  This work presents a detailed analysis of the combinatorics of modular
operads. These are operad-like structures that admit a contraction operation as
well as an operadic multiplication. Their combinatorics are governed by graphs
that admit cycles, and are known for their complexity. In 2011, Joyal and Kock
introduced a powerful graphical formalism for modular operads. This paper
extends that work. A monad for modular operads is constructed and a
corresponding nerve theorem is proved, using Weber's abstract nerve theory, in
the terms originally stated by Joyal and Kock. This is achieved using a
distributive law that sheds new light on the combinatorics of modular operads.
