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!