Neighborhood properties of complex networks

A concept of neighborhood in complex networks is addressed based on the criterion of the minimal number of steps to reach other vertices. This amounts to, starting from a given network R-1, generating a family of networks R-l,l=2,3,... such that, the vertices that are l steps apart in the original R-1, are only 1 step apart in R-l. The higher order networks are generated using Boolean operations among the adjacency matrices M-l that represent R-l. The families originated by the well known linear and the Erdos-Renyi networks are found to be invariant, in the sense that the spectra of M-l are the same, up to finite size effects. A further family originated from small world network is identified.