  For every probability $p\in[0,1]$ we define a distance-based graph property,
the $p$TS-distance-balancedness, that in the case $p=0$ coincides with the
standard distance-balancedness, and in the case $p=1$ is related to the
Hamiltonian-connectedness. In analogy with the classical case, where the
distance-balancedness of a graph is equivalent to the property of being
self-median, we characterize the class of $p$TS-distance-balanced graphs in
terms of their equity with respect to certain probabilistic centrality
measures, inspired by the Travelling Salesman Problem. We prove that it is
possible to detect this property looking at the classical distance-balancedness
(and therefore looking at the classical centrality problems) of a suitable
graph composition, namely the wreath product of graphs. More precisely, we
characterize the distance-balancedness of a wreath product of two graphs in
terms of the $p$TS-distance-balancedness of the factors.
