  We discuss the relevance of methods of graph theory for the study of damage
in simple model materials described by the random fuse model. While such
methods are not commonly used when dealing with regular random lattices, which
mimic disordered but statistically homogeneous materials, they become relevant
in materials with microstructures that exhibit complex multi-scale patterns. We
specifically address the case of hierarchical materials, whose failure, due to
an uncommon fracture mode, is not well described in terms of either damage
percolation or crack nucleation-and-growth. We show that in these systems,
incipient failure is accompanied by an increase in eigenvector localization and
a drop in topological dimension. We propose these two novel indicators as
possible candidates to monitor a system in the approach to failure. As such,
they provide alternatives to monitoring changes in the precursory avalanche
activity, which is often invoked as a candidate for failure prediction in
materials which exhibit critical-like behavior at failure, but may not work in
the context of hierarchical materials which exhibit scale-free avalanche
statistics even very far from the critical load.
