IEEE Computational Intelligence Magazine - February 2023 - 54

have been proposed to optimize the expected influence to solve
various IM problems [27], [28]. These methods construct a variety
of cheap proxy models for influence spread from different
perspectives and employ the meta-heuristic paradigm to optimize
them. However, the similarities between these proxymodels
have not been investigated, which may improve the
performance of the IM algorithm. In addition, the work [29]
pointed out that these proxy models induce different search
behaviors with similar characteristics. Therefore, it is difficult to
choose the most appropriate proxy model when dealing with
actual IM problems.
The similarity between optimization tasks can be reflected
by an overall correlation among the fitness landscapes, which
can be quantified by Spearman's rank correlation [30]. One
million seed sets are randomly generated in the search space
with a fixed number ofseeds, and their valuations are calculated
on two popular proxy models, EDV and TIS. Then, the Spearman's
rank correlation oftheir valuations is regarded as the similarity
oftwo optimization tasks [31], namely, the optimizing of
EDV and the optimizing of TIS. Fig. 1 shows the similarity of
the two optimization tasks on four representative real-world
networks. The fitness landscapes ofthese two optimization tasks
are indeed highly similar because these proxy models all consider
the neighbor information of the seed set, leading to a
large amount of available knowledge between the optimization
processes of the different proxy models.
Multi-transformation optimization (MTFO) [32] refers to the
simultaneous optimization of multiple alternative formulas for a
target optimization task, which can be solved by multi-task optimization
(MTO) methods [33]. Due to the implicit parallelism of
EAs, multi-task optimization has attracted significant attention in
the field of evolutionary computation (EC) [33], [34], [35], [36],
[37]. MTFOs solved by multi-task EAs have been applied to
many practical problems, such as high-dimensional optimization
[38], [39] and expensive optimization [40], [41]. Compared with a
single-transformation EA optimizing a certain formula of the
target task alone, a multi-transformation EA can utilize the unique
advantages of different formulas to significantly improve performance
through knowledge transfer in the evolutionary process.
Inspired by the multi-transformation EA, this article presents a
multi-transformation evolutionary framework for IM(MTEFIM)
to utilize the similarities and unique advantages ofmultiple proxy
models to improve the performance of evolutionary-based IM
algorithms and to prevent users fromchoosing proxy models a priori.
Multiple proxy models, called multiple transformations in this
article, are optimized simultaneously. Each transformation is
assigned a population. The degree ofoverlap between individuals
of different transformations provides an indirect estimate of their
potential relationships. Based on the inter-transformation relationship,
a knowledge transfer process across a given transformation
and the most relevant " assisted " transformation is designed to
exchange common information adaptively. Finally, MTEFIM
considers the comprehensive rank on the optimal seed set of
each transformation and then outputs a final seed set containing
the knowledge of all proxy models. Empirical studies are
conducted on a series ofsynthetic benchmarks and real-world
networks to validate the performance of MTEFIM. The
results show that MTEFIM achieves a highly competitive performance
in terms of the influence spread and the running
time by knowledge transfer across transformations compared
to several popular IM methods.
The main research contributions of this article are presented
as follows:
1) Inspired by the similarities across transformations, a multitransformation
evolutionary framework for IM with convergence
guarantees is proposed to implicitly utilize the
common and unique knowledge of multiple transformations.
Apart from the existing methods that optimize one
proxy model alone, the proposed method optimizes multiple
transformations simultaneously in one run to avoid
transformation selection a priori.
2) By considering the degree of overlap between individuals
of different transformations, a novel inter-transformation
relationship estimation strategy is presented to guide the
adaptive knowledge transfer process in a multi-transformation
environment. To yield the final output seed set, a simple
yet effective selection strategy is introduced, which
considers compromise performance on the optimal seed set
ofeach transformation.
The remainder of this article is organized as follows.
Section II provides background knowledge on IM problems,
proxy models, and MTFO. MTEFIM is introduced in detail
in Section III. Section IV describes the experimental results
and discussion on five real-world networks. Finally, conclusions
and future works are presented in Section V.
FIGURE 1 The similarity of two optimization tasks, namely, the
optimizing of EDV and the optimizing of TIS, on four real-world
networks: Email URIV, Hamsterster, Ego-facebook, and Fb-pagespublic-figure,
where K is the size of the seed set.
54 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
II. Preliminary
In this section, background knowledge on IM in social networks
and proxy models for meta-heuristic methods are introduced
first. Then, a briefreview ofthe basics and related work
in the field ofMTFO is provided.
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