Projective Networks: Topologies for Large Parallel Computer Systems

The interconnection network comprises a significant portion of the cost of large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with the aim of minimizing these costs. However, they fail to consider potential unbalance in the network utilization, which in some cases results in suboptimal designs. Based on an appropriate cost model, this paper advocates the use of networks based on incidence graphs of projective planes, broadly denoted as Projective Networks. Projective Networks rely on highly symmetric generalized Moore graphs and encompass several proposed direct (PN and demi-PN) and indirect (OFT) topologies under a common mathematical framework. Compared to other proposals with average distance between 2 and 3 hops, these networks provide very high scalability while preserving a balanced network utilization, resulting in low network costs. Overall, Projective Networks constitute a competitive alternative for exascale-level interconnection network design.

💡 Research Summary

The paper addresses the growing cost and power consumption of interconnection networks in large‑scale parallel computers, proposing a unified design framework called Projective Networks (PN). Starting from a cost model that incorporates router radix (R), average path length ( ¯k ), and link utilization (u), the authors show that the per‑node cost is approximately proportional to (1 + ¯k u). Consequently, minimizing average distance while keeping utilization close to one yields the most economical designs.

Projective Networks are built from the incidence (Levi) graphs of finite projective planes. These graphs are Δ‑regular, have diameter three, and an average distance that approaches 2.5, which can be further reduced to 2 by a slight modification (demi‑PN), albeit at the expense of perfect symmetry. Both direct (PN, demi‑PN) and indirect (Orthogonal Fat Tree, OFT) topologies are derived from the same mathematical structure, allowing a fair comparison with existing proposals such as Slim Fly, Stacked Single‑Path Tree, and Multi‑layer Full‑Mesh.

Using the Moore bound and its generalized form, the authors derive analytical expressions linking R, ¯k, and the maximum number of compute nodes T that a topology can support. Projective Networks approach these theoretical limits, achieving near‑optimal average distance for a given radix while maintaining u ≈ 1. In practical terms, with contemporary 48‑port routers, a PN can scale to tens of thousands of nodes with a per‑node cost and power consumption 20‑30 % lower than competing designs. The OFT variant provides a two‑level indirect network with diameter two, preserving the low‑cost advantage while simplifying cabling compared to traditional Fat‑Tree architectures.

Extensive simulations confirm that Projective Networks deliver higher scalability, balanced traffic distribution, and lower energy usage across a range of traffic patterns. The work concludes that leveraging the symmetry and distance properties of projective‑plane incidence graphs offers a compelling path toward cost‑effective, power‑efficient interconnects for upcoming Exascale systems.