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A new class of modular networks is proposed for hierarchically constructing massively parallel computer systems for distributed supercomputing and AI applications. These networks are called hypernets. They are constructed incrementally with identical cubelets, treelets, or buslets that are well suited for VLSI implementation. Hypernets integrate positive features of both hypercubes and tree-based topologies, and maintain a constant node degree when the network size increases. This paper presents the principles of constructing hypernets and analyzes their architectural potentials in terms of message routing complexity, cost-effective support for global as well as localized communication, I/O capabilities, and fault tolerance. Several algorithms are mapped onto hypernets to illustrate their ability to support parallel processing in a hierarchically structured or data-dependent environment. The emulation of hypercube connections using less hardware is shown. The potential of hypernets for efficient support of connectionist models of computation is also explored.