  The walk distances in graphs have no direct interpretation in terms of walk
weights, since they are introduced via the \emph{logarithms} of walk weights.
Only in the limiting cases where the logarithms vanish such representations
follow straightforwardly. The interpretation proposed in this paper rests on
the identity $\ln\det B=\tr\ln B$ applied to the cofactors of the matrix
$I-tA,$ where $A$ is the weighted adjacency matrix of a weighted multigraph and
$t$ is a sufficiently small positive parameter. In addition, this
interpretation is based on the power series expansion of the logarithm of a
matrix. Kasteleyn (1967) was probably the first to apply the foregoing approach
to expanding the determinant of $I-A$. We show that using a certain linear
transformation the same approach can be extended to the cofactors of $I-tA,$
which provides a topological interpretation of the walk distances.
