Enhancing robustness of community structure in networks against attacks with gray information

In recent years, network science has received significant attention and application. As an important mesoscale structure of networks, community structure reveals the inherent modular structure and potential functionality of networks. Therefore, maintaining the community structure during attacks is crucial. Existing studies on community robustness primarily focus on two attack behaviors: malicious attacks and random failures. However, little is known on the community robustness in the more practical situations where the attackers have limited ability to access precise network information, leading to gray information attack behaviors. Hence, in this paper, we investigate the robustness of network communities under gray information attacks and the enhancing algorithms. Following the community robustness measure for the sequential attacks, we establish a unified evaluation framework for attacks with gray information by introducing a gray attack coefficient, which treats the usual random failures and malicious attacks as the two special situations in the proposed framework. Several enhancing algorithms with local search strategy are exquisitely devised. The efficacy of our framework and algorithms which significantly improve the community robustness against gray attacks is demonstrated by the experimental results on a variety of real-world networks. Our findings have important implications not only in enhancing the community robustness of existing networks but also in designing robust ones from the scratch, which paves the way to further understand and seek strategies and solutions for network community robustness with gray information.