The Vietoris-Rips complex (https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex) is an abstract simplicial complex that can be defined from any metric space M and distance $\delta$ by forming a simplex for every finite set of points that has diameter at most $\delta$. That is, if the distance between each pair of points in a set $S$ is at most $\delta$, then $S$ is included as a simplex.

The Cech complex (or nerve) is defined by having a simplex for every finite subset of balls with nonempty intersection.

It seems that the two definitions above represent two extremes, since the Vietoris-Rips complex only considers pairwise distances, while the Cech complex considers all possible combinations of intersections.

I am looking for any "intermediate" between these two complexes, or any other similar constructions that can construct a simplicial complex from a set of points.

Thanks for any help.