IEEE Computational Intelligence Magazine - February 2023 - 62

FIGURE 8 The average convergence trends of MTEFIMandMTEFIM-NK
on two representative networks with k ¼30 over 20 independent runs,
where thex-axis represents the value of EDV(TIS), and the y-axis
represents the function evaluations. (a) Email URIV-EDV, (b) Email URIVTIS,
(c) Fb-pages-public-figure-EDV, and (d) Fb-pages-public-figure-TIS.
that contains two major components, a problem-specificpopulation
initialization and a local search based on similarity. EDRLLM
is a state-of-the-art evolutionary deep reinforcement learning
framework for the IM to optimize the 2-hop influence spread.
The other parameters of all comparison methods are consistent
with those ofthe original papers.
B.Analysis ofKnowledge Transfer Process
Table II lists the influence spread obtained by MTEFIM and
MTEFIM-NK over 20 independent runs, where the size of
the selected seed set k is set to 3, 12, 21, and 30. The symbols
"  " , " þ " , and " - " indicate that MTEFIM-NK is similar, significantly
better, and worse than MTEFIM on the Wilcoxon
rank-sum test with a confidence level of 0.95 [64]. The best
values are indicated in boldface.
For small-scale social networks such as the GN-Network, Email
ERIV, and Hamsterster, both MTEFIM-NK and MTEFIM
can find a good seed set. For larger social networks such as Egofacebook,
Fb-pages-public-figure, and NetHEPT, MTEFIM outperforms
MTEFIM-NK. In addition, the performance gap across
MTEFIM-NK and MTEFIM becomes larger as k or Nincreases.
These phenomena indicate that the effectiveness of knowledge
transfer becomes increasingly evident as the complex network optimization
difficulty increases, e.g., the number of network nodes
and the dimensionality of the search space. In general, compared
with MTEFIM-NK, MTEFIM performs better on 12, and ties 4
outof16cases in termsofthe influence spread, which shows the
effectiveness of the knowledge transfer process in MTEFIM. By
transferring individuals across the EDV and the TIS, common
knowledge is utilized during the evolutionary process.
The average convergence trends ofMTEFIMand MTEFIMNKon
two representative networks are shown in Fig. 8 to further
illustrate the effectiveness of the knowledge transfer process. As
can be seen from Fig. 8, MTEFIM is superior overall to
62 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
FIGURE 9 Influence spread obtained by different EMTO algorithms on
the Email URIV and the Fb-pages-public-figure network. (a) Email URIV
network, and (b) Fb-pages-public-figure network.
MTEFIM-NK in the average convergence trend oftwo transformations,
EDV and TIS. Compared with MTEFIM-NK, MTEFIM
converges faster within 10000 function evaluations. This
phenomenon is obvious on a more complex Fb-pages-publicfigure
network. Based on the above observations, it can be
inferred that the knowledge transfer process inMTEFIM can significantly
enhance the convergence performance.
In addition, to verify the efficiency ofthe proposed knowledge
transfer process, MTEFIM is compared with MFEA,
MFEAII, EMEA, and SBGA on two representative networks,
the small-scale Email URIV network and the large-scale Fbpages-public-figure
network, as shown in Fig. 9. In these figures,
the x-axis represents the size of seed set k, and the y-axis
represents the value ofthe influence spread.
Overall, MTEFIM shows a consistently superior performance.
The influence spread obtained by all methods steadily
increases as k increases. First, all methods yield similar results in
terms of the increasing range in the Email URIV network.
This may be because it is not difficult to determine the seed set
IEEE Computational Intelligence Magazine - February 2023

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