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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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