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Capacity of Non-Separable Networks with Restricted Adversaries

Christopher Hojny, Altan B. Kılıç, Sascha Kurz, Alberto Ravagnani

TL;DR

This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries, determining the exact one-shot capacity of one of the fundamental families of 2-level networks introduced in [4], and improving the best currently known lower bounds for another such family.

Abstract

This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries. When the adversary is constrained to operate only on a prescribed subset of edges, classical cut-set bounds are no longer tight, and achieving capacity typically requires a joint design of the outer code and the inner (network) code. This stands in sharp contrast with the case of unrestricted adversaries, where capacity can be achieved by combining linear network coding with appropriate rank-metric outer codes. Building on the framework of network decoding, we determine the exact one-shot capacity of one of the fundamental families of 2-level networks introduced in [4], and we improve the best currently known lower bounds for another such family. In addition, we introduce a new family of networks that generalizes several known examples, and derive partial capacity results that illustrate a variety of phenomena that arise specifically in the restricted-adversary setting. Finally, we investigate the concept of separability of networks with respect to both the rank metric and the Hamming metric.

Capacity of Non-Separable Networks with Restricted Adversaries

TL;DR

This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries, determining the exact one-shot capacity of one of the fundamental families of 2-level networks introduced in [4], and improving the best currently known lower bounds for another such family.

Abstract

This paper investigates the problem of single-source multicasting over a communication network in the presence of restricted adversaries. When the adversary is constrained to operate only on a prescribed subset of edges, classical cut-set bounds are no longer tight, and achieving capacity typically requires a joint design of the outer code and the inner (network) code. This stands in sharp contrast with the case of unrestricted adversaries, where capacity can be achieved by combining linear network coding with appropriate rank-metric outer codes. Building on the framework of network decoding, we determine the exact one-shot capacity of one of the fundamental families of 2-level networks introduced in [4], and we improve the best currently known lower bounds for another such family. In addition, we introduce a new family of networks that generalizes several known examples, and derive partial capacity results that illustrate a variety of phenomena that arise specifically in the restricted-adversary setting. Finally, we investigate the concept of separability of networks with respect to both the rank metric and the Hamming metric.
Paper Structure (14 sections, 14 theorems, 46 equations, 6 figures, 1 table)

This paper contains 14 sections, 14 theorems, 46 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement and Communication Model
  3. Two Families of Simple 2-Level Networks
  4. Family B
  5. Family E
  6. A Generalized Family of Simple 2-Level Networks
  7. Separability
  8. Conclusions and Future Research Questions
  9. A MINLP Model for Finding Network Codes
  10. Unambiguous Codes for Proving Lower Bounds
  11. Lower Bound in Proposition \ref{['prop:familyB']} part 1
  12. Lower Bound in Proposition \ref{['prop:familyB']} part 2
  13. Lower and Upper Bound in Theorem \ref{['thm:another_interesting_ex']} part 1
  14. Lower Bound in Theorem \ref{['thm:another_interesting_ex']} part 2

Key Result

Theorem 2.9

Let $\mathcal{N}=(\mathcal{V},\mathcal{E}, S, \bf T)$ be a network, $\mathcal{A}$ an alphabet, $\mathcal{U} \subseteq \mathcal{E}$, and $t \ge 0$ an integer. We have where $\mathcal{E}' \subseteq \mathcal{E}$ ranges over edge-cuts between $S$ and $T$. In particular, if $\mathcal{U} = \mathcal{E}$ then $\textnormal{C}_1(\mathcal{N},\mathcal{A},\mathcal{E},t) \le \max\{0,\mu_{\mathcal{N}}-2t\}$.

Figures (6)

  • Figure 1: An example of a network.
  • Figure 2: Family B
  • Figure 3: Family E
  • Figure 4: A network of the new family $\mathfrak{S}_{a,b,s}$.
  • Figure 5: Network for Proposition \ref{['prop:restr']}.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: Generalized Network Singleton Bound; see ravagnani2018adversarial
  • Example 2.10
  • Definition 2.11
  • Theorem 3.1: Diamond Network; see beemer2022network
  • ...and 29 more