Let $K_1$ and $K_2$ be two simplicial complexes. I am seeking a measure of the distance between $K_1$ and $K_2$ when viewed as combinatorial objects. What I have in mind is something like this.
Remove simplices (of various dimensions) from $K_1$ such that, at each stage, the remaining object is a simplicial complex. Then add simplices, again always remaining a simplicial complex, until a complex isomorphic to $K_2$ is obtained. The fewest number of simplices added or removed in order to "renovate" $K_1$ to $K_2$ is some measure of their distance. Perhaps, in order to accommodate the different dimensions, a simplex of dimension $d$ should have weight $d+1$ in the count. Let us call this the renovation distance between $K_1$ and $K_2$.
For example, below, removal of two triangles from $K_1$, and
adding a triangle and a segment, reaches $K_2$
(with the isomorphism mapping indicated by the vertex labels)
(Example corrected 5Apr13 by Vidit Nanda comment):
So here the renovation distance is at most $11$ (and I don't see a more efficient path). Likely it is not computationally easy to compute this renovation distance. (Update 5Apr13: Vidit Nanda observes that a special case is subgraph isomorphism, an NP-complete problem.)
My definition is not well-grounded in any theory. Have there been definitions of distances between simplicial complexes that capture a similar intuitive notion? I'd appreciate pointers to relevant literature. Thanks!