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Network theory has proven to be a powerful tool in describing and analyzingsystems by modelling the relations between their constituent objects. In recentyears great progress has been made by augmenting `traditional' network theory.However, existing network representations still lack crucial features in orderto serve as a general data analysis tool. These include, most importantly, anexplicit association of information with possibly heterogeneous types ofobjects and relations, and a conclusive representation of the properties ofgroups of nodes as well as the interactions between such groups on differentscales. In this paper, we introduce a collection of definitions resulting in aframework that, on the one hand, entails and unifies existing networkrepresentations (e.g., network of networks, multilayer networks), and on theother hand, generalizes and extends them by incorporating the above features.To implement these features, we first specify the nodes and edges of a finitegraph as sets of properties. Second, the mathematical concept of partitionlattices is transferred to network theory in order to demonstrate howpartitioning the node and edge set of a graph into supernodes and superedgesallows to aggregate, compute and allocate information on and between arbitrarygroups of nodes. The derived partition lattice of a graph, which we denote bydeep graph, constitutes a concise, yet comprehensive representation thatenables the expression and analysis of heterogeneous properties, relations andinteractions on all scales of a complex system in a self-contained manner.Furthermore, to be able to utilize existing network-based methods and models,we derive different representations of multilayer networks from our frameworkand demonstrate the advantages of our representation. We exemplify anapplication of deep graphs using a real world dataset of precipitationmeasurements.

Deep Graphs - a general framework to represent and analyze heterogeneous complex systems across scales