Random Subnetworks of Random Sorting Networks

TL;DR

This paper studies random subnetworks of sorting networks, providing formulas for swap counts and confirming a conjecture, with implications for the limiting behavior of such networks as size grows.

Contribution

It introduces a probabilistic approach to analyze random subnetworks of sorting networks and proves a conjecture related to their structure.

Findings

Expected number of swaps in subnetwork does not depend on total size n.

Provides a formula for the expected swaps in subnetwork.

Confirms a conjecture of Warrington and supports the great circle conjecture.

Abstract

A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbor swaps. For m<=n, consider the random m-particle sorting network obtained by choosing an n-particle sorting network uniformly at random and then observing only the relative order of m particles chosen uniformly at random. We prove that the expected number of swaps in location j in the subnetwork does not depend on n, and we provide a formula for it. Our proof is probabilistic, and involves a Polya urn with non-integer numbers of balls. From the case m=4 we obtain a proof of a conjecture of Warrington. Our result is consistent with a conjectural limiting law of the subnetwork as n->infinity implied by the great circle conjecture Angel, Holroyd, Romik and Virag.