATS University Islamabad, Islamabad 44000, Pakistan

Email: minhal.abbas514@gmail.com, usmaneng9@gmail.com,

Shakira.musabaig1998@gmail.com, arsalanmalik8936@gmail.com, anees.nizami@hotmail.com,

∗Corresponding Author: nadeemjavaidqau@gmail.com; www.njavaid.com

Abstract—In this paper, we aim to increase the robustness

of Scale-Free Networks (SFNs) that are robust against random

attacks; however, they are fragile to malicious attacks. When the

highest-degree nodes are removed from the SFNs, they greatly

affect the connectivity of the remaining nodes. In the proposed

method, we perform two attacks, namely, the high degree link

attacks and the recalculated high degree link attacks. These link

attacks affect the core part of the network. To measure the

extent of damaging the network at a greater portion, we use

the closeness centrality and the eigenvector centrality. The goal

is to construct a topology that shows better robustness against

different malicious attacks with a minimum computational cost.

The Pearson’s correlation coefﬁcient is used to select the measures

that are positively correlated with each other. Based on these

centrality measures, we optimize the network robustness for

both node and link attacks simultaneously. The optimization is

performed based on two efﬁcient measures, which are computa-

tionally less expensive to increase the network robustness.

Index Terms—Scale-free networks, Malicious attack, Robust-

ness, Network topology, Centrality measures, Optimization.

I. INTRODUCTION

The Internet of Things (IoT) has made remarkable progress

in the last decade. With the increasing number of smart

devices, the digital world is growing rapidly. The IoT allows

manual processes to be automated into the digital world. The

IoT generates the necessary data and includes it in real-time

applications. With the evolution of IoT, its applications are

spread in many ﬁelds, including health system [1], transporta-

tion system [2], [3], medical monitoring system [4], etc.

In complex network theory, there are mainly two types of

models: the Small World Networks (SWNs) and the Scale-Free

Networks (SFNs) [5], [6]. The SWNs are constructed by using

heterogeneous nodes. These nodes have different communi-

cation range, energy, and bandwidth. Moreover, the SWNs

contain a small average path length and a high clustering coef-

ﬁcient [7]. Whereas, in SFNs, the degree of nodes is distributed

according to the power law. With this distribution, the high

degree nodes are less in number as compared to low degree

nodes that make connections with similar types of nodes [8].

On the other hand, the SFNs are constructed by homogeneous

network topologies [9]. The homogeneous nodes have the

same bandwidth and communication range. Therefore, the

network containing the SFNs’ property is usually used in IoT

due to the same bandwidth and transmission range. However,

it is challenging to generate a network topology that is robust

against the malicious attacks [9], [10].

Schneider et al. [11] is introduced a metric to calculate the

robustness R of network against the malicious attacks. The

robustness of the network is calculated as,

R= 1/N

N

X

Q=1

s(Q).(1)

In the above equation, s(Q)represents the fraction of

removed nodes, Nrepresents the total number of nodes. The

Rvalue is calculated by analyzing the connectivity of nodes

in a network, which is followed by removing nodes until the

network is damaged. Therefore, we need to make a network

topology that show robustness against the malicious attacks.

Also, the probability of malicious attacks is based on the

intelligent selection of nodes that is growing increasingly [12],

[13].

The authors in [14] consider the degree and Betweenness

Centrality (BC) and perform attack on both nodes and links

to evaluate the nodes’ importance. Therefore, when high

degree nodes are attacked, the network breaks down into

sub-networks. In a network, random and malicious attacks

may occur simultaneously. Among them, the malicious attacks

damage the network to a greater extent because they occur on

the most inﬂuential nodes in the network [9], [15].

Several strategies are used to make the network robust

against malicious attacks. The SFNs have a high ability to

resist random attacks. However, they are weak against mali-

cious attacks [16]. Therefore, an efﬁcient topology is required

to design the network robust against malicious attacks [17].

Several optimization techniques including memetic algorithm

[18], multi-population genetic algorithm [19], enhanced differ-

ential evolution [20], natural connectivity model [21], greedy

model [22], and elephant herding optimization [23] are used

to enhance the network robustness. These techniques make the

network onion-like without changing the degree of nodes. The

structure is robust against both random and malicious attacks.

Our contributions in this paper are as follows.

•Two efﬁcient measures, namely, Closeness Centrality

(CC) and Eigenvector Centrality (EC) are used to ﬁnd

out the most inﬂuential nodes in the network with less

computational cost.

•The Pearson’s correlation coefﬁcient is used to determine

the relationship between robustness measures based on

CC and EC and robust the network for both node and

link attacks.

•Two attacks are performed, High Degree Link Attack

(HDLA) and Recalculated HDLA (RHDLA). These at-

tacks help the attacker to affect the network as the greater

portion of the network is damaged in a less time.

The rest of the paper is structured as follows. Section II

contains the related work. The proposed model and description

are discussed in Section III. Section IV explains the simulation

results of our proposed schemes and Section V concludes the

paper.

II. RE LATE D WOR K

Previous study uses single-objective optimization with

crossover and mutation. The network that is robust against

node attacks is not necessarily robust against the link attacks

[10]. In the real-world problem, multiple malicious attacks are

not simultaneously considered to optimize the robustness of

a network [14]. Traditional Genetic Algorithm (GA) is used

to make the network robust, premature convergence occurs

due to loss of diversity in the population [19]. Simulated

Annealing (SA) attempts to ﬁnd the best solution with the

worst probability and ﬁnds a better solution, leading to global

optima. The algorithm requires unnecessary comparisons to

optimize the network robustness caused by computational

overhead [18].

With the increase of IoT in smart cities, the network is

prone to cyber-attacks that include malicious and random

attacks. The network is required to be robust against cyber-

attacks. However, many redundant operations cause computa-

tional complexity [20]. In [24], authors use the large SFNs

exploring the hidden features in solution space to enhance the

robustness of the network; however, the computational cost

occurs. The network structure is robust on speciﬁc types of

attacks; however, the structured properties are ignored in a

large complex network [21], [25].

Machine Learning (ML) techniques are used to optimize

the network topology against cyber-attacks. SFNs are efﬁcient

regarding robustness against the random attacks; however,

vulnerable to the malicious attacks [26], [27]. The robust

networks are the essential support for the IoT applications such

as smart cities, intelligent vehicles, smart devices [28], etc.

However, resource-constrained IoT devices and various types

of attacks are challenging against the network’s robustness

and communication capability [29]. ML-based algorithms are

designed to optimize the network robustness against malicious

attacks. Modiﬁcation of data regarding the swapping edges has

inﬂuenced the structure of the network. Constructing a network

robust to withstand malicious attacks is still a challenging issue

in the optimization problem [30], [31].

The robust network against node attacks is not necessarily

strong against the links attack simultaneously [14]. In smart

cities, nodes collect data from different domains [32]. There

is an issue to deal with big data with the network’s robustness

against the malicious attack [19]. GA is used to enhance a

scale-free network’s robustness against malicious attacks as

well as random attacks [19]. Multiple methods are used to

overcome malicious attacks and cascading failures separately

[15], [33], [34]. The evolution algorithm is used the two

operators to optimize the network’s tolerance using ﬁxed

probability rates [19]. However, there is an issue in tackling

the large SFN exploring the hidden features in solution space.

Moreover, the improved network structure is based on speciﬁc

types of attacks.

SFNs are tolerant of random attacks; however, prone to

malicious attacks [16]. Fault-tolerant based network topology

evolution scheme is proposed. A new node joins the network

based on preferential attachment and fault probability. Qiu et

al. [12] propose the scheme that makes the network into the

onion-like structure by performing the degree difference and

angle sum operations. The same degree nodes connect with

each other; however, redundant operations are performed due

to edge swapping. The node’s failure with fault and cyber-

attacks affects the communications among the networks [33].

Furthermore, some of the nodes in the network are more

interconnected internally as compared with the rest of the

network, thus forming the community.

It is an important feature of the network structure. There-

fore, the network is optimized in each community by con-

sidering the onion-like structure [36]. Therefore, robustness is

enhanced along with the preservation of the network commu-

nity.

III. SCA LE -FRE E NET WORKS TOPOLOGY AND

ROBUSTNESS OPTIMIZATION

In this section, the proposed system model is designed, as

shown in Fig. 1. In the ﬁgure, three limitations to be addressed

are mapped to the proposed solutions.

A. Construction of Initial Topology

We construct the network using the Barab´

asi Albert (BA)

model [8]. The model generates the SFNs topology by fol-

lowing the power-law distribution. Initially, a network is con-

structed from a small clique. A new node enters in the network

based on the preferential attachment. The node having a high

degree has more probability to be selected by calculating the

degree of the neighboring node through the roulette wheel

mechanism. The communication range and energy resources

of nodes are limited because the high degree nodes consume

more energy, this impacts the overall performance of the

network. Thus, the maximum degree of nodes is limited [37].

B. Centrality Measures

In the proposed system model, we introduce two measures

CC and EC, to ﬁnd the most inﬂuential nodes in the network.

These measures have less computational cost as compared to

BC. Moreover, CC does not have to determine how many

shortest paths pass through speciﬁc nodes that are required

in BC.

b) Topology after

Attack

b) Topology after

Attack

Onion-like Structure

Eigenvector Centrality

Closeness Centrality

i

j

k

l

i

j

kl

i

j

k

l

Smart Rewiring

Mechanism

a) Influential Node

Attack

L1=>S1

L3=>S3

L2=>S2

L1-Betweenness centrality.

L2-Network optimized against

node or link attacks.

L3-Attacker utilizes information

of nodes for attack.

S1-Closeness and eigenvector

centrality.

S2-Optimized network against

both attacks.

S3-HDLA and RHDLA highly

affect the network.

a) High Degree

Link Attack

Fig. 1. Proposed System Model

The CC measure provides the network’s global information.

It ﬁnds nodes’ centrality based on the relative distance between

each pair of nodes. The centrality of a node ﬁnds the broad-

caster i.e., the communication of one node to all other nodes

is efﬁcient In this process, the shortest distance is calculated

between a speciﬁc node to other nodes in the network.

The CC cof a node xis calculated using the equation

deﬁned as:

cx= 1/X

y

d(y, x).(2)

Where drepresents the shortest distance between two ver-

tices yand x[38].

While, EC measures the importance of nodes in the network.

It assigns a relative score of xto each node. For a given

graph, G= (v, e), the relative score xof EC score a vertex v

connected with other vertex with an edge ecan be deﬁned as:

xv= 1/λ X

t∈G

av,tx(t)(3)

Where arepresents the adjacency matrix, vand tare two

vertex, and λis a constant factor [39].

The EC ﬁnds the node inﬂuencing the entire network and

assigns the relative index value to all nodes based on the

neighbor’s connection. The node containing the high indexed

value contributes more to that node having a low index.

Using the above measures, we ﬁnd the most inﬂuential

nodes in the network to perform the attacks.

C. Centrality Based Malicious Attacks

In the proposed model, three types of malicious attacks

based on degree, CC, and EC are performed. These attacks

affect the functionality of the remaining nodes in the network.

These attacks are performed on both nodes and links. However,

the attacker utilizes the information of high degree nodes

in the network and removes the links. Therefore, in the

proposed system model, we consider two links attacks, namely,

HDLA and RHDLA. In HDLA, the links between two high

degree nodes in the network are removed based on the initial

information. Whereas, in RHDLA the links degree between

two nodes is recalculated after every node’s removal.

D. Optimization of Network Robustness

Our focus is to ﬁnd the robustness of the network with the

limited number of attacks using the provided information from

CC and EC measures [38], [39]. Based on the robustness mea-

sure for nodes and links, we ﬁnd out the Pearson’s correlation

coefﬁcient among them. The optimization is performed on two

positively correlated measures. The topology that increases the

robustness for both nodes and link attacks is considered an

optimized network topology with less computational cost as

shown in Fig.1. A smart rewiring mechanism is adopted in the

proposed model, that helps to convert the scale-free topology

into an onion-like structure.

In Table 1, L1 section contains a measure to calculate

a node’s centrality using betweenness to make the network

robust. The computational cost to ﬁnd the inﬂuential node

based on the BC is high as compared to the node’s selection

by considering node degree removal [14], [16]. Two efﬁcient

measures CC and EC ﬁnd the importance of a node in less

time as mentioned in S1 section. Using these measures, we

will construct a network that is efﬁcient against malicious

attacks. Also, it reduces the computational cost and optimizes

the network robustness. In V1 section, the execution time of

CC and EC is validated in Fig. 2. These measures have less

computation time as compared to BC.

The network is optimized on the basis of robustness against

node attack Rnand robustness against link attack Rlin L1

section. Using the proposed centrality measures, we design

a network topology that is fragmented with less number

of attacks. We use Pearson’s correlation coefﬁcient between

Rnand Rl. The positively correlated measures are used to

enhance the network robustness. Various attacks are performed

to strengthen the network, which includes CC, EC, HDLA,

and RHDLA, as explained in S2 section. To enhance network

robustness, we use the smart rewiring mechanism to achieve

the minimum computational cost and faster convergence that

leads to the optimal solution.

TABLE I

MAP PIN G OF LIMITATIONS WITH PROP OS ED SOL UT ION S AN D THE IR VALI DATIO NS .

Limitations Identiﬁed Solutions Proposed Validations

L1: The BC measure increases the compu-

tational cost to ﬁnd the inﬂuential node [14].

S1: CC and EC ﬁnds the inﬂuential node in

less computational cost.

V1: Validate the execution time with two

centrality measures in Fig. 2.

L2: Network is not robust against different

types of malicious attacks [27].

S2: Network is optimized simultaneously

against different types of node and link

attacks based on CC, EC, HDLA, and

RHDLA.

V2: Network performance is validated with

robustness values in Fig. 4.

L3: Attacker utilizes node’s information to

perform attacks [12], [17], [20], [24].

S3: Introduce links as well as node attacks. V3: Evaluate the network connectivity by

performing attacks in Fig. 3.

L3 section contains attacks on important nodes [12], [17],

[20], [24]. Two link attacks, HDLA and RHDLA, based on

the two high degrees nodes are performed on the network, as

explained in S3 section. These attacks will affect the whole

network due to the failure of a high degree edge to make

the network robust. The V3 section evaluates the network

connectivity against different malicious attacks, including BC,

CC, EC, HDLA, and RHDLA.

IV. SIMULATION RESULTS AN D DISCUSSION

SFNs are used to evaluate the efﬁciency of our model with

ROSE and SA algorithms. The simulations are conducted

in MATLAB. In a sensor ﬁeld of 500*500 m2, nodes are

deployed randomly. The modeling process requires each node

to have a limited number of neighbors, and the communication

range r is set at 200m to indicate the communication radius.

In the simulations, the total number of nodes (N) is set to

100, the edge density (m) = 2 and the maximum value of the

node’s degree is 25. We evaluate the results for 100 iterations

in each algorithm. We observe that our model converges to

optimal results within 100 iterations in Fig. 4.

50 100 150 200

Number of Nodes

0

20

40

60

80

100

120

140

160

Execution Time (seconds)

BC

CC

EC

Fig. 2. Comparison of Execution Time between Centrality Measures.

A. Evaluation of Centrality Measures

The comparison of various centrality measures is shown in

Fig. 2, BC the computational overhead of the BC is high. As

compared to BC, CC has a strong ability to ﬁnd the most

important node in less time due to the calculated shortest

distance between two nodes. With the increasing N from 50 to

200 nodes, the closeness measure outperforms the betweenness

measure because CC is not calculated in the shortest paths that

pass only through a speciﬁc node. Furthermore, EC ﬁnds the

0 20 40 60 80 100 120 140 160 180 200

Number of Link Attacks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of Removed Links

BC

CC

EC

HDLA

RHDLA

Fig. 3. Malicious Link Attacks Affect the Network Topology.

inﬂuential node for the attack in less time as compared to BC.

Because, it indicates not just the direct inﬂuence of a given

node, also the inﬂuence of nodes that are more than a hop

away.

B. Malicious Attacks affect the Network Topology

Validation of the proposed model is achieved through the

removal of links with malicious attacks. Results demonstrate

that the performance of the proposed attacks HDLA and

RHDLA affect the network more as compared to the BC and

CC, as shown in Fig. 3.

With the increasing number of attacks, the probability of

removed links using HDLA and RHDLA is greater than that

of BC and CC attacks. After the 120th attack, HDLA and

RHDLA damage, the network at a greater portion compared

with BC and CC attacks. The effectiveness of HDLA and

RHDLA is shown in Fig. 3. HDLA affects the most important

link between the two high degree nodes.

C. Comparison with Existing Algorithms

The R value of the network is evaluated using the different

number of iterations. Our model, which is smart rewiring

mechanism, is compared with ROSE and SA in Fig. 4.

10 50 100

Number of Iterations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Robustness

Smart Rewiring

ROSE

SA

Fig. 4. Robustness value with Different number of Iterations.

With increasing the number of iterations, the proposed smart

rewiring mechanism outperforms both ROSE and SA in terms

of high robustness (i.e., Schneider R-value). Among the best

results, the robustness value for Smart Rewiring, ROSE, and

SA are 0.350 0.289, and 0.259 respectively. As compared to

ROSE and SA schemes, our model’s performance is higher

to make the network robust against malicious attacks. The

process of edge swap is restricted to the high-degree nodes.

In this way, the re-connection of these nodes will change the

overall performance of the network as compared to ROSE and

SA.

V. CONCLUSION

This paper improves the robustness of SFNs. Two efﬁcient

centrality measures called CC and EC, are used to ﬁnd out the

most inﬂuential nodes in the network. Using these measures,

the malicious attacks are performed. These attacks affect the

network quickly and make it robust. A topology’s robustness

is determined using the Schneider R metric. Two link attacks

are performed, the HDLA and the RHDLA. Both of these

attacks target the high degree links of the network. Simulation

results show that these attacks damage the network with less

computational cost. The network is designed to withstand

malicious attacks on nodes as well as links. Through smart

rewiring, the network topology is turned into an onion-like

structure, which is robust against the malicious attacks.

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