Before revision:  In particular, we introduce a generalized effective resistance for simplexes , provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes.
After revision:  In particular, we introduce a generalized effective resistance for simplices , provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes.