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Fully graphic degree sequences and P-stable degree sequences

Péter L. Erdős, István Miklós, Lajos Soukup

TL;DR

The work investigates how P-stability of degree-sequence families relates to the fully graphic property, introducing precise region definitions and a LEG-based machinery to test graphicality. It demonstrates that several known P-stable, simple regions are in fact fully graphic, and strengthens the connection by proving that fully graphic very simple regions are P-stable with explicit growth bounds on perturbations. The authors extend the landscape with almost-fully-graphic results for extremal-degree regions, construct new P-stable families via Tyshkevich products, and show how large fully graphic subregions exist within broad P-stable families. They also propose a universal bound conjecture for the growth of switch-graph realizations and establish the equivalence of all standard P-stability definitions, highlighting the broad applicability to rapid-switch Markov chains and sampling tasks in graph theory and networks.

Abstract

The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for $P$-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of $P$-stable degree sequence families. At first we will show that every known infinite $P$-stable degree sequence set, described by inequalities of the parameters $n, c_1, c_2, Σ$ (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if $Σ$ does not occur in the determining inequality, then the notions of $P$-stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about $P$-stability, describing the maximal $P$-stable sequence set by $n, c_1, c_2$. Furthermore we conjecture that similar equivalences occur in cases if $Σ$ also part of the defining inequality.

Fully graphic degree sequences and P-stable degree sequences

TL;DR

The work investigates how P-stability of degree-sequence families relates to the fully graphic property, introducing precise region definitions and a LEG-based machinery to test graphicality. It demonstrates that several known P-stable, simple regions are in fact fully graphic, and strengthens the connection by proving that fully graphic very simple regions are P-stable with explicit growth bounds on perturbations. The authors extend the landscape with almost-fully-graphic results for extremal-degree regions, construct new P-stable families via Tyshkevich products, and show how large fully graphic subregions exist within broad P-stable families. They also propose a universal bound conjecture for the growth of switch-graph realizations and establish the equivalence of all standard P-stability definitions, highlighting the broad applicability to rapid-switch Markov chains and sampling tasks in graph theory and networks.

Abstract

The notion of -stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for -stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of -stable degree sequence families. At first we will show that every known infinite -stable degree sequence set, described by inequalities of the parameters (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if does not occur in the determining inequality, then the notions of -stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about -stability, describing the maximal -stable sequence set by . Furthermore we conjecture that similar equivalences occur in cases if also part of the defining inequality.
Paper Structure (10 sections, 35 theorems, 115 equations, 1 figure)

This paper contains 10 sections, 35 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Characterization of fully graphic regions $\operatorname{\mathbb D}(n,\Sigma,c_1,c_2)$
  3. The known $P$-stable, simple degree sequence regions are fully graphic
  4. Sequences defined by minimum and maximum degrees
  5. Sequences defined by extremal degrees and degree sums
  6. $P$-stable degree sequences
  7. Every fully graphic very simple degree sequence region is $P$-stable
  8. Construction of $P$-stable families with special properties
  9. Large fully graphic regions in very simple P-stable regions
  10. A common bound of the growing rate in P-stable regions

Key Result

Theorem 1

The largest fully graphic very simple region is $P$-stable, and so the switch Markov chain is rapidly mixing on $\mathbb{D}_{\max}$.

Figures (1)

  • Figure 1: The unique realization of $\mathbf h_m$.

Theorems & Definitions (69)

  • Theorem
  • Theorem
  • Corollary
  • Theorem
  • Theorem
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • ...and 59 more