The critical node detection based on cascade model is a very important way for analyzing network vulnerability and has recently attracted the attention of many researchers in complex network area. Most of existing works aim to design effective attack strategies which lead to maximal damage to the network (i.e. the destructiveness of the attack), while the number of initial attacked nodes (i.e. $k$) should be given by decision makers in advance. In this paper, we transform the critical node detection based on cascade model as a bi-objective optimization problem (named BCVND), where the destructiveness of the attack and the cost of the attack (i.e. $k$) are optimized simultaneously. The main advantage for the problem transformation is that this multi-objective optimization can provide decision makers with a holistic view for analyzing the network vulnerability. To solve BCVND, we propose an effective multi-objective evolutionary approach termed as MO-BCVND. In MO-BCVND, a cost-reduced population initialization strategy is proposed to increase the population diversity and an adaptive local search strategy is also designed to speed up the population convergence. Finally, the experimental results on 12 real-world complex networks clearly demonstrate the effectiveness of the proposed algorithm compared with the state-of-the-art baselines.

中文翻译：

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复杂网络中基于级联模型的关键节点检测多目标优化

基于级联模型的关键节点检测是分析网络脆弱性的一种非常重要的方法，最近引起了复杂网络领域许多研究人员的关注。大多数现有工作旨在设计有效的攻击策略，从而对网络造成最大的破坏（即攻击的破坏性），而初始攻击节点的数量（即 $k$）应由决策者提前给出。在本文中，我们将基于级联模型的关键节点检测转化为双目标优化问题（命名为 BCVND），其中攻击的破坏性和攻击的成本（即 $k$）同时优化。问题转化的主要优点是这种多目标优化可以为决策者提供分析网络脆弱性的整体视图。为了解决 BCVND，我们提出了一种有效的多目标进化方法，称为 MO-BCVND。在 MO-BCVND 中，提出了一种降低成本的种群初始化策略来增加种群多样性，并且还设计了一种自适应局部搜索策略来加速种群收敛。最后，与最先进的基线相比，在 12 个真实世界复杂网络上的实验结果清楚地证明了所提出算法的有效性。提出了一种降低成本的种群初始化策略来增加种群多样性，并且还设计了一种自适应局部搜索策略来加速种群收敛。最后，与最先进的基线相比，在 12 个真实世界复杂网络上的实验结果清楚地证明了所提出算法的有效性。提出了一种降低成本的种群初始化策略来增加种群多样性，并且还设计了一种自适应局部搜索策略来加速种群收敛。最后，与最先进的基线相比，在 12 个真实世界复杂网络上的实验结果清楚地证明了所提出算法的有效性。