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Graph Compression with Side Information at the Decoder

Praneeth Kumar Vippathalla, Mihai-Alin Badiu, Justin P. Coon

TL;DR

This work addresses the fundamental limit of compressing a graph when decoder-side side information is available, under correlated Erdős-Rényi models for both labelled graphs and their structures. It leverages the entropy-spectrum framework to show that the minimum achievable rate is $R^* = H(A|B)$ across both labelled and unlabelled (structural) representations, and provides both a converse and an achievability via random binning and typicality. In the unweighted case, decoding is linked to graph alignment, with a MAP estimator grounded in the alignment statistic, allowing the same rate characterization to hold. The results quantify how much of a graph’s information is captured by its structure and offer a principled basis for privacy-preserving graph compression, with potential extensions to other random graph models like stochastic block models.

Abstract

In this paper, we study the problem of graph compression with side information at the decoder. The focus is on the situation when an unlabelled graph (which is also referred to as a structure) is to be compressed or is available as side information. For correlated Erdős-Rényi weighted random graphs, we give a precise characterization of the smallest rate at which a labelled graph or its structure can be compressed with aid of a correlated labelled graph or its structure at the decoder. We approach this problem by using the entropy-spectrum framework and establish some convergence results for conditional distributions involving structures, which play a key role in the construction of an optimal encoding and decoding scheme. Our proof essentially uses the fact that, in the considered correlated Erdős-Rényi model, the structure retains most of the information about the labelled graph. Furthermore, we consider the case of unweighted graphs and present how the optimal decoding can be done using the notion of graph alignment.

Graph Compression with Side Information at the Decoder

TL;DR

This work addresses the fundamental limit of compressing a graph when decoder-side side information is available, under correlated Erdős-Rényi models for both labelled graphs and their structures. It leverages the entropy-spectrum framework to show that the minimum achievable rate is across both labelled and unlabelled (structural) representations, and provides both a converse and an achievability via random binning and typicality. In the unweighted case, decoding is linked to graph alignment, with a MAP estimator grounded in the alignment statistic, allowing the same rate characterization to hold. The results quantify how much of a graph’s information is captured by its structure and offer a principled basis for privacy-preserving graph compression, with potential extensions to other random graph models like stochastic block models.

Abstract

In this paper, we study the problem of graph compression with side information at the decoder. The focus is on the situation when an unlabelled graph (which is also referred to as a structure) is to be compressed or is available as side information. For correlated Erdős-Rényi weighted random graphs, we give a precise characterization of the smallest rate at which a labelled graph or its structure can be compressed with aid of a correlated labelled graph or its structure at the decoder. We approach this problem by using the entropy-spectrum framework and establish some convergence results for conditional distributions involving structures, which play a key role in the construction of an optimal encoding and decoding scheme. Our proof essentially uses the fact that, in the considered correlated Erdős-Rényi model, the structure retains most of the information about the labelled graph. Furthermore, we consider the case of unweighted graphs and present how the optimal decoding can be done using the notion of graph alignment.
Paper Structure (16 sections, 6 theorems, 71 equations, 2 figures)

This paper contains 16 sections, 6 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Notation
  4. Organization
  5. Graph Compression with Side Information
  6. Preliminaries
  7. Correlated Erdős-Rényi (CER) Random Graphs
  8. Problem Formulation
  9. Minimum Compression Rate
  10. A Candidate for $R^*$
  11. Characterization of the Limit in \ref{['eq:limit']}
  12. Main Results
  13. Unweighted Graphs
  14. Graph Alignment
  15. Unweighted Graph Compression with Side Information
  16. ...and 1 more sections

Key Result

Lemma 1

Let $(G_{ a}, G_{ b})$ be correlated Erdős-Rényi random graphs specified by the distribution $P_{A, B}$. Let $S_{ a}$ and $S_{ b}$ be the respective structures of $G_{ a}$ and $G_{ b}$. Then Moreover,

Figures (2)

  • Figure 1: A toy example
  • Figure 2: Graph compression with side information at the decoder. The graphs $(G_{ a},G_{ b})$ are correlated Erdős-Rényi graphs, and $(S_{ a}, S_{ b})$ are the respective structures. $U_{ a}$ can be $G_{ a}$ or $S_{ a}$, and $U_{ b}$ can be $G_{ b}$ or $S_{ b}$. The encoder compresses $U_{ a}$ into a message $M_n$ of rate $R$. Using the correlated side information $U_{ b}$, the decoder tries to recover $U_{ a}$.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4: cullina_exactcullina_improved
  • Lemma 2
  • ...and 1 more