  We introduce techniques for analysing the structure of quantum states of
many-body localized (MBL) spin chains by identifying correlation clusters from
pairwise correlations. These techniques proceed by interpreting pairwise
correlations in the state as a weighted graph, which we analyse using an
established graph theoretic clustering algorithm. We validate our approach by
studying the eigenstates of a disordered XXZ spin chain across the MBL to
ergodic transition, as well as the non-equilibrium dyanmics in the MBL phase
following a global quantum quench. We successfully reproduce theoretical
predictions about the MBL transition obtained from renormalization group
schemes. Furthermore, we identify a clear signature of many-body dynamics
analogous to the logarithmic growth of entanglement. The techniques that we
introduce are computationally inexpensive and in combination with matrix
product state methods allow for the study of large scale localized systems.
Moreover, the correlation functions we use are directly accessible in a range
of experimental settings including cold atoms.
