  Discrete curvatures are quantities associated to the nodes and edges of a
graph that reflect the local geometry around them. These curvatures have a rich
mathematical theory and they have recently found success as a tool to analyze
networks across a wide range of domains. In this work, we consider the problem
of constructing graphs with a prescribed set of discrete edge curvatures, and
explore the space of such graphs. We address this problem in two ways: first,
we develop an evolutionary algorithm to sample graphs with discrete curvatures
close to a given set. We use this algorithm to explore how other network
statistics vary when constrained by the discrete curvatures in the network.
Second, we solve the exact reconstruction problem for the specific case of
Forman-Ricci curvature. By leveraging the theory of Markov bases, we obtain a
finite set of rewiring moves that connects the space of all graphs with a fixed
discrete curvature.
