Hierarchies and Level of Detail

Synonyms

Abstraction; Acyclic directed graph; Aggregation; Cognition; FilteringGeneralizing; Level of detail; LOD; Partial order

Definition

A hierarchy is an ordered structure. Order can be established between individuals or between classes of individuals. The ordering function may be any function defining a partial order:

$$\displaystyle{ v_{1} <v_{2}\quad \mathrm{iff}\quad \exists f : v_{2} = f\{v_{\mathrm{i}}\};v_{1} \in \{v_{\mathrm{i}}\}\:. }$$

A vertex v₁ in a graph is on a lower level than v₂ if and only if there exists a function f, such that v₂ can be calculated by applying f to a set of v _i of which v₁ is a member. The function f must be reflexive, antisymmetric and transitive.

The partial ordering can be depicted as a tree with the vertices denoting the individuals of the domain and the edges representing the ordering function between individuals (Fig. 1a).

Hierarchies and Level of Detail, Fig. 1

(a ) Partial ordering: tree; (b ) levels in a hierarchy