In view of the important nodes (including overlapping nodes, central nodes and outlier nodes) in overlapping community and the inherent overlapping community structure discovery problem, a new symmetric nonnegative matrix factorization algorithm was proposed. First, the sum of the error approximation and the asymmetric penalty term was used as the objective function. Then the algorithm was derived by using the principle of gradient update and the nonnegative constraint conditions. Simulation experiments were carried out on five real networks. The results show that the proposed algorithm can find the important nodes of the actual networks and their inherent community structures. The average conductance and the algorithm's execution time of the community discovery results are better than those of Community Detection with Nonnegative Matrix Factorization (CDNMF) method;the weighted average of the accuracy and recall rate's harmonic mean value shows that the proposed method is more suitable for the large databases.