Definition
Plain language
A number describing how much one vertex or row matters for the overall structure of a matrix or graph.
As stated in the literature
A statistical importance measure for rows of a matrix (or vertices of a graph), used in randomized linear algebra and spectral graph sparsification; appears as the technique used by RMA on a Spielman ε-light-subset problem.
Why it matters: Leverage scores let algorithms sample 'the rows that matter' instead of touching everything, which is how many large-scale linear algebra and graph methods stay tractable.
For example, a vertex with a high leverage score sits on many short paths through the graph, so removing it would distort the graph's spectrum more than removing a peripheral vertex.