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Change-point detection within random networks is essential for many applications. Generally, the typical methods focus on the Erdös-Rényi random networks, or assume that the anomalous subnetworks only have high link probability with the fixed membership. In this paper, we consider the stochastic block model of random graphs, and study the change-point detection regarding to the scenario that after a change-point, the connectivity of subnetworks becomes denser or sparser while the membership of nodes also changes. Based on local graph features, we explore a local statistic with dynamic vertex selection for detecting the emergence of an abrupt change-point. In addition, we derive an analytic expression with respect to average run length to set detection threshold in a theoretical fashion, and achieve the probability bounds related to the dynamic vertex selection to characterize the performance of the presented algorithm. As a result, the proposed scheme can provide performance improvement as well as reduce the computational complexity. The proposed algorithm can address a more general problem than the typical methods. Numerical experiments are provided to show the effectiveness of our method.