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

Researcher Moses Boudourides has published a new theoretical computer science paper titled 'Two-Path Operators, Triadic Decompositions, and Safe Quotients for Ego-Centered Network Compression' on arXiv. The work addresses a fundamental challenge in network science: how to compress large, complex social graphs while preserving their most important structural properties. Boudourides develops a rigorous mathematical framework centered on 'two-path operators'—mathematical objects that formalize the concept of wedges (or two-paths) in networks. These wedges are the building blocks of key social phenomena like triadic closure, brokerage, and structural holes, concepts famously explored by sociologist Ronald Burt.

The paper introduces a unique decomposition that splits any network's structure into an 'edge-supported (triadic) part' and a 'nonedge-supported (open) part.' This formal separation allows researchers to understand which connections form closed triangles versus which remain open bridges. Most significantly, Boudourides proves a 'safe transfer theorem' for network contraction. This theorem guarantees that when you compress a collection of ego networks (a person's immediate social circle) using his proposed quotient method, the total 'two-walk mass'—a measure of potential connection pathways—is preserved within a known, non-negative error bound. The method was demonstrated on ten benchmark graphs, showing practical utility. This work provides network analysts with a mathematically sound tool to shrink massive datasets for computational efficiency without losing the nuanced structural patterns that define social relationships and information flow.