  Rotation dynamics of eigenvectors of modular network adjacency matrices under
random perturbations are presented. In the presence of $q$ communities, the
number of eigenvectors corresponding to the $q$ largest eigenvalues form a
"community" eigenspace and rotate together, but separately from that of the
"bulk" eigenspace spanned by all the other eigenvectors. Using this property,
the number of modules or clusters in a network can be estimated in an
algorithm-independent way. A general argument and derivation for the
theoretical detectability limit for sparse modular networks with $q$
communities is presented, beyond which modularity persists in the system but
cannot be detected. It is shown that for detecting the clusters or modules
using the adjacency matrix, there is a "band" in which it is hard to detect the
clusters even before the theoretical detectability limit is reached, and for
which the theoretically predicted detectability limit forms the sufficient
upper bound. Analytic estimations of these bounds are presented, and
empirically demonstrated.
