  The theory of community structure is a powerful tool for real networks, which
can simplify their topological and functional analysis considerably. However,
since community detection methods have random factors and real social networks
obtained from complex systems always contain error edges, evaluating the
robustness of community structure is an urgent and important task. In this
letter, we employ the critical threshold of resolution parameter in Hamiltonian
function, $\gamma_C$, to measure the robustness of a network. According to
spectral theory, a rigorous proof shows that the index we proposed is inversely
proportional to robustness of community structure. Furthermore, by utilizing
the co-evolution model, we provides a new efficient method for computing the
value of $\gamma_C$. The research can be applied to broad clustering problems
in network analysis and data mining due to its solid mathematical basis and
experimental effects.
