IEEE Computational Intelligence Magazine - May 2020 - 33

been tested to provide comparisons. In the
The robustness of signed networks may not be
experiment, we set different optimization objectives for MLMSB as follows: the one aimed at
guaranteed in simple balancing processes, and
finding partitions with maximum Rinter is labeled
targeted optimization seems to be necessary.
as MLMSB-Rinter, the one aimed at finding partitions with maximum Rintra is labeled as MLMSBsampled networks present different features in their balanced
Rintra, and the one aimed at finding balanced robustness between
states. For Gr, inter-cluster connections are important to
Rinter and Rintra, which can be approximatively achieved by reducing h defined in Eq. (2), is labeled as MLMSB-h. Corresponding
enhance the performance of Rinter, and more inter-cluster neganumerical results are given in Table II.
tive links (Nb) can reduce the imbalance in the network and
As presented in the table, the proposed algorithm tends to
improve the invulnerability against losing competitive connecperform stably on networks, and the robustness values of samtions. For Gl, intra-cluster connections dominate the evaluation
pled networks only fluctuate in a relatively small range. Comprocess of Rintra, and more intra-cluster positive links (Pi) guarpared with the results of MLMSB, Gr, Gl, and Gm extracted
antee the imbalance to be lessened and promote the tolerance
against losing supportive connections. For Gm, which performs
from the obtained Pareto front of MOEA/D-RSB show similar robustness levels with the results of MLMSB-R inter,
similarly to the method that simply reduces h, as in [5], [6],
[12], the distribution is mediocre between Gr and Gl. The
MLMSB-Rintra, and MLMSB-h. The search ability of MOEA/
D-RSB is also validated here. In addition, the results obtained
results here reflect the features of different balance processes in
by MLMSB-h, which can be taken as a co-optimization prothe three sampled networks. On the other hand, in terms of the
cess on both Rinter and Rintra, have performances similar to Gm
remaining imbalances in networks, Gm tends to perform the
of MOEA/D-RSB. Only one solution that has mediocre perbest and shows fewer imbalances. Gr reaches higher Rinter but
formance on both robustness measures can be obtained in one
causes more imbalanced links between clusters, and Gl reaches
run of MLMSB, which is of low efficiency. In contrast, the prohigher Rintra but causes more imbalanced links inside clusters.
posed algorithm can provide a series of solutions with different
We can see that the robustness of structurally balanced netrobustness levels of the two measures, demonstrating the higher
works can be enhanced at the cost of some balanced links.
efficiency of MOEA/D-RSB over the existing single-objective
optimization methods in finding robust structural balance soluB. Experiments on Real-World Networks
tions. From the results in Figs. 3, 4, and Table II, MOEA/DThe performance of MOEA/D-RSB has also been verified on
RSB is found to be effective in searching for trade-off solutions
several real-world networks, including COW [48], EGFR [5],
between the two robustness measures; meanwhile, both the
and MIM [49], whose information is depicted in Table S5 of
exploration and exploitation performances of the proposed
the Supplementary Materials. The obtained non-dominated
algorithm have been validated. For exploring the solution
solutions on these networks are given in Fig. 5, and the results
space, the algorithm can obtain widely distributed results that
of MOEA/D-SB, MOEA/D-Net and MOGA-Net are also
outperform the existing methods; for exploiting the optimal
shown to provide comparisons. As shown in the figure, the prosolutions, considerable results with extreme performances on
posed algorithm is effective in the three tested real-world netone of the two measures can also be generated.
works and outperforms the existing methods. Similar to Table II,
Focusing on the cluster partitions of the sampled networks,
Gr, Gl, and Gm are extracted from the obtained solutions in Fig.
NMI is taken to estimate the similarity between different parti5, and the robustness values of these solutions and the results of
tions, and the corresponding numerical results on LFR netthe single-objective optimization method MLMSB-Rinter,
works with 200 nodes are shown in Table III. As it can be seen
MLMSB-Rintra, and MLMSB-h are listed in Table S6 of the
from the table, evident differences can be found between the
Supplementary Materials. As shown in the table, the results of
cluster partitions of Gr, Gl, and Gm, which are sample networks
MOEA/D-RSB reach robustness levels similar to those of the
single-objective optimization methods, which reveals the effecfrom three peculiar parts of the Pareto front in Fig. 4. To
achieve the optimal condition on a specific objective, the allotiveness of MOEA/D-RSB in finding robust structurally balcations of nodes follow different strategies, and the similarity
ance partitions in real-world networks.
information in Table III just reflects that fact.
Furthermore, the numbers of balanced and imbalanced links
are analyzed to give a direct description of the distributions of
TABLE III The partition similarity among sampled networks
positive/negative links in these sampled networks, reported in
evaluated by NMI. The results are averaged over ten
independent realizations.
Table S4 of the Supplementary Materials. As shown in the
table, the sampled networks have different distributions of
Gr
Gm
Gl
attributive links in the network. The first two columns (Pi and
Gr
1.0
0.667 ± 0.105
0.392 ± 0.079
Nb) represent the expected states of structural balance, which
0.667 ± 0.105
1.0
0.375 ± 0.082
Gm
means positive links should be located inside clusters and nega0.392 ± 0.079
0.375 ± 0.082
1.0
Gl
tive ones ought to be located between clusters, and the three

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