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

Moses Boudourides

TL;DR

An operator viewpoint is developed 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.

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.

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

TL;DR

An operator viewpoint is developed 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.

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.
Paper Structure (35 sections, 14 theorems, 24 equations, 2 figures, 2 tables)

This paper contains 35 sections, 14 theorems, 24 equations, 2 figures, 2 tables.

Table of Contents

  1. Why wedges as operators matter in network science
  2. Preliminaries and notation
  3. Incidence and the two--walk operator
  4. Two--incidence and Gram identities
  5. The canonical two--walk operator
  6. A unique triadic/open decomposition of the wedge operator
  7. Edge triangle multiplicity and triadic intensity
  8. A sharp extremal characterization via open--wedge mass
  9. Open wedge matrix and redundancy as nonedge common neighbors
  10. Triangle incidence, edge multiplicities, and operator factorizations
  11. Edge--triangle multiplicity matrix
  12. Triangle incidence and a Gram factorization
  13. Spectral bounds and extremal structure
  14. Clustered and traversing vertices, corrected
  15. Combinatorial bounds for clustered structure and ego domination
  16. ...and 20 more sections

Key Result

Proposition 3.1

There exists a diagonal matrix $\mathbf{D}_2=\operatorname{diag}(d_{2,1},\dots,d_{2,n})$ such that Moreover, $d_{2,i}=\sum_{j}( \mathbf{A}^2)_{ij}-d_i$ is the number of two--paths incident to $i$ (counting by endpoints), and $\sum_i d_{2,i}=2m_2$.

Figures (2)

  • Figure 1: ECDF of local clustering coefficients $C(v)$ across vertices, one curve per graph.
  • Figure 2: The cycle $C_4$ with partition $P_1=\{1,3\}$ (blue) and $P_2=\{2,4\}$ (red).

Theorems & Definitions (42)

  • Definition 2.1: Two--paths and wedges
  • Proposition 3.1: Wedge Gram identity
  • proof
  • Definition 4.1: Triadic and open parts of $\mathbf{W}$
  • Theorem 4.2: Canonical wedge decomposition and uniqueness
  • proof
  • Definition 4.3: $P_3$-free graph
  • Theorem 4.4: Sharp openness inequality and equality graphs
  • proof
  • Theorem 5.1: Open wedge mass as a nonedge sum
  • ...and 32 more