  I report on the development of a novel statistical mechanical formalism for
the analysis of random graphs with many short loops, and processes on such
graphs. The graphs are defined via maximum entropy ensembles, in which both the
degrees (via hard constraints) and the adjacency matrix spectrum (via a soft
constraint) are prescribed. The sum over graphs can be done analytically, using
a replica formalism with complex replica dimensions. All known results for
tree-like graphs are recovered in a suitable limit. For loopy graphs, the
emerging theory has an appealing and intuitive structure, suggests how message
passing algorithms should be adapted, and what is the structure of theories
describing spin systems on loopy architectures. However, the formalism is still
largely untested, and may require further adjustment and refinement.
