Finding Hidden Structures, Hierarchies, and Cores in Networks via Isospectral Reduction

The method of isospectral network reduction allows one the ability to reduce a network while preserving the network’s spectral structure. In this paper we describe a number of recent applications of the theory of isospectral reductions. This includes finding hidden structures, specifically latent symmetries, in networks, uncovering different network hierarchies, and simultaneously determining different network cores. We also specify how such reductions can be interpreted as dynamical systems and describe the type of dynamics such systems have. Additionally, we show how the recent theory of equitable decompositions can be paired with the method of isospectral reductions to decompose networks.