IEEE Computational Intelligence Magazine - May 2020 - 29

in a search round and only one measure can be maximized.
Therefore, a multiobjective optimization method is more
suitable to consider both of the measures and deal with the
multiobjective optimization problem (MOP) between Rinter
and Rintra. Meanwhile, a series of different candidates in the
solution space can be generated to provide diversified solutions to the MOP.

similar operations can be found in [16], [35], [36] to solve
different MOPs in networks.
Based on Eq. (2), the two separate objectives in the initialization operator are defined as follows:
Min h + =

A. Initialization

Intuitively, as indicated in [26], a cluster partition that guarantees more negative links and less positive links between
clusters can lead to a better tolerance against inter-cluster
collapses in the network; similarly, a partition that allocates
more positive links and less negative links inside clusters
makes for the resistance toward intra-cluster failures. In this
way, a population, which is composed of diversified partition candidates emphasizing different strategies for reducing imbalanced links, can provide potentially good
solutions to both of the robustness measures. Another
advantage of this initialization strategy is that it can be easily guided by the subparts in Eq. (2) at a low cost, which is
helpful to reduce the computational cost of MOEA/DRSB. This initial population provides better initial points
for the following multiobjective optimization process, and

e ij+

(5)

i, j ! V and s i ! s j

Min h - =

e mn

(6)

m, n ! V and s m = s n

IV. MOEA/D-RSB

The evolutionary algorithms (EAs) have been proved to be
effective in solving different types of MOPs in past decades [4],
[16], [25], [26], [35], [36]. A multiobjective EA is developed
based on the framework of MOEA/D [25] in this study to
optimize Rinter and Rintra of the signed networks simultaneously,
termed MOEA/D-RSB. In this algorithm, each chromosome
Ci in the population represents a certain cluster partition in the
form of an integer vector with N dimensions (N is the number
of nodes in the network) as C i = {c i1, c i2, f, c iN }, where cij is
the cluster allocation of node j in the network. In this way,
nodes are divided into several partitions, and those with the
same value in the chromosome are in the same cluster. The
details of the algorithm are given in this section.
Generally, a randomly initialized population is suggested
when utilizing EAs to solve optimization problems [25]. However, we focus on an MOP related to network structures,
which is different from numerical optimization problems in
[25]. Specifically, the coding method should properly represent
network structures, and more information is maintained compared with numerical optimization problems. As shown in [18],
[35], even networks with a small size can manifest numerous
structural varieties; therefore, the optimization of networks
tends to have a larger solution space. Meanwhile, considering
that the evaluation process of proposed robustness measures is
relatively computationally costly, the simple random population
initialization seems to be deficient for the enhancement task
and contributes little to the convergence of the whole search
process. Therefore, a problem-directed initial strategy is necessary for solving the designed MOP.

where h+ represents the number of positive links between clusters, and h− represents the number of negative links inside clusters. In the initialization process, search operations are
implemented taking h+ or h− as the objectives to generate cluster candidates. In addition, some randomly generated candidates are also necessary to further promote the diversity of the
initial population. The details of the initialization operator are
given in Algorithm S2 of the Supplementary Materials.
B. Genetic Operator

In solving the MOP between Rinter and Rintra, some genetic
operators are necessary to generate more potential candidates
and search for better solutions in the local area.
As defined in [25], Tn neighbors for each individual in the
population should be determined for conducting genetic
operators. The selection of neighbors is based on the closeness
of the weight vectors. In MOEA/D-RSB, we mainly focus
on searching for partitions that promote the robustness of
signed networks, and the chromosome information for each
individual is a specific partition for all the nodes in the network instead of numerical coding information as in [25]. As
shown in [12], [26], the information exchange between individuals with a larger partition difference may contribute to

(a)

(b)
FIGURE 2 Two possible partitions and corresponding network structures for robustness evaluation of the network given in Figure 1. In
the figure, nodes allocated into different clusters are labeled in different shapes and colors, and those imbalanced links (positive links
between clusters and negative ones inside clusters) are marked in
grey. In the robustness evaluation process, imbalanced links, which
are redundant to the structural balance, should be removed from the
network. Structural differences can be found in the two network
structures for robustness evaluation. We can see a close impact of
nodal partitions on the robustness of signed networks. (a) A possible
partition and its network structure for robustness evaluation.
(b) Another scenario for robustness evaluation.

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