The threshold model can be used to generate random networks of arbitrary size with given local properties such as degree distribution, clustering, and degree correlation. We summarize properties of networks created using the threshold model and present an alternative deterministic construction. These networks are threshold graphs, and therefore contain a highly-compressible layered structure and allow computation of important network properties in linear time. We show how to construct arbitrarily large, sparse, threshold networks with (approximately) any prescribed degree distribution or Laplacian spectrum. Control of the spectrum allows careful study of synchronization properties of threshold networks including the relationship between heterogeneous degrees and resistance to synchrony.