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.

The Vietoris–Rips 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 Čech 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 Čech 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.