Deep graphs—A general framework to represent and analyze heterogeneous complex systems across scales
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 timeevolution 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 selfcontained manner. Furthermore, to be able to utilize existing networkbased 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 selfexplanatory) software package that integrates into the PyData ecosystem and offers interfaces to popular network packages, making it a powerful, generalpurpose data analysis toolkit. We exemplify an application of deep graphs using a real world dataset, comprising 16 years of satellitederived global precipitation measurements. We deduce a deep graph representation of these measurements in order to track and investigate local formations of spatiotemporal clusters of extreme precipitation events.
 Publication:

Chaos
 Pub Date:
 June 2016
 DOI:
 10.1063/1.4952963
 arXiv:
 arXiv:1604.00971
 Bibcode:
 2016Chaos..26f5303T
 Keywords:

 Physics  Data Analysis;
 Statistics and Probability;
 Computer Science  Social and Information Networks;
 Physics  Atmospheric and Oceanic Physics;
 Physics  Physics and Society
 EPrint:
 27 pages, 6 figures, 4 tables. For associated Python software package, see https://github.com/deepgraph/deepgraph/ . Due to length limitations the abstract appearing here is shorter than that in the PDF file. To be published in "Chaos: An Interdisciplinary Journal of Nonlinear Science"