The threshold model can be used to generate random networks of arbitrary size with given local properties such as the degree distribution, clustering, and degree correlation. We summarize the 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 the synchronization properties of threshold networks including the relationship between heterogeneous degrees and resistance to synchrony.

With the goal of understanding the intricate behavior and dynamics of collections of neurons, we present superconducting circuits containing Josephson junctions that model biologically realistic neurons. These "Josephson junction neurons" reproduce many characteristic behaviors of biological neurons such as action potentials, refractory periods, and firing thresholds. They can be coupled together in ways that mimic electrical and chemical synapses. Using existing fabrication technologies, large interconnected networks of Josephson junction neurons would operate fully in parallel. They would be orders of magnitude faster than both traditional computer simulations and biological neural networks. Josephson junction neurons provide a new tool for exploring long-term large-scale dynamics for networks of neurons.

We study the synchronization of identical oscillators diffusively coupled through a network and examine how adding, removing, and moving single edges affects the ability of the network to synchronize. We present algorithms which use methods based on node degrees and based on spectral properties of the network Laplacian for choosing edges that most impact synchronization. We show that rewiring based on the network Laplacian eigenvectors is more effective at enabling synchronization than methods based on node degree for many standard network models. We find an algebraic relationship between the eigenstructure before and after adding an edge and describe an efficient algorithm for computing Laplacian eigenvalues and eigenvectors that uses the network or its complement depending on which is more sparse.

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