  Community structure analysis is a powerful tool for complex networks, which
can simplify their functional analysis considerably. Recently, many approaches
were proposed to community structure detection, but few works were focused on
the significance of community structure. Since real networks obtained from
complex systems always contain error links, and most of the community detection
algorithms have random factors, evaluate the significance of community
structure is important and urgent. In this paper, we use the eigenvectors'
stability to characterize the significance of community structures. By
employing the eigenvalues of Laplacian matrix of a given network, we can
evaluate the significance of its community structure and obtain the optimal
number of communities, which are always hard for community detection
algorithms. We apply our method to many real networks. We find that significant
community structures exist in many social networks and C.elegans neural
network, and that less significant community structures appear in
protein-interaction networks and metabolic networks. Our method can be applied
to broad clustering problems in data mining due to its solid mathematical basis
and efficiency.
