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Recent algorithmic developments have enabled computers to automaticallydetermine and prove the capacity regions of small hypergraph networks undernetwork coding. A structural theory relating network coding problems ofdifferent sizes is developed to make best use of this newfound computationalcapability. A formal notion of network minimality is developed which removescomponents of a network coding problem that are inessential to its corecomplexity. Equivalence between different network coding problems underrelabeling is formalized via group actions, an algorithm which can directlylist single representatives from each equivalence class of minimal networks upto a prescribed network size is presented. This algorithm, together with rateregion software, is leveraged to create a database containing the rate regionsfor all minimal network coding problems with five or fewer sources and edges, acollection of 744119 equivalence classes representing more than 9 millionnetworks. In order to best learn from this database, and to leverage it toinfer rate regions and their characteristics of networks at scale, a hierarchybetween different network coding problems is created with a new theory ofcombinations and embedding operators.

On Multi-source Networks: Enumeration, Rate Region Computation, and Hierarchy