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The Goulden-Jackson cluster method is a powerful tool for obtaininggenerating functions for counting words in a free monoid by occurrences of aset of subwords. We introduce a generalization of the cluster method for monoidnetworks, which generalize the combinatorial framework of free monoids. As asample application of the generalized cluster method, we compute bivariate andmultivariate generating functions counting Motzkin paths---both with heightbounded and unbounded---by statistics corresponding to the number ofoccurrences of various subwords, yielding both closed-form and continuedfraction formulae.

A generalized Goulden-Jackson cluster method and lattice path enumeration