The vulnerability of communities in complex networks: A causal perspective on dynamic retentivity

Community structure is a key characteristic of complex systems, and the vulnerability of communities has a significant impact on their functioning. Some researchers have begun investigating methods to measure this vulnerability using concepts from complex network theory. However, most existing approaches for evaluating the vulnerability of communities primarily focus on quantifying their importance by directly using static structural measures of networks. The structural features do not necessarily coincide with the dynamic or functional properties exerted on the networks. To tackle this issue, we propose a novel measure from the perspective of network analysis, aimed at quantifying the vulnerability of communities through targeted interventions, specifically by removing edges. When an edge is specifically targeted or attacked, it can result in not only the collapse of community structure but also the disruption of network function. We introduce the concept of causal dynamic retentivity, which evaluates the response of community structure and function to interventions using normalized mutual information and effective information, respectively. Then, a compressive vulnerability measure for communities is designed by integrating these two factors. The proposed method can evaluate the vulnerability of communities effectively. Furthermore, it can identify edges that significantly influence vulnerability, allowing resources to be prioritized for protecting these connections. The effectiveness of this method is demonstrated through an illustrated example and three real networks.