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Research Article

A multi-factorial evolutionary algorithm concerning diversity information for solving the multitasking Robust Influence Maximization Problem on networks

Minghao ChenSchool of Intelligent Systems Engineering, Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong, People’s Republic of China

https://orcid.org/0000-0001-7363-7983 View further author information

Shuai WangSchool of Intelligent Systems Engineering, Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong, People’s Republic of ChinaCorrespondence[email protected]

https://orcid.org/0000-0002-1796-7280 View further author information

Jiazhong ZhangSchool of Intelligent Systems Engineering, Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong, People’s Republic of China

https://orcid.org/0000-0002-0138-6668 View further author information

Article: 2275534 | Received 13 Jun 2023, Accepted 21 Oct 2023, Published online: 16 Nov 2023

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https://doi.org/10.1080/09540091.2023.2275534
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Figure 1. Illustration of IC model diffusion, with initial activated nodes S₀= a, b. The blue colour is the seed node in the activated state. In each round, a node in the activation state activates a neighbouring node with probability p (each edge has a propagation probability p).

Figure 1. Illustration of IC model diffusion, with initial activated nodes S0 = a, b. The blue colour is the seed node in the activated state. In each round, a node in the activation state activates a neighbouring node with probability p (each edge has a propagation probability p).

Figure 2. Performance trends on the SF network.

Figure 3. Performance trends on the WS network.

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Figure 4. An intuitive framework of DMFEA. A network containing nine nodes is taken as input and two tasks ( $R_{S}^{L}$ and $R_{S}^{N}$ ) are processed. The algorithm consists of six parts: including initialisation, crossover, mutation, task-oriented learning, gene-oriented learning, and selection. Guided by the two performance metrics and the best information of $R_{S}^{L}$ and $R_{S}^{N}$ , strong seeds are expected to be the output results.

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Table 1 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

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Table 2 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

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Figure 5. Convergence analysis of DMFEA with other single-objective optimisation methods using SF network as an example with seed size K = 20. (a) Convergence curves of different algorithms when $R_{S}^{N}$ is taken as the optimisation task. (b) Convergence curves of different algorithms when $R_{S}^{L}$ is taken as the optimisation task.

Figure 6. Convergence analysis of DMFEA with other multitasking optimisation methods using SF network (N = 200) as an example with seed size K = 20. (a) Convergence curves of different algorithms when $R_{S}^{N}$ is used as the optimisation task (b) Convergence curves of different algorithms when $R_{S}^{L}$ is used as the optimisation task.

Table 3 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

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Table 4 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

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Figure 7. Convergence analysis of DMFEA with other optimisation methods using SF network (N = 500) as an example with seed size K = 20. (a) Convergence curves of different algorithms when used as the optimisation task (b) Convergence curves of different algorithms when taken as the optimisation task.

Table 5 The values of the seeds obtained by different algorithms.

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Figure 8. Ablation experiment with K = 20. (a) Convergence curves of different algorithms when taken as the optimisation task (b) Convergence curves of different algorithms when taken as the optimisation task.

Figure 9. Algorithm convergence curves for different parameter settings on SF network (N = 500) with K = 20 for example. (a) p_t (b) p_g.

Table 6 Performance of different algorithms for different numbers of evaluations.

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Table 7 Comparison of time used by different optimisation algorithms in hours.

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Figure 10. Convergence analysis of DMFEA with other optimisation methods using Frid network as an example with seed size K = 20 (a) Convergence curves of different algorithms when $R_{S}^{N}$ is taken as the optimisation task (b) Convergence curves of different algorithms when $R_{S}^{L}$ is taken as the optimisation task.

Figure 11. Convergence analysis of DMFEA with other optimisation methods using Robot network as an example with seed size K = 20 (a) Convergence curves of different algorithms when $R_{S}^{N}$ is taken as the optimisation task (b) Convergence curves of different algorithms when $R_{S}^{L}$ is taken as the optimisation task.

Figure 12. The distribution of seeds in the Frid network.

Figure 13. The distribution of seeds in the Robot network.

Table 8 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

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Table 9. The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

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Data availability statement

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

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A multi-factorial evolutionary algorithm concerning diversity information for solving the multitasking Robust Influence Maximization Problem on networks

Table 1 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 2 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

Table 3 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 4 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

Table 5 The values of the seeds obtained by different algorithms.

Table 6 Performance of different algorithms for different numbers of evaluations.

Table 7 Comparison of time used by different optimisation algorithms in hours.

Table 8 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 9. The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

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A multi-factorial evolutionary algorithm concerning diversity information for solving the multitasking Robust Influence Maximization Problem on networks

Figures & data

Table 1 The RSNvalues of the seeds obtained by different algorithms when RSN is the optimisation task.

Table 2 The RSL values of the seeds obtained by different algorithms when RSL is the optimisation task.

Table 3 The RSN values of the seeds obtained by different algorithms when RSN is the optimisation task.

Table 4 The RSL values of the seeds obtained by different algorithms when RSL is the optimisation task.

Table 5 The values of the seeds obtained by different algorithms.

Table 6 Performance of different algorithms for different numbers of evaluations.

Table 7 Comparison of time used by different optimisation algorithms in hours.

Table 8 The RSN values of the seeds obtained by different algorithms when RSN is the optimisation task.

Table 9. The RSL values of the seeds obtained by different algorithms when RSL is the optimisation task.

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Table 1 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 2 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

Table 3 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 4 The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.

Table 8 The $R_{S}^{N}$ values of the seeds obtained by different algorithms when $R_{S}^{N}$ is the optimisation task.

Table 9. The $R_{S}^{L}$ values of the seeds obtained by different algorithms when $R_{S}^{L}$ is the optimisation task.