IEEE Computational Intelligence Magazine - August 2023 - 46

TABLE II The summary of difference values of RS between the
obtained results and the corresponding best one in Table I.
TABLE III The summary of difference values of RS tested on
ER networks.
PER
Difference
0.1
0.22
0.3
0.19
0.5
0.48
0.7
0.52
0.9
0.72
Nevertheless, some key questions remain to be addressed for
this intuitive solution. First of all, setting Per as 0.3 may not be
effective for other kinds of network data and other experimental
setups [21]. A possible but time-consuming option is to find the
suitable Per for each network, which is oflow efficiency and causes
additional computational costs. Also, only one solution seems to
be inadequate to fully solve the RIM problem. A clear performance
degeneration can be detected in Table I when there exists
difference in the parametric settings for optimizing and testing processes.
This phenomenon reveals that a series ofsolutions with different
emphases could be considerable for the RIM problem.
Here, the optimization guided by RS with different Per shows
some correlations; the multi-objective technique is not applicable.
In allusion to addressing these questions, the multi-factorial optimization
theory is employed to solve the RIM problem, the
designed algorithm is presented in the next section.
IV. MFEARIM
Similar to the traditional EAs, an MFEA performs the search
process using a single population consisting of P individuals,
and L optimization tasks {T1, T2, ..., TL} are tackled simultaneously.
Here each task Tl indicates a specific optimization target
RS with a certain Per. Two definitions are requisite for the
multitasking optimization process [25].
1) Skillfactor: The skill factor ti ofindividual pi is the one task
selected from all L tasks, on which pi is the most effective.
2) Scalarfitness: The scalar fitness of pi is given by the fitness
rank of this individual in the multitasking environment,
i.e., 'i ¼ 1/rti
i , where rti
i is the rank ofpi on task ti.
With the skill factor, an L-factorial environment is constructed
to leverage the implicit parallelism between different
tasks. All individuals are encoded in a unified space X, and a
complementary genetic material transfer is considered between
individuals with disparate optimization tasks. The transfer operation
is commonly via crossover operations, where a mating
probability may work as the criterion. This algorithmic parameter
is prescribed in [25], and an improved version introduces a
learning procedure on it based on the probabilistic model [27].
Most existing work focuses on solving optimization problems
with numerical decision variables. But the RIM problem
requires discrete data, which are labels of seeds selected from
nodes. This distinct divergence hinders a direct application of
current MFEAs to find seeds with robust influential ability.
A. The Framework
The low efficiency can be found in the intuitive solution in
Section III-B, and MFEARIM is proposed to tackle this deficiency
and manages to find the best seeds under different damage
percentages. Tasks for the RIM problem can be
46 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2023
PER
Difference
0.1
0.36
represented as fR1
S; R2
0.3
0.31
S; ... ; RL
0.5
0.39
0.7
0.35
Sg, where every task Rl
0.9
0.37
S has a
specific Per setting in the evaluation process. A series of seed
sets is expected to be the final output ofMFEARIM when solving
these tasks. An intuitive description of the algorithm is
given in Fig. S3 ofthe supplementary materials.
Algorithm 1. MFEARIM
Input:
G0: The input network; P0: The initial population size;
MaxGen: The maximum number of genetic iterations;
K: The size of seed set;
Output:
{S1
, S2
, ..., SL
}: The best solution found for each task;
______________________________________________________
Conduct the initialization operator (G0, P0, K) to generate Pop0,
set g ¼ 0;
while g < MaxGen do:
Conduct the crossover operator on Pop0 to generate Pop
(shown in Algorithm S1);
Conduct the mutation operator on Pop;
Conduct the local search operator on Pop (shown in
Algorithm S2);
Conduct the selection operation on Pop to generate Pop0;
Update the best solution (S1
g ¼g þ 1;
, S2
endwhile;
P individuals are generated first to initialize the whole population,
and each of them has K seeds as the coding information.
Both random selection and degree related selection have been
adopted in the seed selection operation. The performance of all
individuals is evaluated on L tasks (using corresponding measure
Rl
S) to get the rank information of each individual on every task
Tj
i ,wherej 2 [1, L]and ri
Tj
based on their performance. Each individual pi has L scalar fitness
'i, j ¼ 1=r
is the rank ofpi on task Tj;
then the task Tj that 'i, j reaches the best competence is selected as
the skill factor ti. Conducted on the initial population, the crossover
operator intends to exchange genetic information between
two selected individuals (p1 and p2) and generate more potential
candidates. Here, if p1 and p2 have the same t, then a uniform
crossover operation is implemented. Otherwise, ifp1 and p2 have
diverse t,aseed c in p2 is selected, where the distance information
between seeds in p2 to those inp1 is considered. Then, a seed in p1
is replaced by c.
In this way, several new candidates are generated and included
into the whole population. Followed by the mutation operator,
this operator randomly mutates part ofthe genetic information of
individuals at a low probability. Also, a three-stage local search
, .. ., SL
) for each task;
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