*Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by basic concepts of graph theory.*

In the recent post we defined 4 cases of subsumption. In order to get from these basic cases to real world situations, we see the 4 cases as the corners of a ‘continuum’ A, B, C, D, and thus we get figure (subsumption continuum).

Here the following areas – well known from graph theory – can be identified:

**CC Connected Component**

A connected component is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph.* This is what we get when we reduce the density of A, i.e. **components become less obvious to abstract/model just by their internal composition alone.**

**CY Connectivity**

The connectivity asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other. A higher degree of connectivity is what we get when we increase the strength of A, i.e. **components become less obvious to abstract/model just by their external relationships alone.**

**MO Module**

In a module all members have the same set of neighbors among nodes not in the module. This is what we get when we add edges to C, i.e. **the abstraction/model of the class structure becomes a multi-level tree.**

**CL Concept Lattice**

All formal concepts of a bi-partite graph constitute a lattice. This is what we get when we reduce edges from C, i.e. **the abstraction/model of the class structure becomes a multi-level lattice.**

**EX Expander**

An expander is a sparse graph that has strong connectivity properties. **It is ‘quite’ hard to decompose by subsumption into classes or components.** For this reason expanders are used to design robust IT networks etc.

Finally, **we get E by reducing the edges in A and C and then joining them** to a ‘cube’ that can not reasonably be structured by classes or components.

### Thus …

we can take parts of graph theory for describing typical problem situations of abstraction. **However, most of these parts of graph theory* do not address the issues of the actual abstracting mapping from original to model, as required for ‘modelling for the purpose of understanding’. This is what a theory of modelling should be about.**

So long

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* apart from formal concept analysis