  We generalize the determinant representation of the KP $\tau$ functions to
the case of the 2D Toda $\tau$ functions. The generating functions for the
weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions;
for which we give a determinant representation of weighted Hurwitz numbers.
Then we can get a finite-dimensional equation system for the weighted Hurwitz
numbers $H^d_{G}(\sigma,\omega)$ with the same dimension $|\sigma|=|\omega|=n$.
Using this equation system, we calculated the value of the weighted Hurwitz
numbers with dimension $0,\,1,\,2$ and give a recursion formula to calculating
the higher dimensional weighted Hurwitz numbers. For any given weighted
generating function $G(z)$, the weighted Hurwitz number degenerates into the
Hurwitz numbers when $d=0$. We get a matrix representation for the Hurwitz
numbers. The generating functions of weighted paths in the Cayley graph of the
symmetric group are a parametric family of 2D Toda $\tau$ functions; for which
we obtain a determinant representation of weighted paths in the Cayley graph.
