## 1 Introduction
In this paper, an algorithm called MNG-SC that considers the hidden existing knowledge in concept lattice and accurately calculates the exact value of concept stability is developed, which pave the way for the downstream tasks of the application of FCA in social network analysis. Specifically, a novel perspective of stability calculation by linking it to maximal non-generator is first provided. Then the equivalence between maximal non-generator and lower neighbor concept is proved, which makes it sufficient to calculate the concept stability of a concept by only exploring its lower neighbor concepts. Furthermore, the performed experiments show that the MNG-SC outperforms the pioneering approaches of the literature. It demonstrated that MNG-SC algorithm can directly and effectively deal with accurate stability calculation of formal concepts. In addition, a case study  
Figure 1: An illustration of the role of FCA and stability in identifying illegal users in social networks. (**a**) a social network including a normal user group and an illegal user group. (**b**) the network structure identified by FCA without distinction. (**c**) an illegal user group accurately identified by stability measure. of identifying abnormal users in social networks is presented, which demonstrates the effectiveness and potential application of our algorithm. In this paper, we focus on solving the theoretical NP-complete problem of stability calculation that hinders the application of FCA to SNA's downstream tasks. The main contributions of this paper are summarized as follows:  
* **Equivalence Relation between Maximal Non-generator and Lower Neighbor Concept**: To best of our knowledge, it is the first proved that the equivalence between maximal non-generator and lower neighbor concept. In contrast to the previous algorithm that needs traverse the entire concept lattice to calculate the stability value, the equivalence makes it possible that only exploring the lower neighbor of a formal concept is sufficient to compute its stability, which greatly improves scalability and reduces computational complexity. * **A Novel Algorithm MNG-SC for Calculating the Exact Stability of Formal Concepts**: Stability calculation is an NP-complete problem. Unlike the approximate calculations of most previous algorithms, we introduce a maximal non-generator-based stability calculation algorithm called MNG-SC for accurately calculating the stability value of the concept. * **Evaluation**: Experimental results show that our algorithm can directly and effectively deal with accurate stability calculation of formal concepts and it sharply outperforms by many orders of magnitude the pioneering approach of the literature. In addition, it is worth noting that we further present a case study to show the usefulness of our research in improving structural security, such as improving the accuracy of abnormal user identification. The rest of this paper is organized as follows: Section 2 overviews the related work. The preliminary knowledge and problem statement are provided in Section 3. Section 4 thoroughly describes the MNG-SC algorithm for computing the exact stability values of formal concepts. The experimental results and analysis is presented in Section 5. Finally, Section 6 concludes this paper and discusses future work.