Rank-matching for multihop multiflow

Seeking fundamental insights into multi-hop multi-flow networks we study the simplest non-trivial setting, a 2 × 2 × 2 MIMO interference network comprised of two sources, two relays and two destinations, wherein all nodes have M antennas, all first-hop channels are of rank D₁, and all second hop channels are of rank D₂. For this setting, we show that the optimal sum DoF is min(4D₁, 4D₂, 2M - |D₁ - D₂|). While 4D₁, 4D₂ are the obvious min-cut bottlenecks that are active when either hop is severely rank-deficient, what is remarkable is that under moderate rank-deficiencies the DoF are limited not by the higher or the lower of the two ranks D₁, D₂, but only by the difference of the two ranks |D₁ - D₂|. This suggests an interesting "rank-matching" design principle for multi-hop networks, reminiscent of "impedance matching", wherein the goal is not necessarily to increase or decrease the rank of each hop, but rather to use linear processing at intermediate hops to create effectively a two-hop setting with matching ranks.