## 6 Conclusions  
FCA has been widely used in the identification of social network structures, and stability measures can further improve the accuracy of structure identification. However, stability calculation is an NP-complete task which is the primary challenges in practical. In this paper, we focused on stability calculation and proposed an efficient algorithm MNG-SC to pave the way for the downstream tasks of the application of FCA in social network analysis, such as enhancing the structural security of social networks. Specifically, a novel perspective of stability calculation by linking it to maximal non-generator is first presented. Then, the equivalence between maximal non-generator and the lower neighbor concept is first proved, which makes it sufficient to calculate the concept stability by only exploring its lower neighbor concepts. Experiments demonstrated that the proposed algorithm significantly outperforms several comparative approaches for addressing the concept stability calculation problem and provides better theoretical support for the downstream tasks of SNA, such as abnormal users identification.  
Future work for further exploration is as follows: (1) explore the relationship between the minimal generator and maximal non-generator to find the inner connection of stability calculation algorithms. Then consider the special properties of social networks, such as homogeneity, to develop an efficient stability calculation algorithm specifically suitable for social networks. (2) learn the accurate relationship between topological structure and stability value to improve the accuracy of structure recognition. Generally speaking, the more cohesive structure has a higher stable value. (3) conduct more practical applications of stability measure, such as community detection, social network security, etc.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Datasets**} & \multicolumn{2}{c}{**Running Time(s)**} & \multicolumn{2}{c}{**F1-Measure**} \\ \cline{2-5}  & \(COIN_{12}\) & \(COIN_{12}^{*}\) & \(COIN_{12}\) & \(COIN_{12}^{*}\) \\ \hline Karate & 2.687 & **1.635** & 0.610 & **0.886** \\ Terrorist & 2.920 & **1.792** & 0.591 & **0.850** \\ Football & 58.7321 & **33.722** & 0.651 & **0.835** \\ Neural & 385.818 & **189.813** & 0.622 & **0.804** \\ \hline \hline \end{tabular}
\end{table}
Table 4: The running time and F1-score of \(COIN_{12}\) and \(COIN_{12}^{*}\) on the experimental datasets.  
Figure 5: The running time ratio and F1 ratio on Terrorist between \(COIN_{12}\) and \(COIN_{12}^{*}\).  
**Author Contributions:** Conceptualization, J.G. and F.H.; methodology, J.G.; software, J.G.; validation, J.G. and F.H.; formal analysis, F.H.; investigation, [PERSON]; resources, F.H.; data curation, J.G.; writing--original draft preparation, J.G.; writing--review and editing, F.H. and D.-S.P.; visualization, [PERSON]; supervision, [PERSON] and D.-S.P.; project administration, [PERSON]; funding acquisition, F.H. and D.-S.P. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded in part by the National Natural Science Foundation of China (Grant No. 61702317), the European Union's Horizon 2020 research and innovation programme under the [PERSON] grant agreement No 840922, the National Research Foundation of Korea (No. NRF-2020R1A2B5B01002134), and the Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shaanxi Province (Grant No. 2017024).  
**Conflicts of Interest:** The authors declare no conflict of interest.