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The path toward realizing next-generation petascale and exascale computing is increasingly dependent on building supercomputers with unprecedented numbers of processors. To prevent the interconnect from dominating the overall cost of these ultrascale systems, there is a critical need for scalable interconnects that capture the communication requirements of ultrascale applications. It is, therefore, essential to understand high-end application communication characteristics across a broad spectrum of computational methods, and utilize that insight to tailor interconnect designs to the specific requirements of the underlying codes. This work makes several unique contributions toward attaining that goal. First, we conduct one of the broadest studies to date of high-end application communication requirements, whose computational methods include: finite difference, lattice Boltzmann, particle-in-cell, sparse linear algebra, particle-mesh ewald, and FFT-based solvers. Using derived communication characteristics, we next present the fit-tree approach for designing network infrastructure that is tailored to application requirements. The fit-tree minimizes the component count of an interconnect without impacting application performance compared to a fully connected network. Finally, we propose a methodology for reconfigurable networks to implement fit-tree solutions. Our Hybrid Flexibly Assignable Switch Topology (HFAST) infrastructure, uses both passive (circuit) and active (packet) commodity switch components to dynamically reconfigure interconnects to suit the topological requirements of scientific applications. Overall, our exploration points to several promising directions for practically addressing the interconnect requirements of future ultrascale systems.