Topology reveals universal features for network comparison

The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path provides a notion of scale, which is vitally important in characterizing dominant modes of system behavior. Here, by combining topology with scale, we prove the existence of universal features which reveal the dominant scales of any network. We use these features to compare several canonical network types in the context of a social media discussion which evolves through the sharing of rumors, leaks and other news. Our analysis enables for the first time a universal understanding of the balance between loops and tree-like structure across network scales, and an assessment of how this balance interacts with the spreading of information online. Crucially, our results allow networks to be quantified and compared in a purely model-free way that is theoretically sound, fully automated, and inherently scalable. Across the sciences, complex physical and biological systems are represented by networks. A fundamental challenge is to understand the structure of such networks, and to compare them irrespective of their sizes and origins. Closed paths in a network (Fig. 1) are crucial to this understanding. They determine the topology of any network through its mathematical symmetries, and hence its behavior as a dynamical system. The shapes of these paths capture the full range of scales intrinsic to any network: from local features reflecting small-scale properties at the level of individual nodes, to global features revealing large-scale aspects of system behavior such as diffusion and information flows. Shapes that are over-represented, termed motifs, play a key functional role in networks. They are equally fundamental to mathematical representations: In the theory of large graph limits which has emerged over the past decade, motif densities correspond directly to moments of probability distributions. However, the question of precisely which shapes are essential to a unified understanding of all networks has long remained open. Here we show that the simplest shapes 1 ar X iv :1 70 5. 05 67 7v 1 [ st at .M E ] 1 6 M ay 2 01 7