IEEE Computational Intelligence Magazine - February 2023 - 60

The degree of overlap between two
seed sets can directly reflect their
correlation.
t). Thus, in the t-th iteration ofMFEAIM, the offspring population
generated for the s-th transformation is considered to be
drawn from the followingmixed probability distribution.
Lemma 1. Assuming that a parent-centered genetic operator
is employed in the MTEFIM, the mixed probability distribution
ps(x, t) of the offspring for the s-th transformation at
time step t can be expressed as follows,
psðx; tÞ¼½1 r2
j ¼ arg max
j2f1;2;...;sg;j 6¼s
s;j ðtÞ psðx; tÞþ r2 ðtÞ pj
PN PN
s;j
rs;jðtÞ¼
m¼1
ðx; tÞ
n¼1 Iðxm
s ðtÞ; xn
NNK
(11)
Proof. In a multi-transformation environment, an offspring
solution x in sub-population s is drawn from either the probability
distribution ps(x,t)or pj(x,t). The probability P(x
ps(x,
t) j x2s) that x is drawn from ps(x,t) occurs under two cases (see
Algorithm 1):
Case 1: When a random number rand() is greater than or equal
to rs,j
(t), all individuals ofthe offspring x2s are derived from
ps(x,t). The probability ofthe case is,
P case1ðÞ¼ 1 rs;j
ðÞt :
Case 2: When a random number rand() is less than rs,j
ðÞt 1 rs;j
ðÞt
:
(12)
(t), the
probability that the offspring x2s is derived from ps(x,t) is,
P case2ðÞ¼ rs;j
expressed as,
Px
ps x; tðÞ x 2 s
from pj(x,t) is,
Px
pj
ðÞx; t jx 2 s ¼ r2
s;j ðÞt :
(15)
According to Algorithm 2, the inter-transformation relationship
rs,j(t) can be calculated by the degree of overlap
between two seed sets,
PN PN
rs;j tðÞ ¼
where xs
step t. I(xs
and xj
m¼1
n¼1 IxmðÞt ; xn
NNK
s
j ðÞt
;
(16)
i(t) is the i-th seed set in the s-th sub-population at
m(t), xj
n(t)) is the number of identical seeds in xs
m(t)
n(t). Combining (14), (15), and (16), the mixed probability
distribution ps(x,t) can be obtained.
Furthermore, an asymptotic global convergence analysis of
MTEFIM is presented.
60 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
(13)
Therefore, the probability P(x
ps(x,t) j x2s) can be
ðÞj
¼ P case1ðÞþ P case2ðÞ¼ 1 r2
s;j ðÞt
(14)
Similarly, the probability that the offspring x2s is derived
j ðtÞÞ :
Assumption 1. It is assumed that: 1) the population size is
very large (N!1), 2) a parent-centered genetic operator and
truncation selection with truncation parameter a are
employed, 3) ps(x, 0) is positive and continuous in search space
X, s ¼ 1, .. ., S, and 4) truncation parameter a < 1 r2
s;j ðtÞ.
In the proof, a more complex truncation selection is considered
instead ofelitism because evolutionary algorithms with
the elitism strategy trivially have asymptotic global convergence
properties.
Theorem 1. Assuming that Assumption 1 is satisfied, then
MTEFIM has asymptotic global convergence on each transformation
in the multi-transformation environment,
lim ½	¼
t!1
E ss xðÞ
Z
ss xðÞ ps x; tðÞ dx ¼ s
X
s ¼ 1; ... ; S:
(17)
Proof. The proof is in Appendix A, which is similar to a
theoretical analysis ofmost estimations ofthe distribution algorithms
[53] and MFEA [36]. This theorem states that, as the
number of iterations increases, the population distribution of
each transformation moves towards the position of the global
optimum in the search space.
IV. Experimental Study
In this section, an analysis of the knowledge transfer process,
inter-transformation relationship estimation, and selection of
the output seed set in MTEFIM are first given. Then, the efficacy
ofthe proposed MTEFIM on a range ofnetworks is presented.
For the case of synthetic benchmarks and real-world
networks, several methods are adopted as baselines for comparison
in terms of the influence spread and the running time.
The Monte Carlo spread simulation with 10000 runs is
employed to evaluate the influence spread of the seed sets
obtained by all methods. All the experiments are performed
on a PC with a 3.00-GHz Intel Core i7-9700 CPU and 16GB
RAM, and source codes are implemented in Python. The
following observations are explained in this section:
1) The knowledge transfer strategy in MTEFIM can improve
performance by leveraging common knowledge across
transformations (see Section IV.B).
2) For the IM problem, the knowledge transfer mechanism in
MTEFIM outperforms those in other EMTO methods
(see Section IV.B).
3) Parameter r can explicitly reflect the relationship across the
transformations (see Section IV.C).
4) The output seed set selection strategy, SOSS, is effective
(see Section IV.D).
5) The combination of EDV and TIS provides the best performance
in a multi-transformation environment, which
illustrates the rationale for choosing EDV and TIS (see
Section IV.E).
6) Compared with popular IM methods, MTEFIM demonstrates
a highly competitive performance by utilizing unique
knowledge ofdifferent transformations (see Section IV.F).
s ;
IEEE Computational Intelligence Magazine - February 2023

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