Excerpted from Wikipedia
Source: https://en.wikipedia.org/wiki/Hierarchy
Date: 12-Oct-2015
A hierarchy (from the Greek ἱεραρχία hierarchia, “rule of a high priest”, from ἱεράρχης hierarkhes, “leader of sacred rites”) is an arrangement of items (objects, names, values, categories, etc.) in which the items are represented as being “above,” “below,” or “at the same level as” one another.
A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one’s immediate superior or to one of one’s subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Indirect hierarchical links can extend “vertically” upwards or downwards via multiple links in the same direction, following a path. All parts of the hierarchy which are not linked vertically to one another nevertheless can be “horizontally” linked through a path by traveling up the hierarchy to find a common direct or indirect superior, and then down again. This is akin to two co-workers or colleagues; each reports to a common superior, but they have the same relative amount of authority. Organizational forms exist that are both alternative and complementary to hierarchy. Heterarchy is one such form.
Nomenclature
Hierarchies have their own special vocabulary. In an organizational context, the following terms are often used related to hierarchies:
Object: one entity (e.g., a person, department or concept or element of arrangement or member of a set
System: the entire set of objects that are being arranged hierarchically (e.g., an administration)
Dimension: another word for “system” from on-line analytical processing (e.g. cubes)
Member: an (element or object) in a (system or dimension) at any (level or rank)
Rank: the relative value, worth, complexity, power, importance, authority, level etc. of an object
Level: a set of objects with the same rank OR importance
Ordering: the arrangement of the (ranks or levels)
Hierarchy: the arrangement of a particular set of (ranks or levels) i.e. multiple hierarchies are possible per (dimension or system)
Collection: all of the objects at one level
Superior: a higher level or an object ranked at a higher level (parent or ancestor)
Subordinate: a lower level or an object ranked at a lower level (child or descendent)
Hierarch, the top level of the hierarchy, usually consisting of one object or member of a dimension
Peer: an object with the same rank (and therefore at the same level)
Neighbor: the adjacent level/ranking (the immediate superior and immediate inferior)
Interaction: the relationship between an object and its direct superior or subordinate (i.e. a superior/inferior pair)
a direct interaction occurs when one object is on a level exactly one higher or one lower than the other (i.e., on a tree, the two objects have a line between them)
Distance: the minimum number of connections between two objects, i.e., one less than the number of objects that need to be “crossed” to trace a path from one object to another
Span: a qualitative description of the width of a level when diagrammed, i.e., the number of subordinates an object has
In a mathematical context (in graph theory), the general terminology used is different.
Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, with data structures, objects are known as nodes, superiors are called parents and subordinates are called children. In a business setting, a superior is a supervisor/boss and a peer is a colleague.
Subtypes
Nested Hierarchy
Matryoshka dolls, also known as nesting dolls or Russian dolls. Each doll is encompassed inside another until the smallest one is reached. This is the concept of nesting. When the concept is applied to sets, the resulting ordering is a nested hierarchy.
A nested hierarchy or inclusion hierarchy is a hierarchical ordering of nested sets. The concept of nesting is exemplified in Russian matryoshka dolls. Each doll is encompassed by another doll, all the way to the outer doll. The outer doll holds all of the inner dolls, the next outer doll holds all the remaining inner dolls, and so on. Matryoshkas represent a nested hierarchy where each level contains only one object, i.e., there is only one of each size of doll; a generalized nested hierarchy allows for multiple objects within levels but with each object having only one parent at each level. The general concept is both demonstrated and mathematically formulated in the following example:
square C quadrilateral C polygon C shape
A square can always also be referred to as a quadrilateral, polygon or shape. In this way, it is a hierarchy. However, consider the set of polygons using this classification. A square can only be a quadrilateral; it can never be a triangle, hexagon, etc.
Nested hierarchies are the organizational schemes behind taxonomies and systematic classifications. For example, using the original Linnaean taxonomy (the version he laid out in the 10th edition of Systema Naturae), a human can be formulated as:
H. sapiens C Homo C Primates C Mammalia C Animalia
Taxonomies may change frequently (as seen in biological taxonomy), but the underlying concept of nested hierarchies is always the same.
In many programming taxonomies and syntax models (as well as fractals in mathematics), nested hierarchies, including Russian dolls, are also used to illustrate the properties of Self-similarity and Recursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.
Containment Hierarchy
A containment hierarchy is a direct extrapolation of the nested hierarchy concept. All of the ordered sets are still nested, but every set must be “strict”—no two sets can be identical. The shapes example above can be modified to demonstrate this:
\text{square} \subsetneq \text{quadrilateral} \subsetneq \text{polygon} \subsetneq \text{shape} \,
The notation x \subsetneq y \, means x is a subset of y but is not equal to y.
A general example of a containment hierarchy is demonstrated in class inheritance in object-oriented programming.
Two types of containment hierarchies are the subsumptive containment hierarchy and the compositional containment hierarchy. A subsumptive hierarchy “subsumes” its children, and a compositional hierarchy is “composed” of its children. A hierarchy can also be both subsumptive and compositional.
Subsumptive Containment Hierarchy
A subsumptive containment hierarchy is a classification of objects from the general to the specific. Other names for this type of hierarchy are “taxonomic hierarchy” and “IS-A hierarchy”. The last term describes the relationship between each level—a lower-level object “is a” member of the higher class. The taxonomical structure outlined above is a subsumptive containment hierarchy. Using again the example of Linnaean taxonomy, it can be seen that an object that is part of the level Mammalia “is a” member of the level Animalia; more specifically, a human “is a” primate, a primate “is a” mammal, and so on. A subsumptive hierarchy can also be defined abstractly as a hierarchy of “concepts”. For example, with the Linnaean hierarchy outlined above, an entity name like Animalia is a way to group all the species that fit the conceptualization of an animal.
Compositional Containment Hierarchy
A compositional containment hierarchy is an ordering of the parts that make up a system—the system is “composed” of these parts. Most engineered structures, whether natural or artificial, can be broken down in this manner.
The compositional hierarchy that every person encounters at every moment is the hierarchy of life. Every person can be reduced to organ systems, which are composed of organs, which are composed of tissues, which are composed of cells, which are composed of molecules, which are composed of atoms. In fact, the last two levels apply to all matter, at least at the macroscopic scale. Moreover, each of these levels inherit all the properties of their children.
In this particular example, there are also emergent properties—functions that are not seen at the lower level (e.g., cognition is not a property of neurons but is of the brain)—and a scalar quality (molecules are bigger than atoms, cells are bigger than molecules, etc.). Both of these concepts commonly exist in compositional hierarchies, but they are not a required general property. These level hierarchies are characterized by bi-directional causation. Upward causation involves lower-level entities causing some property of a higher level entity; children entities may interact to yield parent entities, and parents are composed at least partly by their children. Downward causation refers to the effect that the incorporation of entity x into a higher-level entity can have on x’s properties and interactions. Furthermore, the entities found at each level are autonomous.
Contexts and Applications
Almost every system within the world is arranged hierarchically. By their common definitions, every nation has a government and every government is hierarchical. Socioeconomic systems are stratified into a social hierarchy (the social stratification of societies), and all systematic classification schemes (taxonomies) are hierarchical. Most organized religions, regardless of their internal governance structures, operate as a hierarchy under God. Many Christian denominations have an autocephalous ecclesiastical hierarchy of leadership. Families are viewed as a hierarchical structure in terms of cousinship (e.g., first cousin once removed, second cousin, etc.), ancestry (as depicted in a family tree) and inheritance (succession and heirship). All the requisites of a well-rounded life and lifestyle can be organized using Maslow’s hierarchy of human needs. Learning must often follow a hierarchical scheme—to learn differential equations one must first learn calculus; to learn calculus one must first learn elementary algebra; and so on. Even nature itself has its own hierarchies, as numerous schemes such as Linnaean taxonomy, the organization of life, and biomass pyramids attempt to document. Hierarchies are so infused into daily life that they are viewed as trivial.
While the above examples are often clearly depicted in a hierarchical form and are classic examples, hierarchies exist in numerous systems where this branching structure is not immediately apparent. For example, most postal code systems are hierarchical. Using the Canadian postal code system, the top level’s binding concept is the “postal district”, and consists of 18 objects (letters). The next level down is the “zone”, where the objects are the digits 0–9. This is an example of an overlapping hierarchy, because each of these 10 objects has 18 parents. The hierarchy continues downward to generate, in theory, 7,200,000 unique codes of the format A0A 0A0. Most library classification systems are also hierarchical. The Dewey Decimal System is regarded as infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.
Organizations
Organizations can be structured as a dominance hierarchy. In an organizational hierarchy, there is a single person or group with the most power and authority, and each subsequent level represents a lesser authority. Most organizations are structured in this manner, including governments, companies, militia and organized religions. The units or persons within an organization are depicted hierarchically in an organizational chart.
In a reverse hierarchy, the conceptual pyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This model represents the idea that members of the higher rankings are responsible for the members of the lower rankings.