Mixed-SCORE is a recent approach for mixed membership community detection proposed by Jin et al. (2017) which is an extension of SCORE (Jin, 2015). In the note Jin et al. (2018), the authors propose SCORE+ as an improvement of SCORE to handle with weak signal networks. In this paper, we propose a method called Mixed-SCORE+ designed based on the Mixed-SCORE and SCORE+, therefore Mixed-SCORE+ inherits nice properties of both Mixed-SCORE and SCORE+. In the proposed method, we consider K+1 eigenvectors when there are K communities to detect weak signal networks. And we also construct vertices hunting and membership reconstruction steps to solve the problem of mixed membership community detection. Compared with several benchmark methods, numerical results show that Mixed-SCORE+ provides a significant improvement on the Polblogs network and two weak signal networks Simmons and Caltech, with error rates 54/1222, 125/1137 and 94/590, respectively. Furthermore, Mixed-SCORE+ enjoys excellent performances on the SNAP ego-networks.

To solve the community detection problem, substantial approaches, such as Snijders and Nowicki (1997); Nowicki and Snijders (2001); Daudin et al. (2008); Bickel and Chen (2009); Rohe et al. (2011); Amini et al. (2013), are designed based on the standard framework, the stochastic block model (SBM) (Holland et al. (1983)), since it is mathematically simple and relatively easy to analyze (Bickel and Chen (2009)). However, the assumptions of SBM are too restrictive to implement in real networks. It is assumed that the distribution of degrees within the community is Poisson, that is, the nodes within each community have the same expected degrees. Unfortunately, in many natural networks, the degrees follow approximately a power-law distribution (Kolaczyk (2009); Goldenberg et al. (2010); Jin(2015)). The corrected-degree stochastic block model (DCSBM) (Karrer and Newman (2011)) is developed based on the power-law distribution which allows the degree of nodes varies among different communities.