Community detection has been well studied in network analysis, and one popular technique is spectral clustering which is fast and statistically analyzable for detect-ing clusters for given networks. But the more realistic case of mixed membership community detection remains a challenge. In this paper, we propose a new spectral clustering method Mixed-SLIM for mixed membership community detection. Mixed-SLIM is designed based on the symmetrized Laplacian inverse matrix (SLIM) (Jing et al. 2021) under the degree-corrected mixed membership (DCMM) model. We show that this algorithm and its regularized version Mixed-SLIM {\tau} are asymptotically consistent under mild conditions. Meanwhile, we provide Mixed-SLIM appro and its regularized version Mixed-SLIM {\tau}appro by approximating the SLIM matrix when dealing with large networks in practice. These four Mixed-SLIM methods outperform state-of-art methods in simulations and substantial empirical datasets for both community detection and mixed membership community detection problems.
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