1 General description (informal)

A precise, mathematical definition of hierarchy will be given below. This section will try to explore the ideas behind that more compact definition.

A hierarchy is based on an asymmetrical relationship, such as "is the boss of", "is part of", or "is better than". Such relationships are "asymmetrical" in the sense that if they "work one way", they don't "work the other". For example, if Sally is the boss of Jim, then Jim is not the boss of Sally. When two nodes are related, one is designated the "superior" (or sometimes the "parent") and the other the "subordinate" (or sometimes the "child"). In the intuitive case of the "is the boss of" relation, the boss is the superior and the employee is the subordinate.

A hierarchy's asymmetrical relationship can link entities in one of three ways: directly, indirectly, or not at all. The illustration shows a direct link between the craft and culture sections; the craft section is directly linked to the culture section by the "contains" relationship. This is akin to how your boss is directly in charge of you. In contrast, the illustration shows an indirect link between craft and encyclopedia; the craft section is only "contained" by the encyclopedia as a whole by virtue of being "contained" by the culture section. This is akin to how the CEO of a company is in charge of a factory worker only via middle management. Finally, there is effectively no link between the art and the craft sections; neither section contains the other. This is akin to two co-workers, neither of whom is the other's boss.

Every member is reachable from any other by following the relationship in either direction, but there is no way of coming back to a particular member by always following the relationship in the same direction.

2 General description (formal)

A hierarchy can be represented as a connected directed acyclic graph with a designated initial node (the root). Such structures are also commonly named trees (since they look like upside-down trees, with the root at the top).

3 Examples of reasoning with hierarchies

Many aspects of the world are analyzed, arguably fruitfully, from a hierarchical perspective. The concept of hierarchy thus qualifies as interdisciplinary. Science provides the following examples:

• In biology, organisms are commonly described as an assembly of parts (organs) which are themselves assemblies of yet smaller parts, and so on.
• In physics, the standard model decomposes bodies down to their smallest particle components.
• In linguistics, words or sentences are often broken down into hierarchies of parts and wholes.
• In ethics, various virtues are enumerated and sometimes organized hierarchically according to certain brands of virtue theory .

In all of these examples, the asymmetric relationship is "is composed of".

4 Social hierarchies

Many human organizations, such as businesses, churches, armies and political movements are structured hierarchically, at least officially; commonly superiors, called bosses, have more power than their subordinates. Thus the asymmetrical relationship might be "has power over". (Some analysists question whether power "really" works as the traditional organizational chart indicates, however.) See also Chain of command.

Feminists talk about a hierarchy of gender, in which a culture sees males or masculine traits as superior to females or feminine traits. In the terms above, these feminists present us a hierarchy of only two nodes, "masculine" and "feminine", connected by the asymmetrical relationship "is valued more highly by society". An example of this usage:

The hierarchical nature of the dualism - the systematic devaluation of females and whatever is metaphorically understood as "feminine" - is what I identify as sexism. (Nelson 1992, p. 106)

Note that in the contexts of feminism and other social criticism, the word hierarchy usually is used as meaning power hierarchy or power structure. Feminists do not take issue with hierarchy in the most general definition of the word, but rather with the specific asymmetrical relationship of unequal value between men and women. Various other social critics take issue with different types of power hierarchies that they see as being unjust.

5 Hierarchical nomenclatures in the arts and sciences

Hierarchies are important for categorization and organization of large numbers of objects. Taxonomies, for example, such as biological taxonomies, are built on hierarchies. Hierarchy is also often used to control complexity in engineering endeavors. For example, large electronic devices such as computers are usually composed of modules, which are themselves created out of smaller components ( integrated circuits), which in turn are internally organized using hierarchial methods (e.g. using standard cells).

Hierarchies are used very extensively in computer science and information theory; here are a few examples. Computer files in a file system are stored in a hierarchy of directories in most operating systems. In object-oriented programming, classes are organized hierarchically; the relationship between two related classes is called inheritance. In the Internet, IP addresses are increasingly organized in a hierarchy (so that the routing will continue to function as the Internet grows).

The pitches and form of Tonal music are organized hierarchically, all pitches deriving their importance from their relationship to a tonic key, and secondary themes in other keys are brought back to the tonic in a recapitulation of the primary theme. Susan McClary connects this specifically in the sonata-allegro form to the feminist hierarchy of gender (see above) in her book Feminine Endings, even pointing out that primary themes were often previously called "masculine" and secondary themes "feminine."

6 Wikis

The Wikipedia community is noteworthy for being not overtly hierarchically structured, as no contributor possesses inherently higher standing than another, excepting certain limited "admin" and "developer" powers restricted to a few. However, some would counter that although there is no explicit hierarchy there are social norms which make contributions unequal, as some contributors have more influence because their edits command higher respect.

Those who frequent Wikis might label Wikipedia's organization "wikiarchical".

Note that the content of the Wikipedia (as opposed to the community) is also organized in a fashion that is not overtly hierarchival. For the most part, the relationship between the individual articles does not form any sort of structure.

7 Criticism and alternatives

Hierarchies and hierarchical thinking has been criticized by many people, including Susan McClary as above. Possible alternatives include:

• Enumerative organization, a list
• Retiary organization, a web or network

8 External links

• Principles and annotated bibliography of hierarchy theory
• Summary of the Principles of Hierarchy Theory - S.N. Salthe

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Generalizations: Structure

Specializations:

• purpose: hierarchical organization
• structural properties: hierarchical tree structure, rooted hierarchy
• nature of the hierarchical relationship: part-whole hierarchy , containment hierarchy, inheritance hierarchy , responsibility hierarchy , control hierarchy

Relevant examples:

• Biology: Linnaean taxonomy
• Computer science: Hierarchy_%28object-oriented_programming%29 , Classes
• Genealogy: genealogy tree
• Linguistics: Chomsky hierarchy
• Psychology: Maslow's hierarchy of needs

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9 See also

• Unity of command

10 References

• Julie Nelson (1992). "Gender, Metaphor and the Definition of Economics". Economics and Philosophy, 8:103-125.

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