  A combinatorial trade is a pair of sets of blocks of elements that can be
exchanged while preserving relevant subset intersection constraints. The class
of balanced and swap-robust minimal trades was proposed in [1] for exchanging
blocks of data chunks stored on distributed storage systems in an access- and
load-balanced manner. More precisely, data chunks in the trades of interest are
labeled by popularity ranks and the blocks are required to have both balanced
overall popularity and stability properties with respect to swaps in chunk
popularities. The original construction of such trades relied on computer
search and paired balanced sets obtained through iterative combining of smaller
sets that have provable stability guarantees. To reduce the substantial gap
between the results of prior approaches and the known theoretical lower bound,
we present new analytical upper and lower bounds on the minimal disbalance of
blocks introduced by limited-magnitude popularity ranking swaps. Our
constructive and near-optimal approach relies on pairs of graphs whose vertices
are two balanced sets with edges/arcs that capture the balance and potential
balance changes induced by limited-magnitude popularity swaps. In particular,
we show that if we start with carefully selected balanced trades and limit the
magnitude of rank swaps to one, the new upper and lower bound on the maximum
block disbalance caused by a swap only differ by a factor of $1.07$. We also
extend these results for larger popularity swap magnitudes.
