IEEE Computational Intelligence Magazine - February 2023 - 65

optimization method, EDVEA, and TISEA, MTEFIM produced
better results. This implies that MTEFIM can simultaneously
utilize all transformations ofunique knowledge in the
search process to significantly improve performance. This
observation is even more evident in large-scale social networks
(see Fig. 12(e)). In Fig. 12(b) and (c), for the Hamsterster and
the Ego-facebook, CELFþþ, EDRL-LM, and MTEFIM
achieve similar results in terms of the influence spread. When
k > 15, the performances of MTEFIM, EDRL-LM, and
CELFþþ are better than those of all other methods. In
Fig. 13(e), for the large-scale Fb-pages-public-figure network,
MTEFIM is similar to CELFþþ. The influence spreads of
Degree and EDVEA increase slowly with k. This might be
because these methods ignore overlapping second-degree
neighborhoods. In Fig. 12(f), for the NetHEPT network,
MTEFIM outperforms all the other baseline methods, which
further illustrates the effectiveness of MTEFIM on large-scale
networks. Overall, the proposed MTEFIM has obtained
highly competitive results on social networks ofdifferent sizes.
GN networks have a clear community structure without
obvious heterogeneous structural information. Compared
with GN networks, real networks present more heterogeneous
structural information. Fig. 12 shows that MTEFIM outperforms
greedy methods (CELFþþ) across the board in synthetic
GN networks. In most real-world networks, MTEFIM
has a highly competitive performance compared to CELFþþ.
This is because the main greedy principle of CELFþþ is tailored
based on the heterogeneous information ofthe network,
while MTEFIM applied in this article does not depend on specific
network structural properties. In addition, real-world networks
show the scale-free property of nodes, which makes it
possible to obtain a good influence spread through a small
number ofnodes with high node degrees.
Fig. 13 shows a runtime analysis of the comparative methods
on both the GN benchmark network and the real-world social
networks, where computational time refers to the sum ofthe runtime
ofa method in ten configurations with k ¼ 3, 6, 9, .. .,30.
First, since three heuristic methods, Degree, SDD, and PageRank,
only need to sort nodes according to the indicator and then select
the top k nodes as the seed set, their computational cost is
extremely meager. In addition, the running time of PageRank
In different stages of evolution, the
estimated r12 adaptively guides
different degrees of knowledge
exchange to accelerate the MTEFIM
convergence.
increases rapidly with the scale ofthe network. Second, the singletransformation
optimization methods, EDVEA and TISEA, perform
better than MTEFIM in terms ofcomputational cost. However,
MTEFIM can find a higher quality seed set by optimizing
EDV and TIS at the same time. Third, the computational cost of
MA-IM and EDRL-LM is higher than that of the other evolution-based
methods because the local search in MA-IM and the
model learning in EDRL-LM are computationally expensive.
Finally, the proposed MTEFIM achieves a highly competitive
performance with CELFþþ at a far lower computational cost.
V. Conclusion
Inspired by the similarities and unique advantages of different
transformations for influence maximization in social networks,
a novel multi-transformation evolutionary framework is introduced
in this article. By considering the degree of overlap
across seed sets of different transformations, the proposed
method can capture the inter-transformation potential relationship,
which controls the frequency and degree of the
knowledge transfer process. In addition, an output seed set
selection strategy is provided to avoid manual selection by
users. Several experiments on benchmarks and real-world
social networks are employed to illustrate the reasonability of
the above claims. Compared with EMTO and the IM-specific
methods, MTEFIM can obtain a high-quality seed set with a
lower computational cost.
Because ofthe broad application potential ofIM, the following
topics deserve further research. First, it is still worthwhile to
theoretically explore the relationships across IM transformations.
Second, a series ofswarm intelligence-based methods for IM have
shown extraordinary performances [21], [22], [24], [25], [46], [47],
[48]. The knowledge transfer process design across multiple transformations
in swarm intelligence optimization methods to
enhance performance for IM can be taken into consideration.
Finally, applying MTEFIM to solve more complex IM problems
such as the competitive IM [28] and the multi-round IM [65],
[66] also serves as a promising research topic.
FIGURE 13 Runtime analysis of the compared methods on both the
GN benchmark network and real-world social networks.
Acknowledgment
This work was supported in part by the Key Scientific Technological
Innovation Research Project through Ministry of
Education; in part by the National Natural Science Foundation
of China Innovation Research Group Fund under Grant
61621005; in part by the State Key Program and the Foundation
for Innovative Research Groups of the National Natural
Science Foundation of China under Grant 61836009; in part
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 65
IEEE Computational Intelligence Magazine - February 2023

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