Clique topology of real symmetric matrices

The purpose of this supplement is to provide a more complete account of the mathematics underlying our analyses in the main text. In particular, the order complex and clique topology are described more precisely here. The order complex of a matrix is analogous to its Jordan Form, in that it captures features that are invariant under a certain type of matrix transformation. Likewise, the clique topology of a matrix is analogous to its eigenvalue spectrum, in that it provides a set of invariants that can be used to detect structure. While the Jordan Form and eigenvalue spectrum are invariant under linear change of variables, the order complex and clique topology are invariant under monotonic transformations of the matrix entries.