  The spectrum of the adjacency matrix plays several important roles in the
mathematical theory of networks and in network data analysis, for example in
percolation theory, community detection, centrality measures, and the theory of
dynamical systems on networks. A number of methods have been developed for the
analytic computation of network spectra, but they typically assume that
networks are locally tree-like, meaning that the local neighborhood of any node
takes the form of a tree, free of short loops. Empirically observed networks,
by contrast, often have many short loops. Here we develop an approach for
calculating the spectra of networks with short loops using a message passing
method. We give example applications to some previously studied classes of
networks.
