  This survey describes some useful properties of the local homology of
abstract simplicial complexes. Although the existing literature on local
homology is somewhat dispersed, it is largely dedicated to the study of
manifolds, submanifolds, or samplings thereof. While this is a vital
perspective, the focus of this survey is squarely on the local homology of
abstract simplicial complexes. Our motivation comes from the needs of the
analysis of hypergraphs and graphs. In addition to presenting many classical
facts in a unified way, this survey presents a few new results about how local
homology generalizes useful tools from graph theory. The survey ends with a
statistical comparison of graph invariants with local homology.

Algebraic Topology