Mitigate Cascading Failures on Networks using a Memetic Algorithm

Abstract

Research concerning cascading failures in complex networks has become a hot topic. However, most of the existing studies have focused on modelling the cascading phenomenon on networks and analysing network robustness from a theoretical point of view, which considers only the damage incurred by the failure of one or several nodes. However, such a theoretical approach may not be useful in practical situation. Thus, we first design a much more practical measure to evaluate the robustness of networks against cascading failures, termed Rcf. Then, adopting Rcf as the objective function, we propose a new memetic algorithm (MA) named MA-Rcf to enhance network the robustness against cascading failures. Moreover, we design a new local search operator that considers the characteristics of cascading failures and operates by connecting nodes with a high probability of having similar loads. In experiments, both synthetic scale-free networks and real-world networks are used to test the efficiency and effectiveness of the MA-Rcf. We systematically investigate the effects of parameters on the performance of the MA-Rcf and validate the performance of the newly designed local search operator. The results show that the local search operator is effective, that MA-Rcf can enhance network robustness against cascading failures efficiently, and that it outperforms existing algorithms.

Full text

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Scientific RepoRts | 6:38713 | DOI: 10.1038/srep38713
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Mitigate Cascading Failures
on Networks using a Memetic
Algorithm
Xianglong Tang, Jing Liu & Xingxing Hao
Research concerning cascading failures in complex networks has become a hot topic. However, most of
the existing studies have focused on modelling the cascading phenomenon on networks and analysing
network robustness from a theoretical point of view, which considers only the damage incurred by the
failure of one or several nodes. However, such a theoretical approach may not be useful in practical
situation. Thus, we rst design a much more practical measure to evaluate the robustness of networks
against cascading failures, termed Rcf. Then, adopting Rcf as the objective function, we propose a new
memetic algorithm (MA) named MA-Rcf to enhance network the robustness against cascading failures.
Moreover, we design a new local search operator that considers the characteristics of cascading failures
and operates by connecting nodes with a high probability of having similar loads. In experiments, both
synthetic scale-free networks and real-world networks are used to test the eciency and eectiveness
of the MA-Rcf. We systematically investigate the eects of parameters on the performance of the MA-
Rcf and validate the performance of the newly designed local search operator. The results show that
the local search operator is eective, that MA-Rcf can enhance network robustness against cascading
failures eciently, and that it outperforms existing algorithms.
Many man-made infrastructures such as the Internet, transportation networks, and electric power grids can
be represented as complex networks1. Because these complex networks play an important role in society, their
robustness is pivotal2–4. However, most of these infrastructures have been found to be heterogeneous and to have
a power-law degree distribution1,5,6. With their “heavy-tailed” properties, these complex networks have been
found to be robust against random attacks; however, they are rather fragile under malicious attacks, especially
cascade-based attacks7,8.
Cascading failures are common in modern social networks. For example, in electrical power grids, when a
power transmission station or a power line goes down, its power will be shied to the nearby stations (lines). In
most cases, neighbouring stations can manage the extra load. However, in some extreme circumstances, these
neighbouring stations may become overloaded and fail, resulting in a redistribution of their loads to their neigh-
bours. Ultimately, the redistribution eort may lead to a cascading failure in which a large number of power trans-
mission stations (lines) are overloaded and, consequently, malfunction9. Cascading failures may also take place on
the Internet. e load on an Internet router represents data packets that must be transmitted per unit of time, and
overloading corresponds to congestion10. Rerouting data packets from a congested router to another might spread
the congestion to a large fraction of subnetworks. Some Internet collapses have been caused by congestion11.
Another example is a power grid, in which each component is designed to deal with a specic load of power. On
August 14, 2003 in Canada and the northeastern United States, a massive power blackout occurred that led to a
cascading failure12. A similar breakdown occurred in southern Oregon on August 10, 1996 13,14.
Cascading failures in complex networks have been widely studied over the past few decades15–24. Dierent
cascading failure models have been proposed to reproduce cascading phenomena. Motter et al. rst proposed
the “C-L” model in ref. 18, performing experiments on both random and scale-free networks that focused on
cascading triggered by the failure of a single node. e “C-L” model obtained good results that were consistent
with experts’ intuition about how cascading failures occur. Crucitti et al.19 introduced a dynamical model that
considered the dynamical redistribution of ow in networks, in which overloaded nodes obstruct network traf-
c rather than removed. Zhao et al.20 provided a mathematical proof of the “C-L” model in scale-free networks
Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, Xi’an
710071, China. Correspondence and requests for materials should be addressed to J.L. (email: neouma@mail.xidian.
edu.cn)
Received: 18 March 2016
Accepted: 15 November 2016
Published: 09 December 2016
OPEN

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that analysed the cascading breakdown in scale-free networks in terms of phase transitions. Feng et al.23 pro-
posed an approach of simple, self-consistent probability equations to study cascading behaviours in interdepend-
ent networks and showed that this approach can greatly simplify the mathematical analysis of systems ranging
from single-layer networks to various types of interdependent networks. Hu et al.24 used a percolation approach
to study more realistic coupled networks system in which both interdependent and interconnected links exist
and found rich and unusual phase-transition phenomena—including mixed rst- and second-order hybrid
transitions.
Based on dierent cascading failure models, various strategies have been proposed to enhance network
robustness against cascading failures. Koç et al.25 proposed a robust metric for cascading failures on power grid
networks; an entropy-based metric was introduced in ref. 26. Wang et al.27 studied cascading failures on the
Internet. Based on a new cascading edge model, they proposed some methods to protect the Internet from cas-
cading failures. However, all the above methods focused only on cascades triggered by removing one or two
nodes, and such methods cannot evaluate the overall robustness of networks against cascading failures and may
not be useful in many practical applications. Additionally, these methods rarely take the cost involved in updating
the real-world systems into account.
Considering only the cascading failure resulting from removing individual nodes in networks is insucient
because many of the remaining nodes are still connected; therefore, the network still maintains its integrity to a
certain extent. In contrast, in this paper, we rst design a new robustness measure to evaluate the overall robust-
ness of networks against cascading failures. In this robustness measuring scheme, the network is attacked through
cascading failures repeatedly until the entire network collapses. During this process, aer each cascade attack, the
remaining large network components are calculated.
Based on this measure, we propose a memetic algorithm (MA) called MA-Rcf that enhances network robust-
ness against cascading failures. Memetic algorithms form a popular branch of evolutionary algorithms (EAs)
that successfully combine global and local searches and have been shown to be more ecient and more eective
than traditional EAs for many problems28–30. In a previous study, we designed a new memetic algorithm named
MA-RSFMA, which improves the robustness of scale-free networks against malicious attacks that achieved a good
performance31. us, the algorithm proposed here, MA-Rcf, is based on the framework developed for MA-RSFMA
and makes use of the properties of cascading failures to design new operators; that is, a new local search operator
that considers the characteristic of cascading failures is designed for MA-Rcf. In MA-Rcf, the degree distribution
of networks is also kept unchanged to minimize the costs of updating real-world systems. Both synthetic and
real-world networks are used to validate the performance of the MA-Rcf. e experimental results show that the
MA-Rcf can enhance the network robustness against cascading failures eciently. Moreover, some properties of
robust networks are also analysed.
Methods
Robustness Measure for Cascading Failures. A network can be modelled as a graph, G = (V, E), where
V = {1, 2, … , N} is a set of N nodes and E = { ejk| j, k∈ V and j ≠ k} is a set of M links. In ref. 18, Motter et al. pro-
posed the “C-L” model for cascading failures in which, for a given network, at each time step, one unit of the
relevant quantity (such as energy or goods) is exchanged between each pair of nodes and transmitted along the
shortest connecting path. e “load” at a node consists of the total number of shortest paths passing through
it32,33. Each node carries the maximum load that it can handle, and in man-made networks, node capacity is lim-
ited by economic costs. e capacity Ci of node i and its initial load Li have the following proportional relation:
α=+ =…
CLiN(1 ), 1, 2, ,, (1)
ii
0
where the constant α is a tolerance parameter, and
Li
0
is the initial load of the ith node. Initially, the network
operates in a free-ow state insofar as α ≥ 0. However, the failure of a node for any reason triggers the dynamics
of the redistribution of loads. When the load at a node becomes larger than the node’s capacity, the node fails. is
forces the load previously carried by that node to shi to its neighboring nodes, which in turn, can cause them to
fail. Consequently, subsequent failures can occur, and this step-by-step process is a cascading failure18.
In ref. 34, Schneider et al. proposed an eective robustness measure, R, to evaluate networks’ ability to resist
targeted attacks on individual nodes. e R measure is based on the “giant component,” namely, the largest con-
nected component le in the network aer each node removal. To calculate R, the network must be attacked until
only separated nodes are le. us, R can evaluate the robustness of entire networks. erefore, we combine the
“C-L” model with R to design a new measure, Rcf, which can evaluate the overall robustness of networks against
cascading failures. With the original property of “C-L” model in mind, the process for calculating Rcf is described
be low.
Step 1. Ssum ← 0 and t ← 1, where Ssum is the accumulated size of the giant components and t is the index of
cascaded attack rounds;
Step 2. Calculate the initial load
Li
0
and Ci of each node;
Step 3. Remove the node with the maximum load and all the edges connected to it;
Step 4. If the number of remaining nodes is equal to 1, go to Step 6;
Step 5. Recalculate the
Li
t
of each remaining node:
If a remaining node i is overloaded, namely, Li > Ci, then remove node i and the edges connected to it,
then, go to Step 4;
If no remaining node is overloaded, calculate the relative size of the giant component St, Ssum ← Ssum+St,
t ← t + 1, then, go to Step 3;
Step 6. Calculate the robustness measure Rcf against cascading failures as follows,

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∑
==
=
R
S
NN
S
1
(2)
cf sum
t
T
t
1
where T is the total number of rounds that a cascading failure-based attack needs to destroy the entire network,
reducing it to only one node. Obviously, T may vary even for networks of the same size; therefore, the normaliza-
tion factor 1/N ensures comparability for the robustness of networks with dierent sizes.
Memetic Algorithm to Enhance Rcf. Memetic algorithms have been shown to be highly capable of search-
ing for the optimal solution in optimization problems28–30. In ref. 31, we designed a new memetic algorithm
named MA-RSFMA to improve the robustness of scale-free networks against malicious attacks, and it achieved
a good performance31. us, based on the framework of MA-RSFMA, in this paper, we propose a new memetic
algorithm, MA-Rcf, to enhance the overall robustness of networks against cascading failures. By considering the
intrinsic property of cascading failures, we design a new local search operator for MA-Rcf that takes Rcf as its
objective function while keeping the degree of each node unchanged. Next, we introduce the representation of
chromosomes and the initialization process. en, we describe the evolutionary operators, including the newly
designed local search operator. Finally, we summarize the entire framework of the MA-Rcf algorithm.
Representation and initialization. In the MA-Rcf algorithm, each chromosome represents a network. Initially,
MA-Rcf has a population with W chromosomes. e initialization process for MA-Rcf is the same as that used for
MA-RSFMA31. During the initialization, because we need to preserve the number of links and the degree of each
node, each chromosome is generated by randomly adjusting a fraction of the edges in the initial network, G0—
that is, the connections of two randomly chosen edges that have no common nodes are swapped in the network.
During the initialization, the goal is to generate dierent networks with the same degree distribution; there-
fore, any edge-swapping operations that can keep the network connected are accepted without checking whether
the swap improves the robustness of the network. e details of this initialization process are summarized in
Algorithm 1 (also refer to ref. 31 for more information).
Algorithm 1 Population Initialization
Input:
W: Population size;
G0: Initial network;
Output:
=...PGG G{, ,, }
W
11
12
11
: Population for the 1st generation;
for i = 1 to W do
Gi
1
←G0;
for j = 1 to M do /*M is the number of nodes in the network; */
Randomly select two edges ekl and emn from
Gi
1
, where m, n are dierent than k, l and ekm and eln do not exist in
Gi
1
;
Remove ekl and emn from
Gi
1
, and add ekm and eln to
Gi
1
;
if (
Gi
1
is not connected)
Remove ekm and eln from
Gi
1
, and add ekl and emn back to
Gi
1
;
end if;
end for;
end for;
Evolutionary operators. In evolutionary algorithms, crossover operators are oen performed to exchange
genetic information among the individuals in the population. In ref. 31, a new crossover operator that operates
on complex networks is designed that exchanges the structures of two networks eectively without changing their
degree distributions. We also employ this crossover operator in this paper. is crossover operator (whose details
are summarized in Algorithm 2) acts on two randomly selected parent chromosomes and generates a pair of child
chromosomes. Please refer to ref. 31 for more information about this crossover operator.
Algorithm 2 Crossover Operator
Input:
Gp1 and Gp2: Two parent chromosomes;
pc: Crossover rate;
Output:
Gc1 and Gc2: Two child chromosomes;
Gc1←Gp1, Gc1←Gp1;
Continued

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for i = 1 to N do/*N is the number of edges in the network; */
if (U(0, 1) < pc)/*U(0, 1) is a uniformly distributed random real number in [0, 1];*/
Determine
Vi
Gc1
and
Vi
Gc2
, which are the sets of neighbours of node i in Gc1 and Gc2, ;respectively,
←−
VV VV(I)
i
Gci
G
c
i
G
c
i
G
c
111
2 and ←−
VV VV(I)
i
Gci
G
c
i
G
c
i
G
c
221
2;
for each node ∈
j Vi
Gc1
do
Randomly select a node ∈
k Vi
G
c
2
;
Remove eij from Gc1 and eik from Gc2;
Add eik to Gc1 and eij to Gc2;
Randomly select another edge ekl that connects to node k in Gc1 but where ejl does not exist in Gc1;
Remove ekl and add ejl in Gc1;
Randomly select another edge ejm that connects to node j in Gc2 but where ekm does not exist in Gc2;
Remove ejm and add ekm in Gc2;
=−VV k{}
i
Gci
Gc22
;
end for;
end if;
end for;
e local search operator is another important operator in MAs. In ref. 35, Tanizawa et al. found that net-
works with an “onion-like” structure, where nodes with almost the same degree are connected to each other, are
more robust under targeted attacks than those without such onion-like structures. Inspired by this, to search
for networks that are the most robust against cascading failures, we design a new local search operator that lets
nodes with similar loads connect to each other. e principle of this operator is simple: if a node with small load
connects to a node with a very large load, the small-load node will crash immediately if the large-load node
fails, because the small-load node has insucient capacity to handle the extra load. erefore, connecting nodes
with similar loads to each other have a high probability of avoiding such situations. Suppose edges eij and epq are
selected and are swapped to eip and ejq. is swap is accepted only if
β−+−<×−+−
()
LL LL LL LL,
(3)
ip jq ij pq
where Li, Lj, Lp and Lq are the loads of the corresponding nodes, and β is a parameter in the range of [0, 1] that
controls the acceptance level for the dierence in loads between nodes. e smaller the value of β is, the stronger
the constraint is and, thus, the larger the reduction in load dierences is. Consequently, this operator can eec-
tively guarantee that nodes with similar loads will be connected to each other, which enhances the search for
load-balanced networks. e details of this local search operator are given in Algorithm 3.
Algorithm 3 Local Search Operator
Input:
G: One chromosome;
pl: Local search probability;
β: Predened parameter;
Output:
G: Chromosome aer performing the local search operator;
for (each edge eij in G) do
if (U(0, 1) < pl)/*U(0, 1) is a uniformly distributed random real number in [0, 1];*/
Randomly select another existing edge epq in G;
if (equation(3) is satised)
G* ← G;
Remove eij and epq from G*;
Add eip and ejq to G*;
if (Rcf(G*) > Rcf (G))
G ← G*;
end if;
end if;
end if;
end for;
MA-Rcf uses binary tournament selection in each generation to select the chromosomes for the next popula-
tion. Binary tournament selection involves a “tournament” between two chromosomes chosen randomly from
the population in which the winner is the chromosome whose tness is better. Binary tournament selection is a
popular method for selecting better chromosomes from a population in an evolutionary algorithm.

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Algorithm 4 MA-Rcf
Input:
W: Population size;
G0: Initial network;
pc: Crossover probability;
pl: Local search probability;
Output:
G*: Chromosome with the highest robustness (Rcf);
P1← Population_Initialization (W, G0) and t ← 1;
while (terminal criteria are not satised) do
←∅Pc
t
//
Pc
t
is the child population of Pt
Repeat
Randomly choose two chromosomes
Gi
t
and
Gj
t
from Pt that have not been selected;
()
GG
,
ci
t
cj
t
← Crossover_Operator
()
GGp
,,
i
t
j
t
c
;
∪
←
{}
GGP P ,
c
t
c
t
ci
t
cj
t
;
Until (all chromosomes in Pt have been selected);
Calculate the robustness of each chromosome in
Pc
t
;
for i = 1 to W do
Select a chromosome Gt from Pt and
Pc
t
using roulette wheel selection based on the robustness of all chromosomes;
Conduct the local search operator on Gt with probability pl;
end for;
Pt+1 ← 2-Tournament_Selection(Pt,
Pc
t
);
t ← t + 1;
end while.
Implementation of MA-Rcf. In MA-Rcf, the initialization operator is rst used to generate an initial population
with W chromosomes. In each generation, the crossover operator is applied to the population rst, and then, the
local search operation is conducted. Aer performing the crossover operator, a new child population is obtained.
Figure 1. e eect of parameter β on the performance of MA-Rcf.
NInitial
Without (Average ± Standard
Deviati on)
With (Average ± Standard
Deviati on)
100 0.0751 0.1617 ± 0.0135 0.1669 ± 0.0128
200 0.0521 0.1202 ± 0.0127 0.1309 ± 0.0119
300 0.0395 0.0924 ± 0.0097 0.1072 ± 0.0085
500 0.0263 0.0783 ± 0.0074 0.0874 ± 0.0067
Table 1. e Rcf values of BA networks obtained by MA-Rcf with and without the local search operator. e
results are averaged over 10 independent runs.

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en, the local search operator and the binary tournament selection operator are applied to both the parent and
child populations to generate the child population for the next generation. Finally, the best chromosome in the
last population is the most robust network found. e framework of MA-Rcf is summarized in Algorithm 4.
Results
In this section, because scale-free networks have become an important type of network, experiments are con-
ducted on both synthetic scale-free networks and real-world networks to validate the performance of MA-Rcf. We
also study some of the network properties of the robust networks obtained by MA-Rcf. e synthetic scale-free
networks were generated using the BA model5, and their average degree was set to 4. In ref. 31, Zhou et al. showed
that MA-RSFMA can improve the robustness of scale-free networks against malicious attacks eectively; conse-
quently, we also compare networks optimized by MA-Rcf with those optimized by MA-RSFMA to investigate the
dierent properties of both optimized networks.
e parameter α in (1) reects the capacity of a node to handle its load. A larger α indicates a stronger node.
e value of α is always assumed to be in the range [0, 1]: a α > 1 is unrealistically large18,19. In this work, we
assume that ability of a node to handle its loads is average (neither very strong nor very weak). us, in the follow-
ing experiments, α is set to a median value, 0.5.
In the local search operator, the tolerance parameter β controls the percentage by which loads can dier
between connected nodes. erefore, we rst conducted an experiment to nd an appropriate value for β. is
experiment used BA networks with 100 nodes. e robustness obtained by MA-Rcf under dierent values of β
is plotted in Fig.1. e results are averaged over ten independent runs on each sampled point. As Fig.1 shows,
MA-Rcf achieves the highest robustness when β equals 0.8. us, β is set to 0.8 in the following experiments.
e other parameters of MA-Rcf were set as follows: the population size W was set to 10, the crossover prob-
ability pc and the local search probability pl were set to 0.8 and 0.5, respectively, and the maximum number of
objective function evaluations was set to 5 × 104.
N Algorithms Best Wor s t Average ± Standard deviation
100
Hill Climbing 0.1231 0.0912 0.1035 ± 0.0113
Simulated Annealing 0.1356 0.1037 0.1187 ± 0.0109
MA-Rcf 0.1797 0.1476 0.1669 ± 0.0128
200
Hill Climbing 0.0926 0.0747 0.0813 ± 0.0066
Simulated Annealing 0.1021 0.0893 0.0922 ± 0.0053
MA-Rcf 0.1427 0.0891 0.1309 ± 0.0119
300
Hill Climbing 0.0821 0.0668 0.0729 ± 0.0059
Simulated Annealing 0.0932 0.0720 0.0864 ± 0.0066
MA-Rcf 0.1157 0.0892 0.1072 ± 0.0085
500
Hill Climbing 0.0782 0.0595 0.0651 ± 0.0054
Simulated Annealing 0.0883 0.0692 0.0793 ± 0.0059
MA-Rcf 0.1021 0.0722 0.0874 ± 0.0067
Table 2. e Rcf of BA networks of dierent sizes obtained by the three tested algorithms. e results are
averaged over 10 independent runs.
Figure 2. A comparison between MA-Rcf and existing algorithms on BA networks of dierent sizes.

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Experiments on Synthetic Networks. In this experiment, scale-free networks with dierent scales were
used to test the performance of MA-Rcf. First, an experiment was carried out to test the eectiveness of the local
search operator. In this experiment, versions of MA-Rcf both with and without the local search operator were
Figure 3. e network topology before and aer optimization by MA-Rcf. e size of each node is
proportional to its degree.
Figure 4. e change in the relative size of the giant component St over a series of cascaded attack circles t.
e BA network has 200 nodes. e Rcf values of the initial BA network, the optimized network obtained by MA-
RSFMA and the optimized network obtained by MA-Rcf were 0.0521, 0.0558 and 0.1319, respectively.

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tested on BA networks with 100, 200, 300, and 500 nodes. e obtained robustness values are listed in Table1,
which shows that the MA-Rcf version with the local search operator always performs better than the version with-
out the local search operator. erefore, the local search operator in MA-Rcf is eective.
Next, some experiments were conducted to test the ability of MA-Rcf to search for the most robust networks.
Network structure optimization is a hard optimization problem. In existing works, the hill climbing algorithm34
and the simulated annealing algorithm36,37 are widely used to address this problem. us, we compared the per-
formance of MA-Rcf with that of the hill climbing algorithm34 and the simulated annealing algorithm36,37. e
maximum number of objective function evaluations for these two algorithms was also set to 5 × 104 to obtain the
results of these three algorithms at the same computational cost.
We tested BA networks with 100, 200, 300, and 500 nodes. e best, worst and average values of Rcf of the
three algorithms over 10 independent runs are reported in Table2. In addition, the corresponding curves of the
average robustness obtained by the dierent algorithms are plotted in Fig.2. As shown, MA-Rcf obtains the high-
est robustness values among these algorithms on all test networks. at is, MA-Rcf always nds more network
structures more robust to cascading failures than do the other algorithms.
It is useful to study the robustness of the network structures obtained MA-Rcf. us, the network topologies
of BA networks before and aer optimized by MA-Rcf are plotted in Fig.3, where the size of each node is pro-
portional to its degree. As shown, before optimization, low degree nodes are oen connected to nodes with high
degrees; consequently, the entire network is composed of numerous star networks with hub nodes. However,
the optimized networks which have higher Rcf, the low degree nodes are more likely to be connected to other
low degree nodes and the high degree nodes are more likely to be connected to other high degree nodes. e
entire network is a hub-node-connected structure. Considering the property of cascading failures, it is easy to
Figure 5. e change in the relative size of the giant component St under high node degree attack circles t.
e BA network has 200 nodes. e Rcf values of the initial BA network, the optimized network obtained by MA-
RSFMA and the optimized network obtained by MA-Rcf were 0.0521, 0.0558 and 0.1319, respectively.
N
A (Average ± Standard
Deviati on)
L (Average ± Standard
Deviati on)
C (Average ± Standard
Deviati on)
100
Before Optimization − 0.1489 ± 0.0000 3.1681 ± 0.0000 0.3563 ± 0.0000
MA-RSFMA 0.6206 ± 0.0242 8.1168 ± 0.2203 0.2186 ± 0.0093
MA-Rcf 0.3956 ± 0.0132 3.8307 ± 0.1841 0.3139 ± 0.0125
200
Before Optimization − 0.2194 ± 0.0000 3.5645 ± 0.0000 0.3124 ± 0.0000
MA-RSFMA 0.3951 ± 0.0176 7.9266 ± 0.2312 0.2057 ± 0.096
MA-Rcf 0.1734 ± 0.0118 3.5726 ± 0.1873 0.3115 ± 0.0103
300
Before Optimization − 0.2344 ± 0.0000 3.6327 ± 0.0000 0.3001 ± 0.0000
MA-RSFMA 0.3480 ± 0.0198 8.2787 ± 0.2241 0.1801 ± 0.0102
MA-Rcf 0.1470 ± 0.0093 3.7308 ± 0.1254 0.2943 ± 0.0089
500
Before Optimization − 0.2507 ± 0.0000 3.8264 ± 0.0000 0.2836 ± 0.0000
MA-RSFMA 0.3651 ± 0.0142 8.1968 ± 0.2101 0.1869 ± 0.0114
MA-Rcf 0.1033 ± 0.0076 3.8304 ± 0.1219 0.2814 ± 0.0081
Table 3. e changes in important network properties of BA networks with dierent sizes before and aer
optimization by MA-Rcf and MA-RSFMA, including the assortative coecients (A), the average shortest
path length (L) and the global communication eciency (C). e values of the optimized networks were
averaged over 10 independent runs.

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understand why hub-node-connected networks have a stronger ability to resist cascade failures: when a hub
node fails, the neighbouring hub nodes can withstand the additional loads eectively, preventing the spread of
cascading failures.
Next, we carried out an experiment to test how well the networks obtained by MA-Rcf resist cascading failures.
We simulated the process of cascaded failures on BA networks with 200 nodes until the size of the giant compo-
nent decreased to 1. e decreasing progress of the size of the giant component St is plotted in Fig.4. As shown,
along with the increasing cascade attack circle t, the MA-Rcf optimized network, which has a higher Rcf value,
protects the giant component more eectively. e area between the curve of the “Initial BA network” and the
curve of the “MA-Rcf optimized network” in Fig.4 represents the amount of cascade failure mitigation, which cor-
responds to improving network robustness against cascade failures by 153%. ese results show that the networks
obtained by MA-Rcf can resist cascading failures eectively.
In ref. 31, Zhou et al. found that the onion-like network in which nodes with similar degree connect to each
other can contribute to resisting high node degree attacks, we plotted the decreasing process of the size of St of
networks optimized by MA-RSFMA under cascading failures in Fig.4, which has 200 nodes. In Fig.5, we sepa-
rately plotted the decreasing process of St of networks optimized by both MA-Rcf and MA-RSFMA under high node
degree attacks. In each attack circle, the node with largest degree and all the edges connected to it are removed.
To perform a fair comparison, the parameters for MA-RSFMA were set to the same as those for MA-Rcf, namely,
the population size was set to 10, the crossover probability and the local search probability were set to 0.8 and
0.5, respectively, and the maximum number of objective function evaluations was set to 5 × 104. In Fig.4, under
cascaded attack circles, the size of the St of the MA-RSFMA optimized networks decreases as fast as that of the
initial BA network—even more sharply aer the rst several attack rounds. Moreover, under high node degree
attack circles, the size of the St of the MA-Rcf optimized network decreases as fast as initial BA networks (see Fig.5
for more details). In other words, the MA-Rcf algorithm cannot contribute to resisting high node degree attacks.
ese two experiments show that although the network structures optimized by these two algorithms have some
similarity, their ability to resist cascading failures is signicantly dierent.
We are also interested in whether other important network properties might have changed because of the
optimization. erefore, in Table3 shows the results of assessing the assortativity coecient38, the average short-
est path length and the global communication eciency39 of networks both before and aer being optimized by
MA-Rcf. As shown, before the optimization, the BA networks have negative assortativity coecient values that
become positive aer the optimization. In other words, the correlation degree of the networks changes from dis-
assortativeness to assortativeness aer the optimization. is occurs because the optimization process promotes
the connection of high degree nodes with other high degree nodes. Aer the optimization, the average shortest
path length of BA networks is slightly increased while the global communication eciency has a slight decrease,
which means that the optimization process has no signicant eect on network communication eciency.
Because the networks obtained by MA-RSFMA are also assortative, it is interesting to study the dierence in
terms of the network properties of networks obtained by both MA-Rcf and MA-RSFMA; these properties are listed
in Table3. We can see that when both algorithms optimize the same network, the network obtained by MA-Rcf is
far less assortative than that obtained by MA-RSFMA. In addition, the average shortest path length of the MA-Rcf
is only half that of MA-RSFMA. Moreover, the networks obtained by MA-Rcf have higher global communication
eciency.
Experiments on Real World Networks. In this section, MA-Rcf is applied to two real-world networks.
One is an electrical power grid in Western Europe (mainly Portugal and Spain)40, labelled the WE Power grid
network. It has 217 nodes and 320 edges. e average degree of the WE Power grid network is 2.95. e other net-
work is the US air network—the US air transportation system41—consisting of 332 airports and 2126 air routes,
in which the nodes represent airports and the edges present ights. e average degree of the US air network is
12.81. ese two real networks are well connected and without any separate component.
e hill climbing algorithm, simulated annealing algorithm and MA-Rcf are used to independently optimize
the above two networks. e obtained robustness values are reported in Table4. As we can see, MA-Rcf always
performs better than the two other algorithms. e network topologies of these two real networks before and aer
optimized by MA-Rcf are shown in Fig.6. Comparing the network structure before and aer optimization, we can
see that, in the optimized networks, nodes with similar degrees are more likely to connect with each other, and
the hub nodes are more closely connected to each other than before. Consequently, even for an existing network,
MA-Rcf can nd a structure more robust against cascading failure. By comparing the robust structure with the
Network Algorithms Best Wor st Average ± Standard deviation
WE Power
Hill Climbing 0.1221 0.1072 0.1135 ± 0.0063
Simulated Annealing 0.1256 0.1097 0.1187 ± 0.0059
MA-Rcf 0.1494 0.1126 0.1330 ± 0.0082
US Air
Hill Climbing 0.0463 0.0378 0.0415 ± 0.0047
Simulated Annealing 0.0481 0.0382 0.0422 ± 0.0051
MA-Rcf 0.0537 0.0401 0.0475 ± 0.0062
Table 4. e robustness of networks aer optimization with dierent algorithms on two real world
networks. e results shown were averaged over 10 independent runs. e initial Rcf values of the WE Power
network and US air network were 0.1022 and 0.0251, respectively.

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initial structure, we can nd several key edges, which—if changed—would increase network robustness signi-
cantly. Considering the cost of optimization, there is no need to change all the edges of the real network; instead,
MA-Rcf can help nd the key edges.
e assortativity coecient, average shortest path length and global communication eciency of these two
real networks both before and aer optimization are reported in Table5. As listed, the WE Power network has a
positive assortativity coecient while the US air network has a negative assortativity coecient. Aer optimiza-
tion by MA-Rcf, the WE Power network has stronger assortativeness, while the disassortativeness of the US air
network gets weaker. is result is the same as the results of the experiments with synthetic networks, further
verifying that networks with more hub nodes connected to each other have a stronger ability to resist cascad-
ing failures. Interestingly, aer optimization by MA-Rcf, the average shortest path length of these two networks
decreases while their global communication eciency increases. In other words, MA-Rcf can not only increase a
network’s global robustness against cascading failures but can also increase its global communication eciency—
even when applied to real networks.
Figure 6. e network topology of two real world networks before and aer optimization by MA-Rcf. e
size of each node is proportional to its degree.
N
A (Average ± Standard
Deviati on)
L (Average ± Standard
Deviati on)
C (Average ± Standard
Deviati on)
WE Power Before Optimization 0.1269 ± 0.0000 6.9381 ± 0.0000 0.1870 ± 0.0000
Aer Optimization 0.2176 ± 0.0097 4.9511 ± 0.1903 0.2313 ± 0.0115
US Air Before Optimization − 0.2078 ± 0.0000 2.7381 ± 0.0000 0.4059 ± 0.0000
Aer Optimization − 0.0819 ± 0.0056 2.4980 ± 0.1410 0.4336 ± 0.0089
Table 5. e changes in some important network properties of real world networks before and aer
optimization by MA-Rcf, including the assortative coecients (A), the average shortest path length
(L) and the global communication eciency (C). e values of optimized networks were averaged over 10
independent runs.

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Discussions
Securing network infrastructure is critical in today’s society. When studying networks subject to cascading fail-
ures, considering only the damage from one or even several nodes is insucient. In this paper, we describe the
design of a more comprehensive index that can evaluate the ability of networks to resist cascading failures. To
enhance networks resistance to cascading failures, we propose a memetic algorithm, MA-Rcf. en, to test the
performance of MA-Rcf, we tested it on both synthetic and real networks. e topologies of the robust networks
obtained by MA-Rcf are shown and some of their network properties are discussed. From experiments comparing
with MA-Rcf other network optimization algorithms, we can conclude that MA-Rcf achieves a better performance,
showing that MA-Rcf is an eective algorithm for enhancing the robustness of networks against cascading failures.
When dealing with complex networks, the large computational complexity of calculating shortest paths limits
the algorithms that rely on such calculations from being applied to large-scale networks. For example, the com-
putation of Rcf in Equation(2) needs to calculate the shortest path of the network under each cascading attack
circle; consequently, MA-Rcf is unable to optimize large networks at a low computational cost. However, studying
the eects of MA-Rcf on cascading failures is still meaningful. MA-Rcf provides an opportunity to explore network
structures that are robust against cascading failures. In this paper, we apply MA-Rcf to networks with specic
sizes and study the topological structure and network properties of robust networks. e experiments show con-
necting hub nodes to each other more closely would be a good strategy when designing networks that are robust
against cascading failures. Moreover, extending this discovery to large-scale networks is not dicult. In contrast
with other traditional algorithms, MA-Rcf is a competitive algorithm for optimizing networks against cascading
failures.
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Acknowledgements
is work is partially supported by the Outstanding Young Scholar Programme of the National Natural Science
Foundation of China (NSFC) under Grant 61522311, the General Programme of NSFC under Grant 61271301,
the Overseas, Hong Kong & Macao Scholars Collaborated Research Programme of NSFC under Grant 61528205,
the Research Fund for the Doctoral Programme of Higher Education of China under Grant 20130203110010, and
the Fundamental Research Funds for the Central Universities under Grant K5051202052.
Author Contributions
X. T. and J. L. designed the study, X. T. and X. H. performed the research, X. T., J. L. and X. H. interpreted the
results, and X. T. and J. L. wrote the manuscript. All authors reviewed the manuscript.
Additional Information
Competing nancial interests: e authors declare no competing nancial interests.
How to cite this article: Tang, X. et al. Mitigate Cascading Failures on Networks using a Memetic Algorithm.
Sci. Rep. 6, 38713; doi: 10.1038/srep38713 (2016).
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