The Redundant Edge

A complex network has many edges. Social networks, neural networks, transportation networks — each connection carries potential influence, information, disease. The density of connections determines how fast something spreads. Remove edges and the spreading slows. Add edges and it accelerates. This is the intuition: more connections, faster dynamics.

The authors remove more than half the edges and the spreading dynamics barely change.

The method is algebraic. An edge is redundant if there exists an alternative path between its endpoints that is shorter — not in the usual metric sense, but under a generalized path-length measure. Specifically, the distance backbone retains only those edges that satisfy a generalized triangle inequality. An edge from A to B is kept only if no intermediate path through C provides a shorter connection under the chosen measure. If such a path exists, the direct edge is redundant — it carries no information the network does not already have through other routes.

The optimal measure turns out to be a specific power mean: the cube root. Under this measure, the backbone removes the most edges while best preserving node centrality rankings and spreading dynamics. The choice is not arbitrary — it emerges from sweeping through all possible measures and finding the one that maximizes fidelity of the sparsified dynamics.

The structural insight is that most edges in empirical networks are topologically redundant for spreading. They provide alternative routes that parallel existing shorter paths. Removing them does not disconnect the network, does not eliminate any shortest path, does not substantially alter which nodes are central. The edges were present but not load-bearing.

The network was denser than it needed to be. The spreading was carried by its skeleton.