Measuring intrinsic significance of community structure

The significance of communities is an important inherent property of the community structure. It measures the degree of reliability of the community structure identified by the algorithm. Real networks obtained from complex systems always contain error links. Moreover, most of the community detecting algorithms usually involve random factors. Thus evaluating the significance of community structure is very important. In this paper, using the matrix perturbation theory, we propose a normalized index to efficiently evaluate the significance of community structure without detecting communities. Furthermore, we find that the peaks of this index can be used to determine the optimal number of communities and identify hierarchical community structure, which are two challenging problems in many community detecting algorithms. Lastly, the index is applied to 16 typical real networks, and we find that significant community structures exist in many social networks and in the C. elegans neural network. Comparatively insignificant community structures are identified in protein-interaction networks and metabolic networks. Our method can be generalized to broad clustering problems in data mining.