Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monotone Property Thresholds

Colton Magnant, Thor Whalen

Abstract

This paper was originally written by the authors circa 2005 but was never submitted for publication. The present version corrects minor errors, adds references to work published since the original draft, and includes a section discussing further research directions. The core framework -- the property function $ψ_G$, the density function $\varphi^ψ_μ$, and all threshold results -- is unchanged from the original.

Monotone Property Thresholds

Abstract

This paper was originally written by the authors circa 2005 but was never submitted for publication. The present version corrects minor errors, adds references to work published since the original draft, and includes a section discussing further research directions. The core framework -- the property function , the density function , and all threshold results -- is unchanged from the original.

Paper Structure

This paper contains 10 sections, 2 theorems, 20 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Property Function
  3. The Density Function
  4. Results
  5. Further Research
  6. Spectral density parameters.
  7. Rupture degree and tenacity.
  8. Edge-removal analogues.
  9. Directed and hypergraph extensions.
  10. Random graph thresholds.

Key Result

Lemma 1

The $\mu$-threshold for $P$ is $T = \mathrm{opt}_{R_P} \varphi^\psi_\mu$.

Figures (2)

  • Figure 1: $y = \psi_G(x)$
  • Figure 2: Charts of $\varphi_e$ and $\varphi_\delta$.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2: Toughness
  • Definition 3: Scattering Number
  • Definition 4: $(\ell)$-connectivity
  • Definition 5: Component Order Toughness
  • Definition 6: Integrity
  • Definition 7: Component Order Connectivity
  • Definition 8
  • proof
  • Definition 9: $(f, \ell)$-connectivity
  • ...and 6 more