  Let $(W,S)$ be a Coxeter system. A $W$-graph encodes a representation of the
Hecke algebra $\mathcal{H}$ of $W$. We construct universal representations of
multi-parameter Hecke algebras on certain quotients of path algebras, and study
their relationships with $W$-graph representations. We also study the quotients
of path algebras on their own, motivated by one example where the quotient path
algebra is isomorphic to an ideal of Lusztig asymptotic Hecke algebra. Finally,
we describe a method to obtain a generating set for the ideals by which we
quotient the path algebras.
