A modularity based spectral method for simultaneous community and anti-community detection
This work addresses the problem of detecting both community and anti-community structures in networks, which is a niche extension of community detection; the method is incremental as it builds on existing spectral modularity approaches.
The paper proposes a spectral method to simultaneously detect communities (high positive modularity) and anti-communities (high negative modularity) in networks by analyzing extreme eigenvalues and invariant subspaces of modularity matrices. The method uses matrix angles based on Frobenius inner products to localize relevant subspaces.
In a graph or complex network, communities and anti-communities are node sets whose modularity attains extremely large values, positive and negative, respectively. We consider the simultaneous detection of communities and anti-communities, by looking at spectral methods based on various matrix-based definitions of the modularity of a vertex set. Invariant subspaces associated to extreme eigenvalues of these matrices provide indications on the presence of both kinds of modular structure in the network. The localization of the relevant invariant subspaces can be estimated by looking at particular matrix angles based on Frobenius inner products.