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

TL;DR

This paper introduces a novel operator framework for analyzing ego-centered networks using two-path structures, providing methods for network compression and theoretical guarantees on the preservation of network properties.

Contribution

It develops a new two-path operator formalism, including a decomposition and contraction techniques with safety guarantees for ego network analysis.

Findings

Validated on ten benchmark graphs with diagnostics and distribution analysis.

Established a safe transfer theorem with explicit error bounds.

Demonstrated the effectiveness of the contraction methods in preserving key network features.

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 safe (inequality) transfer theorem for two--walk mass under contraction, with an explicit nonnegative error 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 and two distribution figures.