PathOverlaps

PathOverlaps is a concept used in graph theory and related fields to describe the extent to which multiple predefined paths share common elements, such as vertices or edges, within a network. It provides a way to quantify redundancy and interaction among routes, sequences, or trajectories that traverse a same underlying graph.

A common way to define PathOverlaps is through overlap measures. For two paths, overlap can be measured by the number of shared edges or vertices, the length of the shared subpath, or a ratio such as the overlap length divided by the total path length. For multiple paths, aggregate measures include pairwise overlap statistics, average overlap, or set-based metrics like the Jaccard similarity of edge sets or the Sørensen-Dice coefficient. In weighted graphs, overlaps can be weighted by edge costs or capacities to reflect resource usage or risk.

Computational aspects vary with context. For two paths, overlap is straightforward to compute by comparing their edge or vertex sequences. For larger collections of paths, one may compute pairwise overlaps and summarize, or apply optimization or clustering techniques to maximize diversity (minimize overlap) or minimize shared resources.

Applications of PathOverlaps span routing and logistics, where reduced overlap can improve resilience and efficiency, to network design and traffic engineering. In robotics and autonomous systems, managing overlaps helps with coordination and collision avoidance. In electronic design and circuit routing, controlling path overlap can reduce congestion and cross-talk.

See also: path diversity, edge-disjoint paths, vertex-disjoint paths, overlap coefficient, Jaccard similarity.