Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximate Sampling of Graphs with Near-$P$-stable Degree Intervals

Péter L. Erdős, Tamás Róbert Mezei, István Miklós

TL;DR

The paper studies approximate sampling of graphs with degree intervals, proving that the degree interval Markov chain is rapidly mixing when the interval set is thin and centered at $P$-stable degree sequences. Building on the degree-interval framework of Rechn and Amanatidis–Kleer, it develops a precursor machinery to reuse the Erdos–Mixing 2022 approach for $P$-stable degree sequences, enabling a polynomial-time mixing bound. The central contributions are (i) a multicommodity-flow construction on the interval state space, (ii) a detailed alternating-trail decomposition and (iii) a two-stage precursor construction that handles closed and open trails, with an extensive case analysis to ensure all possibilities are covered. This yields a broad rapid-mixing result for degree-interval realizations under near-regular, $P$-stable conditions, facilitating robust approximate sampling even when only partially observed networks are available.

Abstract

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller-Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well-studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently Amanatidis and Kleer published a beautiful paper (arXiv:2110.09068), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper we extend substantially their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centred at P-stable degree sequences.

Approximate Sampling of Graphs with Near-$P$-stable Degree Intervals

TL;DR

The paper studies approximate sampling of graphs with degree intervals, proving that the degree interval Markov chain is rapidly mixing when the interval set is thin and centered at -stable degree sequences. Building on the degree-interval framework of Rechn and Amanatidis–Kleer, it develops a precursor machinery to reuse the Erdos–Mixing 2022 approach for -stable degree sequences, enabling a polynomial-time mixing bound. The central contributions are (i) a multicommodity-flow construction on the interval state space, (ii) a detailed alternating-trail decomposition and (iii) a two-stage precursor construction that handles closed and open trails, with an extensive case analysis to ensure all possibilities are covered. This yields a broad rapid-mixing result for degree-interval realizations under near-regular, -stable conditions, facilitating robust approximate sampling even when only partially observed networks are available.

Abstract

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller-Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well-studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently Amanatidis and Kleer published a beautiful paper (arXiv:2110.09068), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper we extend substantially their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centred at P-stable degree sequences.
Paper Structure (29 sections, 20 theorems, 82 equations, 4 figures)

This paper contains 29 sections, 20 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Our contribution.
  4. Constructing and bounding the multicommodity-flow
  5. Constructing and counting the auxiliary matrices
  6. The alternating-trail decomposition
  7. The precursor
  8. Constructing the precursor
  9. Stage 1: closed trails
  10. Stage 2: open trails
  11. Case A1: $\nabla_2=\emptyset$.
  12. Case A2: $\nabla_1\neq\emptyset$ and $\nabla_2\neq \emptyset$.
  13. Case B: $s=v_0v_1\ldots v_{\lambda-1}v_\lambda$ is an open trail of even length and $v_0=v_{\lambda-1}$.
  14. Case C: $s=v_0v_1\ldots v_{\lambda-1}v_\lambda$ is an open trail of even length and $v_0\neq v_{\lambda-1}$.
  15. Proof of \ref{['thm:main']}.
  16. ...and 14 more sections

Key Result

Theorem 2.16

Let $\mathbb{G}$ be an irreducible, symmetric, reversible, and lazy Markov chain. Let $f$ be a multicommodity-flow on $\mathbb{G}$ which sends $\sigma(X)\sigma(Y)$ commodity between any ordered pair $X,Y\in V(\mathbb{G})$, where $\sigma\equiv |V(\mathbb{G})|^{-1}$ is the unique stationary distributi where $\ell(f)$ is the length of the longest path with positive flow and $\rho(f)$ is the maximum l

Figures (4)

  • Figure 1: The three types of operations employed by the degree interval Markov chain. Solid () and dashed () line segments represent edges and non-edges, respectively.
  • Figure 2: \ref{['thm:JMS']} defines pairs of lower and upper bounds ($\delta$ and $\Delta$), such that any degree sequence which obeys these bounds is $P$-stable; the area between these functions is filled with vertical lines. The pairs $(\delta,\Delta)$ of most distant bounds allowed by \ref{['eq:JMS']} are given by intersections with vertical lines. For example, any degree sequence which is (element-wise) between $\delta=\frac{1}{4} n$ and $\Delta=\frac{3}{4} n$ is $P$-stable. In comparison, the solid gray region represents a $\sqrt{r}$-wide region around the regular degree sequences, which corresponds to the domain of \ref{['thm:ak21']}.
  • Figure 3: The functions $s(u,\bullet)$ and $s(v,\bullet)$ pair the edges incident on $u$ and $v$, respectively. The orange arcs () join edges that are pairs in $s(u,\bullet)$. The cyan arcs () join edges that are pairs in $s(v,\bullet)$. The cyan loop () corresponds to the relation $s(v,uv)=uv$.
  • Figure 4: An example for $\nabla=\nabla_{X,Y}$ and an $({X,Y})$-alternating $s\in \Pi(\nabla)$ (see \ref{['def:XYalternating']}). Red edges belong to $X$ and blue edges belong to $Y$. There are $2^4$ different $\pi\in\Pi(\nabla)$ that are $({X,Y})$-alternating. There is one such $\pi$ where $L(\nabla,\pi)$ has 3 components (the two $C_6$'s and a $C_4$ in the middle), and there are 6 cases where $L(\nabla,\pi)$ has 2 components. The black arcs represent an $s$ such that $L(\nabla,s)$ has exactly one component, or, in other words, $s$ defines a closed Eulerian trail on $\nabla$.

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 55 more