Ramanujan Graphs and the Spectral Gap of Supercomputing Topologies

TL;DR

This paper investigates the potential of Ramanujan graphs to serve as optimal supercomputing interconnection networks by comparing their spectral properties with existing topologies.

Contribution

It provides analytic expressions for spectral gap, bisection bandwidth, and diameter of various topologies, highlighting Ramanujan graphs' superior spectral expansion.

Findings

Ramanujan graphs have larger spectral gaps than current topologies.

Spectral gap differences suggest Ramanujan graphs could improve network performance.

Derived new analytic expressions for properties of supercomputing topologies.

Abstract

Graph eigenvalues play a fundamental role in controlling structural properties, such as bisection bandwidth, diameter, and fault tolerance, which are critical considerations in the design of supercomputing interconnection networks. This motivates considering graphs with optimal spectral expansion, called Ramanujan graphs, as potential candidates for interconnection networks. In this work, we explore this possibility by comparing Ramanujan graph properties against those of a wide swath of current and proposed supercomputing topologies. We derive analytic expressions for the spectral gap, bisection bandwidth, and diameter of these topologies, some of which were previously unknown. We find the spectral gap of existing topologies are well-separated from the optimal achievable by Ramanujan topologies, suggesting the potential utility of adopting Ramanujan graphs as interconnection networks.