  Transmission line failures in power systems propagate and cascade
non-locally. This well-known yet counter-intuitive feature makes it even more
challenging to optimally and reliably operate these complex networks. In this
work we present a comprehensive framework based on spectral graph theory that
fully and rigorously captures how multiple simultaneous line failures
propagate, distinguishing between non-cut and cut set outages. Using this
spectral representation of power systems, we identify the crucial graph
sub-structure that ensures line failure localization -- the network
bridge-block decomposition. Leveraging this theory, we propose an adaptive
network topology reconfiguration paradigm that uses a two-stage algorithm where
the first stage aims to identify optimal clusters using the notion of network
modularity and the second stage refines the clusters by means of optimal line
switching actions. Our proposed methodology is illustrated using extensive
numerical examples on standard IEEE networks and we discussed several
extensions and variants of the proposed algorithm.
