Network structure is often considered one of the most important features of a network, and various models exist to generate graphs having one of the most studied types of structures, such as blocks/communities or spatial structures. In this article, we introduce a framework for the generation of random graphs with a controlled size-number of nodes, edges-and a customizable structure, beyond blocks and spatial ones, based on node-pair rank and a tunable probability function allowing to control the amount of randomness. We introduce a structure zoo-a collection of original network structuresand conduct experiments on the small-world properties of networks generated by those structures. Finally, we introduce an implementation as a Python library named Structifynet.