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Simple Constructions of Unique Neighbor Expanders from Error-correcting Codes

Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf

TL;DR

The paper addresses explicit, unbalanced $(\delta,\alpha)$-UN expanders by deriving them from simple outer expander graphs with mild expansion. Using the routed product, the authors reinterpret the SS2 LDPC code construction as a parity-check graph, enabling an elementary, self-contained analysis. Two instantiations yield explicit UN expanders: (i) outer spectral expanders via edge-vertex incidence and (ii) outer combinatorial expanders; both with constant $\delta,\alpha$ and with a constant fraction of unique neighbors. Compared with prior work requiring Ramanujan or high-dimensional objects, the approach is simpler and suggests further avenues for simple biregular outer expanders.

Abstract

In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by Sipser-Spielman [SS96] (and Tanner [Tan81]), and their analysis. We also show how to obtain expanders with many unique neighbors using similar ideas. There were many exciting results on this topic recently, starting with Asherov-Dinur [AD23] and Hsieh-McKenzie-Mohanty-Paredes [HMMP23], who gave a similar construction of unique neighbor expander graphs, but using more sophisticated ingredients (such as almost-Ramanujan graphs) and a more involved analysis. Subsequent beautiful works of Cohen-Roth-TaShma [CRT23] and Golowich [Gol23] gave even stronger objects (lossless expanders), but also using sophisticated ingredients. The main contribution of this work is that we get much more elementary constructions of unique neighbor expanders and with a simpler analysis.

Simple Constructions of Unique Neighbor Expanders from Error-correcting Codes

TL;DR

The paper addresses explicit, unbalanced -UN expanders by deriving them from simple outer expander graphs with mild expansion. Using the routed product, the authors reinterpret the SS2 LDPC code construction as a parity-check graph, enabling an elementary, self-contained analysis. Two instantiations yield explicit UN expanders: (i) outer spectral expanders via edge-vertex incidence and (ii) outer combinatorial expanders; both with constant and with a constant fraction of unique neighbors. Compared with prior work requiring Ramanujan or high-dimensional objects, the approach is simpler and suggests further avenues for simple biregular outer expanders.

Abstract

In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by Sipser-Spielman [SS96] (and Tanner [Tan81]), and their analysis. We also show how to obtain expanders with many unique neighbors using similar ideas. There were many exciting results on this topic recently, starting with Asherov-Dinur [AD23] and Hsieh-McKenzie-Mohanty-Paredes [HMMP23], who gave a similar construction of unique neighbor expander graphs, but using more sophisticated ingredients (such as almost-Ramanujan graphs) and a more involved analysis. Subsequent beautiful works of Cohen-Roth-TaShma [CRT23] and Golowich [Gol23] gave even stronger objects (lossless expanders), but also using sophisticated ingredients. The main contribution of this work is that we get much more elementary constructions of unique neighbor expanders and with a simpler analysis.
Paper Structure (19 sections, 15 theorems, 6 equations)

This paper contains 19 sections, 15 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. History: Unique Neighbor Expanders and Codes
  3. SS1:
  4. SS2:
  5. Our result: Unique Neighbor Expanders from Codes
  6. Preliminaries
  7. Spectral expander
  8. Combinatorial expander
  9. Unique neighbor expanders
  10. The routed product
  11. Unique Neighbor Expanders from Spectral Expanders
  12. One unique neighbor
  13. Constant fraction of unique neighbors
  14. Instantiation
  15. Unique Neighbor Expanders from Combinatorial Expanders
  16. ...and 4 more sections

Key Result

Theorem 2.2

Suppose that $G = (V,E)$ is a $d$-regular graph that is a $\lambda$-spectral expander. Then for any pair of subsets $S, T \subseteq V$,

Theorems & Definitions (30)

  • Definition 2.1: Spectral expander
  • Theorem 2.2: Expander Mixing Lemma, AC06
  • Theorem 2.3: Explicit spectral expanders
  • Definition 2.4: Edge-vertex incidence graph
  • Definition 2.5: Combinatorial expander
  • Definition 2.6: Unique neighbor (UN) expander
  • Lemma 2.7: Non-explicit unique neighbor expanders
  • Definition 2.8: The routed product AC02AD23
  • Lemma 3.1
  • Lemma 3.2
  • ...and 20 more