  We introduce and analyse ensembles of 2-regular random graphs with a tuneable
distribution of short cycles. The phenomenology of these graphs depends
critically on the scaling of the ensembles' control parameters relative to the
number of nodes. A phase diagram is presented, showing a second order phase
transition from a connected to a disconnected phase. We study both the
canonical formulation, where the size is large but fixed, and the grand
canonical formulation, where the size is sampled from a discrete distribution,
and show their equivalence in the thermodynamical limit. We also compute
analytically the spectral density, which consists of a discrete set of isolated
eigenvalues, representing short cycles, and a continuous part, representing
cycles of diverging size.

Disordered Systems and Neural Networks