Abstract
Network theory has proven to be a powerful tool in describing and analyzing systems by modelling the relations between their constituent objects. Particularly in recent years, a great progress has been made by augmenting “traditional” network theory in order to account for the multiplex nature of many networks, multiple types of connections between objects, the time-evolution of networks, networks of networks and other intricacies. However, existing network representations still lack crucial features in order to serve as a general data analysis tool. These include, most importantly, an explicit association of information with possibly heterogeneous types of objects and relations, and a conclusive representation of the properties of groups of nodes as well as the interactions between such groups on different scales. In this paper, we introduce a collection of definitions resulting in a framework that, on the one hand, entails and unifies existing network representations (e.g., network of networks and multilayer networks), and on the other hand, generalizes and extends them by incorporating the above features. To implement these features, we first specify the nodes and edges of a finite graph as sets of properties (which are permitted to be arbitrary mathematical objects). Second, the mathematical concept of partition lattices is transferred to the network theory in order to demonstrate how partitioning the node and edge set of a graph into supernodes and superedges allows us to aggregate, compute, and allocate information on and between arbitrary groups of nodes. The derived partition lattice of a graph, which we denote by deep graph, constitutes a concise, yet comprehensive representation that enables the expression and analysis of heterogeneous properties, relations, and interactions 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 framework and demonstrate the advantages of our representation. On the basis of the formal framework described here, we provide a rich, fully scalable (and self-explanatory) software package that integrates into the PyData ecosystem and offers interfaces to popular network packages, making it a powerful, general-purpose data analysis toolkit. We exemplify an application of deep graphs using a real world dataset, comprising 16 years of satellite-derived global precipitation measurements. We deduce a deep graph representation of these measurements in order to track and investigate local formations of spatio-temporal clusters of extreme precipitation events.
The main focus of this paper is to provide a formal framework that enables a mathematically accurate description of any given system in a self-contained fashion. In addition, the purpose of this framework is to facilitate the utilization of existing methods and models supporting a practical data analysis. Network theory serves as the mathematical foundation of our framework. A network models the elements of a system as nodes, and their relations (or interactions) as edges. Particularly in the recent past—certainly also due to the deluge of available data—one could notice a large number of publications attempting to augment “traditional” networks, in order to accommodate the increased heterogeneity of data, and to assign labels and values to nodes and edges (e.g., networks of networks and multilayer networks (MLN)). The framework proposed here entails and unifies these approaches, but also generalizes them with two main aspects in mind: (1) Any node and any edge may be assigned possibly distinct types of properties (e.g., a node representing a human being may have “age” as a type of property whose value is a number, and “blood values” as another type of property whose value is a table of labels and numbers) and (2) integration of properties of groups of nodes and their respective interrelations within the same framework. Together, these objectives make it possible to combine different datasets (e.g., climatological and socioecological data or (electro) physiological records of different organs), integrate a priori knowledge of groups of objects and their relations, and carry out an analysis of potential relationships of the respective systems within the same network representation. On the basis of the mathematical work we provide here, the existing network measures can be generalized and new measures developed. Yet, in order to practically conduct data analysis, we also provide a rich software implementation of our framework that integrates into the PyData ecosystem (which comprises various libraries for scientific computing) and offers interfacing methods to popular network packages, making it a considerable general-purpose data analysis toolkit
