# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
###### Abstract  
Community structure is one of the most important features of complex networks. Modularity-based methods for community detection typically rely on heuristic algorithms to optimize a specific community quality function. Such methods have two major limits: (1) the resolution limit problem, which prohibits communities of heterogeneous sizes being simultaneously detected, and (2) divergent outputs of the heuristic algorithm, which make it difficult to differentiate relevant and irrelevant results. In this paper, we propose an improved method for community detection based on a scalable community \"fitness function.\" We introduce a new parameter to enhance its scalability, and a strict strategy to filter the outputs. Due to the scalability, on the one hand, our method is free of the resolution limit problem and performs excellently on large heterogeneous networks, while on the other hand, it is capable of detecting more levels of communities than previous methods in deep hierarchical networks. Moreover, our strict strategy automatically removes redundant and irrelevant results; it selectively but inartificially outputs only the best and unique community structures, which turn out to be largely interpretable by the a priori knowledge of the network, including the implanted community structures within synthetic networks, or metadata observed for real-world networks. community detection; resolution limit problem; modularity; multilevel community; Louvain algorithm +
Footnote †: journal: Applied Sciences  
structure of the whole network is often multilevel or hierarchical [8; 20; 21]. Community detection methods generally turn to heuristic algorithms for acceptably good solutions [13]. A good many community detection algorithms came out, among which the most well-known ones include spectral clustering [22; 23; 24], stochastic block models [25; 26], modularity-based [13; 27; 28; 29; 30] or Hamiltonian-based [31; 32; 33; 34; 35; 36] optimization methods, and information-based approaches such as Infomap [37]. Within the scope of this paper, we focus on the modularity-based methods. In a broad sense, modularity-based methods assess the validity of each potential network division with a specific community quality function; a heuristic algorithm is then employed to optimize the communities by maximizing this community quality function. Modularity-based methods, and also Hamiltonian-based methods, have been argued to have two major limits. The first one is the well-known \"resolution limit problem\" raised by [PERSON] and [PERSON] in 2007 [3]. Depending on the numbers of intra connections of communities and the total number of connections within the whole network, a modularity optimization method tends to merge small communities (even if they are well-defined clusters as complete graphs) into larger but sparser ones. This reveals a fact that the modularity-based method cannot find communities of small sizes, much as a microscope cannot find microbes beyond its resolution range. For quality functions other than the modularity, the same phenomenon has also been observed [38]. On the other hand, modularity-based methods may detect unreasonable community structures due to inappropriate resolutions, for example, they detect communities for random graphs [39; 40]. In order to overcome the resolution limit, a variety of \"multiresolution\" methods were suggested. They use a tunable parameter to alter the resolution and detect communities of different levels within different resolution scales [5; 30; 33; 35; 41]. However, a further study by [PERSON] and [PERSON] [42] pointed out that the resolution limit problem is actually induced by two opposite tendencies: the tendency of merging small communities, and the tendency of breaking large ones. If the communities within the same network have very different sizes, it becomes impossible for an optimization approach to avoid both biases simultaneously.