title: Eigenvector dynamics under perturbation of modular networks.
abstract: 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.
authors.affiliation: Design Lab, Faculty of Architecture, Design and Planning, University of Sydney, Australia NSW 2006 and ARC Centre of Excellence for Integrative Brain Function.
authors.forename: Somwrita
authors.initials: S
authors.lastname: Sarkar
authors.affiliation: Qatar Computing Research Institute, Hamad Bin Khalifa University (HBKU), Qatar Foundation, Doha, Qatar and Faculty of Engineering and IT, University of Sydney, Australia NSW 2006.
authors.forename: Sanjay
authors.initials: S
authors.lastname: Chawla
authors.affiliation: School of Physics, University of Sydney, Australia NSW 2006 and ARC Centre of Excellence for Integrative Brain Function.
authors.forename: P A
authors.initials: PA
authors.lastname: Robinson
authors.affiliation: Department of Computer Science, Aalto University, Finland.
authors.forename: Santo
authors.initials: S
authors.lastname: Fortunato