  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.
