Abstract
Module or community structures widely exist in complex networks, and optimizing statistical measures is one of the most popular approaches for revealing and identifying such structures in real-world applications. In this paper, we focus on critical behaviors of (Quasi-)Surprise, a type of statistical measure of interest for community structure, accompanied by a series of comparisons with other measures. Specially, the effect of various network parameters on the measures is thoroughly investigated. The critical number of dense subgraphs in partition transition is derived, and a kind of phase diagrams is provided to display and compare the phase transitions of the measures. The effect of "potential well" for (Quasi-)Surprise is revealed, which may be difficult to get across by general greedy (agglomerative or divisive) algorithms. Finally, an extension of Quasi-Surprise is introduced for the study of multi-scale structures. Experimental results are of help for understanding the critical behaviors of (Quasi-)Surprise, and may provide useful insight for the design of effective tools for community detection.