In the context of Algebraic Topology, an analogous notion to the one of size function has been developed under the name of size homotopy group (Frosini and Mulazzani, 1999).
Size homotopy groups for computation of natural size distances.
1] is a discrete homotopy group, to be defined below.
It was also shown by Fadell and Neuwirth  that the higher homotopy groups of the complement are trivial.
Instead of being defined on the topological space of a geometric realization of a simplicial complex, the discrete homotopy groups are defined in terms of the combinatorial connectivity of the complex.
It had been shown previously by Babson  and independently by Bjorner that that the discrete fundamental group of the permutahedron is isomorphic to the classical homotopy group of the real complement of the k-equal arrangement, [M.
n] in the same way the classical homotopy group gives us information about a topological space.
Since MacPherson's work, some progress on this question has been made, most notably by Anderson [And99], who obtained results on homotopy groups of the matroid Grassmannian, and by Anderson and Davis [AD02], who constructed maps between the real Grassmannian and the matroid Grassmannian--showing that philosophically, there is a splitting of the map from topology to combinatorics--and thereby gained some understanding of the mod 2 cohomology of the matroid Grassmannian.
As mentioned in the introduction, Anderson [And99], and Anderson and Davis [AD02] made some progress on this question, obtaining results on the homotopy groups and cohomology of the matroid Grassmannian.
Behrens (mathematics, Massachusetts Institute of Technology) describes the relationship between two machines for computing the 2-primary unstable homotopy groups
of spheres: the EHP spectral sequence and the Goodwillie tower of the identity.
Ranging from the later 1950s and into the later 1960s, these papers and include the "exotic spheres," including a procedure for killing homotopy groups
of differentiable manifolds; expository lectures on topology, differentiable structures, and smooth manifolds with boundary based on "Variedades diferenciables con frontera", papers on relations with algebraic topology, and a series on cobiordism that is evidence of a staggering level of work done in a very short time.
The papers, in English and French, include such subjects as invariants of combinatorial line arrangements and Rybnikov's example, time averaged optimization of dynamic inequalities on a circle, Thom polynomial computing strategies, quasi-convex decomposition in o-minimal structures, homotopy groups
of complements to ample divisions, Massey products of complex hypersurface complements, weighted homogeneous polynomials and blow-analytic equivalence, an infinitesimal criterion for topological triviality of families of sections of analytical variants, valuations and local uniformization, and finite Dehn surgery along A'Campo's divide knots.