Two-Path Operators, Triadic Decompositions, and Majorized Quotients for Ego-Centered Network Compression

Abstract

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.