  Redundancy needs more precise characterization as it is a major factor in the
evolution and robustness of networks of multivariate interactions. We
investigate the complexity of such interactions by inferring a connection
transitivity that includes all possible measures of path length for weighted
graphs. The result, without breaking the graph into smaller components, is a
distance backbone subgraph sufficient to compute all shortest paths. This is
important for understanding the dynamics of spread and communication phenomena
in real-world networks. The general methodology we formally derive yields a
principled graph reduction technique and provides a finer characterization of
the triangular geometry of all edges -- those that contribute to shortest paths
and those that do not but are involved in other network phenomena. We
demonstrate that the distance backbone is very small in large networks across
domains ranging from air traffic to the human brain connectome, revealing that
network robustness to attacks and failures seems to stem from surprisingly vast
amounts of redundancy.
