IEEE Computational Intelligence Magazine - February 2023 - 57

FIGURE 3 The outline of MTEFIM for bi-transformation IM.
methods to construct low-dimensional optimization transformations,
such as the unsupervised neural network and the random
embedding [38]. Feng et al. [39] proposed a multi-variation search
to optimize high-dimensional and low-dimensional transformations
simultaneously. Multi-variation collaborative work not only
utilizes the unique advantages of each alternative transformation,
but also avoids transformation selection for users a priori. However,
due to the specificity of individual representation for IM
problems, the existing MTFO methods cannot directly deal with
a multi-transformation IM.
III. The Problem Form and MTEFIM
A. Outline ofMTEFIM
Given S proxy models {s1, s2, ..., sS} of the IM problem s
on a social network G ¼ (V, E) under an influence spread
model, the multi-transformation IM can be expressed as,
max
AV
si AðÞ; i ¼ 1; 2; ... ; S
s:t: Ajj ¼ k;
(6)
where the size of the seed set selected from the network is
constrained to k. All transformations share the same search
space. The intention is to optimize all transformations simultaneously,
exploiting their potential knowledge by evolutionary
multitasking.
The outline of MTEFIM for a bi-transformation IM is
shown in Fig. 3. As observed, each transformation is assigned a
population. The individuals, i.e., seed sets, in all populations
are initialized in the same search space A2V. Each population
is only evaluated on the corresponding transformation. In the
main loop, for any two transformations si and sj, the intertransformation
relationship rij is estimated by the degree of
overlap across individuals of different transformations online.
Each population is employed to generate the offspring population
based on the designed genetic operator. Then, for each
transformation, based on the inter-transformation relationship
R ¼ {rij j 1 i, j S, i 6¼j, rij ¼ rji}, the top individuals from
the most related transformation are transferred to randomly
replace some individuals in the current transformation, as
shown in Fig. 3, where the transferred individuals are viewed
as the carriers of common knowledge across transformations.
Then, the next population from the parent and offspring populations
are selected by elitism. The final output solution is
selected from the optimal solution of each transformation,
which contains all the proxy model knowledge. In MTEFIM,
each transformation is distributed with an evolutionary solver.
Individuals that may carry shared knowledge flow across transformations.
The framework of MTEFIM for multi-transformation
IM is described in Algorithm 1. Next, the details of
MTEFIM are given in this section.
B. Population Initialization
In MTEFIM, each individual p in all populations represents a
seed set with k nodes, which can be indicated as,
p ¼ p1;p2; ... ;pk
;
(7)
where pk is the node index in {1, 2, ..., v}, and v is the number
of nodes in the social network G. Fig. 4 shows an illustration
ofthe representation.
Since random initialization often leads to a slower convergence
speed, MTEFIM utilizes a warm-starting method commonly
used in IM, that is, the degree discount heuristic [21]. It
is worth noting that other heuristics are still applicable. The
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 57
IEEE Computational Intelligence Magazine - February 2023

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