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We discuss spatially distributed networks that exhibit a diffusive coupling structure, common in biomolecular networks and multi-agent systems. We first review conditions that guarantee spatial homogeneity of the solutions of these systems, referred to as “synchrony.” We next point to structural system properties that allow diffusion-driven instability - a phenomenon critical to pattern formation in biology - and show that an analogous instability mechanism exists in multi-agent systems. The results reviewed in the paper also demonstrate the role played by the Laplacian eigenvalues in determining the dynamical properties of diffusively coupled systems. We conclude with a discussion of how these eigenvalues can be assigned with a design of node and edge weights of a graph, and present a formation control example.