  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.
