Two–path operators, triadic decompositions, and majorized quotients for ego–centered network compression

Two–paths (wedges) are the elementary combinatorial objects behind clustering, triadic closure, redundancy, and brokerage. Motivated by a two–path formalism that links Burt’s structural holes to node–centered ego networks, we develop an operator viewpoint in which wedge incidence induces a canonical “two–walk” matrix and a unique decomposition into an edge–supported (triadic) part and a nonedge–supported (open) part. We then study quotient/contraction constructions designed to compress collections of dominating ego networks together with selected “traversing” nodes, and we prove a two–walk transfer theorem under contraction, establishing an inequality with an explicit nonnegative error term and an equality characterization in terms of a wedge–equitable partition. Finally, we illustrate the theory on ten benchmark graphs and their ego–traversing contractions using table–driven diagnostics.