In real-world social networks, global information (e.g., the number of nodes and the connections between them) is incomplete or expensive to acquire; therefore, local community detection becomes especially important. Local community detection is used to identify the local community to which the given starting node belongs according to local information. For a given node, most existing local community detection methods can only find single scale local communities but not those of variable sizes. However, local communities with different scales are often required. Therefore, it is necessary and meaningful to find local communities of the given starting node with different scales; we call this multiscale local community detection. In this paper, we propose a new local modularity inspired by the global modularity and prove the equivalence of the proposed local modularity with two other typical local modularities. Furthermore, to detect local communities with different scales, we present a method based on the proposed local modularity. We test this method on several synthetic and real datasets, and the experimental results indicate that the detected community is meaningful and its scale can be changed reasonably.

Most of the community detection methods require the global information of the original network to be available, however, it is often expensive (even no way) to obtain the global information of the network in many real-world networks. So, the local community detection, only based on the local information, becomes especially important. The local community is the community in the network to which a given starting node belongs. Some local community detection methods have been proposed. However, these methods did not consider the characteristics of the local community during the local community formation. In this paper, we analyze the formation of the local community and propose two local community detection algorithms based on the dynamic membership function. Each of the algorithms is divided into three stages: 1) the initial stage, 2) the middle stage, and 3) the closing stage. At the initial stage, we design a dynamical membership function to detect local community and nodes with the greatest neighborhood intersect rate could be added to the local community. At the middle stage, we design another dynamical membership function, and the goal of this stage is to make the connection of the node in the local community closest. At the closing stage, the third dynamical membership function is provided, and the local community is further improved by collecting some nodes that should not be omitted. We test our algorithms on several synthetic datasets and real datasets; the results show that the local communities detected by our method are closer to the real local communities.