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Power law random graphs with infinite variance degree distribution are shown to possess a fascinating architecture with a softly hierarchical core network. The core provides loglog-scalable connectivity properties for the whole giant component of the graph. The core is also shown to be robust against attacks to the very heart of the architecture, the top layers of the hierarchy. The structure's similarities and dissimilarities with the Internet's autonomous system graph are discussed. The main significance of a soft hierarchy is seen, however, in the context of structural aspects of large future networks.