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A novel approach to network’s topology evolution

and robustness optimization of scale free networks

Muhammad Usman1, Nadeem Javaid2∗, Syed Minhal Abbas2, Muhammad Mohsin Javed2,

Muhammad Aqib Waseem3, Muhammad Owais2

1Department of Electrical and Computer Engineering, COMSATS University Islamabad, Islamabad 44000, Pakistan

2Department of Computer Science, COMSATS University Islamabad, Islamabad 44000, Pakistan

3Department of Computer Science, Bahauddin Zakariya University, Multan 60000, Pakistan

Email: usmaneng9@gmail.com, minhal.abbas514@gmail.com, mmohsinjaved007@gmail.com,

owaisbhutta98@gmail.com, ansariaqib786@gmail.com

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

Abstract—Internet of Things (IoT) is rapidly increasing day by

day due to its involvement in many applications such as electric

grids, biological networks, transport networks, etc. In complex

network theory, the model based on Scale Free Networks (SFNs)

is more suitable for IoT. The SFNs are robust against random

attacks; however, vulnerable to malicious attacks. Furthermore,

as the size of a network increases, its robustness decreases.

Therefore, in this paper, we propose a novel topology evolution

approach to enhance the robustness of SFNs. Initially, we divide

the network area into upper and lower parts. The nodes are

deployed equally in both parts and connected via one-to-many

correspondence. The distribution is made because small sized

networks are more robust against malicious attacks. Moreover,

we use k-core decomposition to calculate the hierarchical changes

in the nodes’ degree. In addition, the core-based and degree-

based attacks are performed to analyze the robustness of SFNs.

For the network optimization, we compare the Genetic Algorithm

(GA) with Artiﬁcial Bee Colony (ABC) and Bacterial Foraging

Algorithm (BFA). In the optimization process, the node’s distance

based edge swap is performed to draw long links in the network

because these links make the network more robust.

Index Terms—scale free networks, robustness, optimization,

malicious attack, network topology, edge swap

I. INTRODUCTION

The Wireless Sensor Networks (WSNs) have various appli-

cations in electric grids [1, 2], transportation [3], military [4,

5], healthcare [6, 7], smart homes [8] etc. In the WSNs, sensor

nodes take environmental information, including temperature,

moisture, radiations, air quality, etc., [9]. This information is

then transmitted to the central control unit i.e., a base station

or a hub to make important decisions about the network.

When the sensor nodes join the network, the WSNs become

an integrated part of the Internet of Things (IoT).

The sensor nodes in the IoT network are properly arranged

in the form of a topology [10]. The construction of network

topology is based on graph theory where nodes are considered

as vertices and the connections between them as edges. In

complex networks, Small World Networks (SWNs) [11] and

Scale Free Networks (SFNs) [12] are two models mostly used

in the IoT [13]. In SWNs the nodes have average shortest path

length and high clustering coefﬁcient, whereas in SFNs nodes’

degree follows a power-law distribution. The power-law proves

that the SFNs have a small number of high degree nodes

and a large number of low degree nodes. The nodes in the

network can be classiﬁed into two categories: homogeneous

and heterogeneous. The homogeneous nodes have the same

communication range, bandwidth and energy, whereas these

characteristics vary in heterogeneous nodes. Homogeneous

nodes are generally considered in the study of SFNs. The

SFNs are generated by following the famous Barab´

asi Albert

(BA) model [12]. The model consists of two processes: growth

and preferential attachment process. The network grows with

the addition of nodes asynchronously and the preferential

attachment deﬁnes that a new node connects with the high

degree nodes in the network.

Recently, there has been an exponential increase in the

number of IoT devices therefore, the IoT networks are be-

coming dense. Consequently, these networks are becoming

vulnerable to failure or attacks [14]. By considering attacks for

SFNs, they are mainly classiﬁed as random and malicious. In

random attacks, a node is removed randomly from the network.

However, malicious attacks happen on the important nodes

[15]. The importance of the nodes is usually measured based

on their degree and the attack on them termed as High Degree

Adaptive (HDA) attacks. The SFNs are robust against random

attacks; however, they are vulnerable to malicious attacks.

The robustness is the ability of a network to resist attacks.

Many methods are available to measure the robustness such

as conditional connectivity, Laplacian matrix [16], Natural

Connectivity (NC) [17] etc. However, these measures have

high computational cost; therefore, Schneider et al. [18] pro-

posed a robustness measure based on the percolation theory. It

states when the attacks happen on the high degree nodes, the

network fragments into multiple subgraphs. The robustness R

is calculated as,

R=1

N−1

N−1

X

N=0

MCSn

N(1)

Where, Nis the total number of nodes in a network, 1

N−1

is a normalization factor, MC Snis maximum connected

subgraphs after nth high degree node removed and summation

means nodes removal after each attack is considered.

The robustness is greatly reduced after the removal of

important nodes or edges [19] from the network. Their impor-

tance is usually based on degree and betweenness centrality.

The edge degree is calculated as,

dij =pki∗kj(2)

Here, dij is the degree of edge, kiand kjare degree of nodes

iand j, respectively.

To increase the robustness of a network, the easy approach

is to add edges between nodes [20], [21]. However, it is prac-

tically not possible to add edges among all the nodes, because

of the cost constraint. The network should be optimized to

achieve high robustness without adding cost. Therefore, the

independent edges are required in the process of optimization.

The two edges are considered to be independent if they are in

the communication range of each other and they are adjacent

to each other. The robustness of the network is calculated after

performing the edge swap. If the robustness increases, then the

edge swap is accepted and the topology is updated. Otherwise,

the next pair of independent edges are searched and the same

process is repeated.

For the SFNs, the onion-like structure is proved to be

robust against malicious attacks [22]. Its core consists of high

degree nodes surrounded by the rings of nodes whose degree

decreases hierarchically. Each ring presents the same degree

nodes in the network. A perfect onion-like structure proves

that the network robustness is enhanced. However, the long

tail of low degree nodes is formed that affects the robustness

of the network.

By considering the importance of SFNs, the following

contributions are presented in this paper:

•to increase the robustness, small sized networks are

evolved because they are more robust against malicious

attacks,

•networks are connected by one-to-many correspondence

i.e., a node in a network A is linked with more than one

node of the network B and vice versa. These links’ degree

distribution follow power-law,

•the nodes’ degree in an onion-like structure changes

hierarchically. Therefore, it is calculated by k-core de-

composition,

•the edges are swapped to make long links in the network

because the existence of these links make the network

robust against malicious attacks. A high degree node

connects with the node having a low degree and present

far as compared to other neighboring nodes,

•Genetic Algorithm (GA), Artiﬁcial Bee Colony (ABC)

and Bacterial Foraging Optimization (BFO) algorithms

are used and we select one that has better performance.

The rest of the paper is organized as follows: related work

studies are presented in Section II. Section III describes

the edge swap mechanism to enhance the robustness. Scale

free networks topology evolution and robustness optimization

are demonstrated in Section IV. In the last Section V the

conclusion and future work are explained.

II. RE LATE D WO RK

The SFNs are vulnerable to malicious attacks and a lot

of real world networks have scale free nature. Therefore, to

resist attacks these networks need to be optimized to have

a proper structure. Global edge swap based Hill Climbing

(HC) algorithm in [22] enhances the robustness; however, HC

traps into local optima. Moreover, the local optima problem is

solved by introducing the local edge swap [23]. For the WSNs

construction, the node’s communication range is limited that is

addressed in [24]. For SFNs, onion-like structure considers the

importance of node’s degree in rings therefore, same degree

nodes need to be connected with each other. Furthermore,

when a node removes, its respective edges should be used

to enhance the topological parameters of the network. By

increasing the nodes in Maximam Connected Subgraph (MCS)

robustness can be increased [25].

The optimization of SFNs is an NP-hard problem because of

the large number of edges in a network. Heuristic algorithms

are used to solve these problems, GA is among them. Classical

GA stops convergence to a sub-optimal solution that is called

premature convergence. It happens due to less population di-

versity. In optimization, the global optimal solution is required.

Therefore, [26, 27] deals with the premature convergence

problem of GA. Multi-population is used to achieve high

diversity. However, the computation cost is increased due

to operations on multi-population. Therefore, [28] solves the

premature convergence with less computation cost by self-

competition among individuals. The same problem is solved

using the local search operation by the authors in [29]. On the

other hand, to get the optimal solution quickly an algorithm

is proposed in [30]. During the evolution of the network

against attacks, the fault probability is not considered. The

fault probability and preferential attachment based network

evolution is proposed in [31].

In optimizing SFNs, a single objective is considered so far.

However, a network that is optimized for nodes attack collapse

when the links attack happens and vice versa. The SFNs are

vulnerable to malicious attacks; therefore, to optimize them

according to the attacks on high degree nodes and links,

Multi-Objective Optimization (MOO) is required. Therefore,

the authors in [19] proposed a MOO algorithm. It consists of

two phases: sampling and optimization. The sampling phase is

used to generate the diverse population and the optimization

phase is used to enhance the robustness.

Furthermore, the robustness of an undirected network is

discussed so far; however, directedness is also an important

network feature. By considering the directedness of the net-

work two important variables emergence of cooperation and

controllability robustness are used to deﬁne the resilience of

a network against different attacks.

Moreover, no practical approach is available to understand

the correlation between a network’s topology features and their

robustness by considering controllability. From the theoretical

analysis, it is impossible to deﬁne this relation at that time.

Therefore, to control networks for better utilizing them a

practical approach is proposed by [32].

All of the above algorithms although improve the robustness

of the networks; however, have high computation costs. Due to

this, self-optimization is not possible. Therefore, an Artiﬁcial

Intelligence (AI) based robustness optimization technique is

proposed in [33]. The back propagation is used to ﬁnd the

optimal solution. However, it is not suitable for different size

of networks and edge densities.

III. EDGE SWAP

The SFNs topology is represented as a graph G=

(V, E )where set of Nnodes represented as vertices V=

{1,2, ..., N }and the set of M edges are shown as E=

{emp|m, p ∈Vand m6=p}. The edge swap is used because

nodes’ degree remains the same. To perform the edge swap,

the edges emp and eno should be independent. To prove the

independency of edges, they should follow two conditions:

•nodes m, n, o and pshould be in the communication

range of each other,

•there is no extra edge between nodes except emp and eno.

The edge swap is performed in the Fig. 1. In the orig-

inal network topology, as shown in Fig. 1(a), the nodes

m, n, o, and pfulﬁlls the independent edges conditions. We

have two alternative connections as shown in Fig. 1(b) and

Fig. 1(c). The idea behind this edge swap is to enhance the

robustness of the network. After performing the edge swap,

robustness is calculated. If the robustness is improved then the

edge swap is accepted; otherwise, a new independent edge pair

is found. When edges are considered in edge swap, they are

marked. So that they can not be considered in the next edge

swap process. By marking edges, the number of redundant

operations can be reduced.

Considering the importance of edge swap, two types of edge

swaps are performed.

1) Random edge swap

2) Degree based edge swap

A. Random edge swap

In random edge swap, edges are randomly selected in the

network and the swap is made. Since, the network structure

consists of different topological features and some nodes are

more densely connected than the rest of the network. So, the

random edge swap may affect the network structure. Moreover,

in the SFNs there is more number of low degree nodes;

therefore, the probability of edges selection of low degree

nodes is high.

m

p

n

o

m

p

n

o

m

p

n

o

(a) (b) (c)

Fig. 1. Edge swap mechanism

B. Degree based edge swap

In a degree based edge swap, two high degree nodes are

selected in the network. Afterward, their neighboring nodes

that have a low degree are selected. These nodes must be

different so that they follow the independent edge conditions.

The degree based edge swap makes the similar degree nodes

connect. Since, the onion-like structure consists of rings that

have same degree of nodes. Therefore, the existence of edges

between same degree nodes enhances the robustness.

IV. SCA LE F RE E NE TW OR KS TOPOLOGY EVO LU TI ON A ND

ROB US TN ES S OP TI MI ZATI ON

In this section, the complete process of network topology

evolution is discussed. The proposed network topology pro-

vides the solutions to the limitations that are discussed in the

Table. I. Moreover, the degree of the node based on k-core

decomposition is found to know the hierarchically changes

in the network degree. Furthermore, the network optimization

against the attacks is discussed.

A. Network topology evolution

A network having a small number of nodes is more robust

against the malicious attacks [24], [27], [29]. Therefore, to

generate a robust network topology, nodes are distributed

equally into two parts. In each part, the networks are evolved

by considering the power-law distribution. There are two

ways to connect both parts of the network: one-to-one cor-

respondence and one-to-many correspondence. In one-to-one

correspondence each node of network A is connected with a

node of network B. However, in one-to-many correspondence,

a node in network A is connected with more than one node

of network B and vice versa. One-to-many correspondence is

preferred because it makes the network more robust [34].

In Fig. 2, the network topology evolution is shown. The

dotted line shows the division of the network. In both parts,

equal number of nodes are randomly deployed. The blue nodes

are used to denote network A (NA), black nodes are used for

network B (NB)and NMdenotes the mutual nodes of both

networks. The black solid lines represent the connectivity links

(CL) while the dotted lines are used for mutual links (ML).

Both upper and lower parts form a network by synchronously

adding edges for each node.

NA

NB

NM

CL

ML

C1

C2

C3

C4

Fig. 2. Network topology evolution

B. Nodes degree distribution based on k-core

In the network, the nodes’ have different degrees. Due to

the power-law distribution in SFNs, there are more low degree

nodes as compared to high degree nodes. Therefore, to ﬁnd

the hierarchical change of nodes’ degree in SFNs, k-core

decomposition as shown in Fig. 2 is performed.

Different rings represent the existence of different degree

nodes. In k-core decomposition, nodes’ removal starts with

low degree nodes and these nodes are placed in C4. After

that, the degree of the nodes is recalculated and low degree

nodes are placed in C3. The process continues until the highest

degree nodes are removed from the network. In Fig. 2 C1 is

the highest core consisting of the most important nodes based

on degree.

C. Attacks on the designed topology

The attackers have complete information about the network

and can make new attacks to paralyze it. So, the defenders

should take measures against these attacks to make the network

robust. Therefore, malicious attacks based on inner core nodes

and nodes’ degree based on rings are considered. At ﬁrst,

the inner core nodes are removed because they have more

inﬂuence on the network. In Fig. 3, the attack on inner core

nodes is shown and red color nodes (NR) are removed from

the network. Due to the better topology evolution, initially, by

removing high degree nodes the network is still connected.

So, multiple attacks are required to fragment the network.

After attacks are made on the core nodes, the network is

divided into multiple subgraphs. High degree nodes present

in the subgraphs as shown in Fig. 4 are removed and the

robustness is calculated. Red nodes represent the removed

nodes of the inner core nodes and white shaded area represents

the MCS. Yellow nodes are the highest degree nodes in their

respected subgraphs.

To increase the number of nodes in MCS, edge swap is

made at the outer core nodes because after the removal of high

degree nodes long tails of low degree nodes are created. Swap

NA

NB

NM

NR

CL

ML

Fig. 3. Attack based on inner core nodes

of low degree nodes’ edges increases the number of nodes

in MCS. In the proposed model edges are swapped without

changing nodes degree. So, the cost remains the same in that

operation.

Random edge swap increases the computational cost due to

redundant operations. Therefore, the edge swap should be kept

minimum. So, to increase the robustness it is more important

that the topology evolved better than by improved it through

edges swap.

The nodes’ degree attack based on rings enables the attack-

ers to remove a speciﬁc portion of the network. Due to the

existence of long tails, the nodes removal using this approach

has less computational cost.

D. Robustness optimization of the network by heuristic algo-

rithms

Due to the premature convergence of GA and high com-

putational cost required to solve the problem, two heuristic

algorithms Artiﬁcial Bee Colony (ABC) and Bacterial Forag-

ing Optimization (BFO) are used. In both of these algorithms,

a random position change is required to ﬁnd the global optimal

solution in the search space. In SFNs, a random position

change is not possible; therefore, a degree based edge swap

and a random edge swap are made. When the operators in

these algorithms improve the robustness then degree based

edge swap is performed. However, when they trap into local

optima random edge swap is performed [22].

V. CONCLUSION

The SFNs have become attractive due to their property

to resist random attacks. However, they are vulnerable to

malicious attacks. Therefore, this paper studies the importance

of topology design to enhance the robustness of SFNs. The

small sized networks are more robust against malicious attacks

as compared to large scale networks; therefore, the network is

evolved by dividing the total number of nodes equally into two

parts. The power-law distribution is followed in both parts.

After that, the networks are connected using one-to-many

TABLE I

MAPPING OF THE IDENTIFIED LIMITATIONS WITH PROPOSED SOLUTIONS AND THEIR VALIDATIONS

Limitations identiﬁed Proposed solutions Validations

L1: Large sized networks are more vulnerable

to malicious attacks

S1: Proposed topology evolution technique

makes small networks

V1: Network is divided into two parts; in each

part, the topology evolved [12]

L2: The interdependent links do not following

power-law [34]

S2: Using the interdependent links concept,

networks are connected by power-law

V2: Through the degree distribution of the

mutual nodes, the power-law is validated

L3: There are no predeﬁned criteria to know

how nodes’ degree changes in onion-like

structure

S3: k-core decomposition is used to ﬁnd the

same degree nodes in the network

V3: After the network deployment, nodes are

removed based on the degree and same degree

nodes are connected with each other

L4: Edge swap using the Degree Difference

Operation (DDO) increases redundant opera-

tion

S4: Edge swap is done on the basis of the

distance between nodes because the long links

make the network more robust

V4: Through performing distance based edge

swap the robustness of the network is calcu-

lated

L5: Premature convergence of GA S5: Optimization is done by ABC and BFO V5: For different network structures, robust-

ness is calculated

NA

NB

NM

NR

CL

ML

Fig. 4. Attack based on high degree nodes that are part of MCS

correspondence. Then the random and malicious attacks are

performed on both parts. The network becomes robust because

the nodes are removed in one part; however, the second part

of the network is still connected. The effect of attacks on the

proposed network is considered and the network is optimized

by GA, ABC and BFO. The experimental results prove that

the network robustness is increased against malicious attacks.

Furthermore, the onion-like structure consists of high degree

nodes that are at the center of the network are removed using

the k-core decomposition. The high degree attack is more

vulnerable to the network as compared to the core based attack.

In the future, the proposed scheme will be validated through

simulations on the synthetic and real-world networks.

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