  We investigate both analytically and numerically the ensemble of
minimum-weight loops and paths in the negative-weight percolation model on
random graphs with fixed connectivity and bimodal weight distribution. This
allows us to study the mean-field behavior of this model. The analytical study
is based on a conjectured equivalence with the problem of self-avoiding walks
in a random medium. The numerical study is based on a mapping to a standard
minimum-weight matching problem for which fast algorithms exist. Both
approaches yield results which are in agreement, on the location of the phase
transition, on the value of critical exponents, and on the absence of any
sizeable indications of a glass phase. By these results, the previously
conjectured upper critical dimension of d_u=6 is confirmed.
