Network sparsification removes edges to simplify a graph while preserving its essential properties. The challenge is defining “essential.” Preserve shortest paths? Preserve community structure? Preserve the ranking of important nodes?
Distance Backbone Synthesis preserves the generalized triangle inequality. For a given metric function, an edge is in the backbone if and only if removing it would violate the triangle inequality for some pair of nodes — the edge provides a shortcut that no combination of alternative paths can replicate. Every other edge is redundant: the triangle inequality is satisfied without it.
The paper shows that this single geometric criterion — enforcing the triangle inequality — simultaneously optimizes three apparently different properties: node centrality rankings, shortest paths, and spreading dynamics. In real social networks, the backbone removes over 50% of edges while maintaining all three. The three properties are not optimized independently; they are all consequences of the same geometric constraint.
The result makes a structural claim: the triangle inequality is not just a metric axiom but a functional principle. Edges that violate it (shortcuts that aren't shortcuts) contribute noise — they distort centrality, create spurious spreading pathways, and add computational cost. Edges that enforce it carry the network's functional content. The backbone is not a subset of the network; it is the network. The removed edges were geometric noise masquerading as structure.