# Codes for Graph Erasures

## TL;DR

This paper introduces a new class of codes over graphs designed to efficiently recover from node failures in systems like neural networks and distributed systems, with constructions that reduce field size requirements.

## Contribution

It presents novel constructions of codes over graphs that correct multiple node failures using smaller fields, including optimal codes over binary fields for prime node counts.

## Key findings

- Optimal codes over graphs correcting two node failures over binary fields.
- Construction of codes correcting multiple node failures with reduced field size.
- Improved code constructions for specific failure scenarios.

## Abstract

Motivated by systems where the information is represented by a graph, such as neural networks, associative memories, and distributed systems, we present in this work a new class of codes, called codes over graphs. Under this paradigm, the information is stored on the edges of an undirected graph, and a code over graphs is a set of graphs. A node failure is the event where all edges in the neighborhood of the failed node have been erased. We say that a code over graphs can tolerate $\rho$ node failures if it can correct the erased edges of any $\rho$ failed nodes in the graph. While the construction of such codes can be easily accomplished by MDS codes, their field size has to be at least $O(n^2)$, when $n$ is the number of nodes in the graph. In this work we present several constructions of codes over graphs with smaller field size. In particular, we present optimal codes over graphs correcting two node failures over the binary field, when the number of nodes in the graph is a prime number. We also present a construction of codes over graphs correcting $\rho$ node failures for all $\rho$ over a field of size at least $(n+1)/2-1$, and show how to improve this construction for optimal codes when $\rho=2,3$.