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'ssubordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following apath. All parts of the hierarchy that 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 twoco-workers orcolleagues; 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.
Level or Tier: 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 members into (ranks or levels). Multiple hierarchies are possible per (dimension taxonomy or Classification-system), in which selected levels of the dimension are omitted to flatten the structure
Terms about Placement
Hierarch, the apex of the hierarchy, consisting of one single orphan (object or member) in the top level of a dimension. The root of aninverted-tree structure
Member, a (member or node) in any level of a hierarchy in a dimension to which (superior and subordinate) members are attached
Orphan, a member in any level of a dimension without a parent member. Often the apex of a disconnected branch. Orphans can be grafted back into the hierarchy by creating a relationship (interaction) with a parent in the immediately superior level
Leaf, a member in any level of a dimension without subordinates in the hierarchy
Neighbour: a member adjacent to another member in the same (level or rank). Always a peer.
Superior: a higher level or an object ranked at a higher level (A parent or an ancestor)
Subordinate: a lower level or an object ranked at a lower level (A child or a descendant)
Collection: all of the objects at one level (i.e. Peers)
Peer: an object with the same rank (and therefore at the same level)
Interaction: the relationship between an object and its direct superior or subordinate (i.e. a superior/inferior pair)
adirect interaction occurs when one object is on a level exactly one higher or one lower than the other (i.e., on atree, 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 apath from one object to another
Span: aqualitative description of the width of a level when diagrammed, i.e., the number of subordinates an object has
Terms about Nature
Attribute: a heritable characteristic of (members and their subordinates) in a level (e.g.hair-colour)
Attribute-value: the specific value of a heritable characteristic (e.g.Auburn)
Most hierarchies use a more specific vocabulary pertaining to their subject, but the idea behind them is the same. For example, withdata structures, objects are known asnodes, superiors are calledparents and subordinates are calledchildren. In a business setting, a superior is asupervisor/boss and a peer is acolleague.
Degree ofbranching refers to the number of directsubordinates or children an object has (in graph theory, equivalent to the number of othervertices connected to via outgoing arcs, in a directed graph) a node has. Hierarchies can be categorized based on the "maximum degree", the highest degree present in the system as a whole. Categorization in this way yields two broad classes:linear andbranching.
In alinear hierarchy, the maximum degree is 1.[2] In other words, all of the objects can be visualized in a line-up, and each object (excluding the top and bottom ones) has exactly one direct subordinate and one direct superior. This is referring to theobjects and not thelevels; every hierarchy has this property with respect to levels, but normally each level can have an infinite number of objects.
In abranching hierarchy, one or more objects has a degree of 2 or more (and therefore the minimum degree is 2 or higher).[2] For many people, the word "hierarchy" automatically evokes an image of a branching hierarchy.[2] Branching hierarchies are present within numerous systems, includingorganizations andclassification schemes. The broad category of branching hierarchies can be further subdivided based on the degree.
Aflat hierarchy (also known for companies asflat organization) is a branching hierarchy in which the maximum degree approaches infinity, i.e., that has a wide span.[3] Most often, systems intuitively regarded as hierarchical have at most a moderate span. Therefore, a flat hierarchy is often not viewed as a hierarchy at all. For example,diamonds andgraphite are flat hierarchies of numerouscarbon atoms that can be further decomposed into subatomic particles.
Anoverlapping hierarchy is a branching hierarchy in which at least one object has two parent objects.[2] For example, agraduate student can have twoco-supervisors to whom the student reports directly and equally, and who have the same level of authority within theuniversity hierarchy (i.e., they have the sameposition ortenure status).
Possibly the first use of the English wordhierarchy cited by theOxford English Dictionary was in 1881, when it was used in reference to the three orders of three angels as depicted byPseudo-Dionysius the Areopagite (5th–6th centuries). Pseudo-Dionysius used the relatedGreek word (ἱεραρχία,hierarchia) both in reference to thecelestial hierarchy and theecclesiastical hierarchy.[4] The Greek termhierarchia means 'rule of a high priest',[5] fromhierarches (ἱεράρχης, 'president of sacred rites, high-priest')[6] and that fromhiereus (ἱερεύς, 'priest')[7] andarche (ἀρχή, 'first place or power, rule').[8] Dionysius is credited with first use of it as an abstract noun.
Since hierarchical churches, such as theRoman Catholic (seeCatholic Church hierarchy) andEastern Orthodox churches, had tables of organization that were "hierarchical" in the modern sense of the word (traditionally withGod as the pinnacle or head of the hierarchy), the term came to refer to similar organizational methods insecular settings.
Maslow's hierarchy of human needs. This is an example of a hierarchy visualized with a triangle diagram. The hierarchical aspect represented here is that needs at lower levels of the pyramid are considered more basic and must be fulfilled before higher ones are met.
A hierarchy is typically depicted as apyramid, where the height of a level represents that level's status and width of a level represents the quantity of items at that level relative to the whole.[9] For example, the fewDirectors of a company could be at theapex, and thebase could be thousands of people who have no subordinates.
These pyramids are oftendiagrammed with atriangle diagram which serves to emphasize the size differences between the levels (but not all triangle/pyramid diagrams are hierarchical; for example, the 1992USDA food guide pyramid). An example of a triangle diagram appears to the right.
More recently, as computers have allowed the storage and navigation of ever larger data sets, various methods have been developed to represent hierarchies in a manner that makes more efficient use of the available space on a computer's screen. Examples includefractal maps,TreeMaps andRadial Trees.
In the design field, mainly graphic design, successful layouts and formatting of the content on documents are heavily dependent on the rules ofvisual hierarchy. Visual hierarchy is also important for proper organization of files on computers.
An example of visually representing hierarchy is through nested clusters. Nested clusters represent hierarchical relationships using layers of information. The child element is within the parent element, such as in aVenn diagram. This structure is most effective in representing simple hierarchical relationships. For example, when directing someone to open a file on a computer desktop, one may first direct them towards the main folder, then the subfolders within the main folder. They will keep opening files within the folders until the designated file is located.
For more complicated hierarchies, the stair structure represents hierarchical relationships through the use of visual stacking. Visually imagine the top of a downward staircase beginning at the left and descending on the right. Child elements are towards the bottom of the stairs and parent elements are at the top. This structure represents hierarchical relationships through the use of visual stacking.
In plain English, a hierarchy can be thought of as aset in which:[2]
No element is superior to itself, and
One element, the (apex orhierarch), is superior to all of the other elements in the set.
The first requirement is also interpreted to mean that a hierarchy can have nocircular relationships; the association between two objects is alwaystransitive.The second requirement asserts that a hierarchy must have a leader orroot that is common to all of the objects.
Mathematically, in its most general form, a hierarchy is apartially ordered set orposet.[10] Thesystem in this case is the entire poset, which is constituted of elements. Within this system, each element shares a particular unambiguous property. Objects with the same property value are grouped together, and each of those resultinglevels is referred to as aclass.
"Hierarchy" is particularly used to refer to a poset in which the classes are organized in terms of increasing complexity. Operations such as addition, subtraction, multiplication and division are often performed in a certain sequence or order. Usually, addition and subtraction are performed after multiplication and division has already been applied to a problem. The use of parentheses is also a representation of hierarchy, for they show which operation is to be done prior to the following ones. For example:(2 + 5) × (7 - 4).In this problem, typically one would multiply 5 by 7 first, based on the rules of mathematical hierarchy. But when the parentheses are placed, one will know to do the operations within the parentheses first before continuing on with the problem. These rules are largely dominant in algebraic problems, ones that include several steps to solve. The use of hierarchy in mathematics is beneficial to quickly and efficiently solve a problem without having to go through the process of slowly dissecting the problem. Most of these rules are now known as the proper way into solving certain equations.
Matryoshka dolls, also known asnesting dolls orRussian dolls. Each doll is encompassed inside another until the smallest one is reached. This is the concept ofnesting. When the concept is applied tosets, the resulting ordering is anested hierarchy.
A nested hierarchy orinclusion hierarchy is a hierarchical ordering ofnested sets.[11] The concept of nesting is exemplified in Russianmatryoshka 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:
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 canonly be a quadrilateral; it can never be atriangle,hexagon, etc.
Nested hierarchies are the organizational schemes behindtaxonomies and systematic classifications. For example, using the originalLinnaean taxonomy (the version he laid out in the 10th edition ofSystema Naturae), a human can be formulated as:[12]
Taxonomies may change frequently (as seen inbiological 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 ofself-similarity andrecursion. Recursion itself is included as a subset of hierarchical programming, and recursive thinking can be synonymous with a form of hierarchical thinking and logic.[13]
A diagram illustrating a containment hierarchy. The set of all squares is completely contained in the larger set of quadrilaterals, and so on.
A containment hierarchy is a direct extrapolation of thenested 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:
The notation meansx is a subset ofy but is not equal to y.
Two types of containment hierarchies are thesubsumptive containment hierarchy and thecompositional containment hierarchy. A subsumptive hierarchy "subsumes" its children, and a compositional hierarchy is "composed" of its children. A hierarchy can also be both subsumptiveand compositional[example needed].[14]
Asubsumptive containment hierarchy is a classification of object classes from the general to the specific. Other names for this type of hierarchy are "taxonomic hierarchy" and "IS-A hierarchy".[10][15][16] 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 a member of the levelMammalia "is a" member of the levelAnimalia; 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".[16] For example, with the Linnaean hierarchy outlined above, an entity name likeAnimalia is a way to group all the species that fit theconceptualization of an animal.
Acompositional containment hierarchy is an ordering of the parts that make up a system—the system is "composed" of these parts.[17] 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 thehierarchy of life. Every person can be reduced toorgan systems, which are composed oforgans, which are composed oftissues, which are composed ofcells, which are composed ofmolecules, which are composed ofatoms. In fact, the last two levels apply to allmatter, at least at themacroscopic scale. Moreover, each of these levels inherit all the properties of theirchildren.
In this particular example, there are alsoemergent properties—functions that are not seen at the lower level (e.g.,cognition is not a property ofneurons but is of thebrain)—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. Theselevel hierarchies are characterized by bi-directionalcausation.[11]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 entityx into a higher-level entity can have onx's properties and interactions. Furthermore, the entities found at each level areautonomous.
While the above examples are often[quantify] 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, mostpostal-code systems are hierarchical. Using theCanadian postal code system as an example, the top level's binding concept, the"postal district", 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 anoverlapping 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 formatA0A 0A0 (the second and third letter positions allow 20 objects each). Mostlibrary classification systems are also hierarchical. TheDewey Decimal System is infinitely hierarchical because there is no finite bound on the number of digits can be used after the decimal point.[24]
In areverse hierarchy, the conceptualpyramid of authority is turned upside-down, so that the apex is at the bottom and the base is at the top. This mode represents the idea that members of the higher rankings are responsible for the members of the lower rankings.
Empirically, when we observe in nature a large proportion of the (complex) biological systems, they exhibit hierarchic structure.[26] On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity.[27]System hierarchies analysis performed in the 1950s,[28][29] laid the empirical foundations for afield that would become, from the 1980s,hierarchical ecology.[30][31][32][33][34]
CGI andcomputer-animationprograms mostly use hierarchies for models. On a3D model of ahuman for example, thechest is aparent of the upper left arm, which is a parent of the lower left arm, which is a parent of thehand. This pattern is used inmodeling andanimation for almost everything built as a 3Ddigital model.
In this system, the three (or four withAlgonquian languages) persons occur in a hierarchy ofsalience. To distinguish which is subject and which object,inverse markers are used if the object outranks the subject.
On the other hand, languages include a variety of phenomena that are not hierarchical. For example, the relationship between a pronoun and a prior noun-phrase to which it refers commonly crosses grammatical boundaries in non-hierarchical ways.
The structure of a musical composition is often understood hierarchically (for example byHeinrich Schenker (1768–1835, seeSchenkerian analysis), and in the (1985)Generative Theory of Tonal Music, by composerFred Lerdahl and linguist RayJackendoff). The sum of all notes in a piece is understood to be an all-inclusive surface, which can be reduced to successively more sparse and more fundamental types of motion. The levels of structure that operate in Schenker's theory are the foreground, which is seen in all the details of the musical score; the middle ground, which is roughly a summary of an essential contrapuntal progression and voice-leading; and the background orUrsatz, which is one of only a few basic "long-range counterpoint" structures that are shared in the gamut of tonal music literature.
Thepitches andform oftonal music are organized hierarchically, all pitches deriving their importance from their relationship to atonic key, and secondary themes in otherkeys are brought back to the tonic in a recapitulation of the primary theme.
In the work of diverse theorists such asWilliam James (1842 to 1910),Michel Foucault (1926 to 1984) andHayden White (1928 to 2018), important critiques of hierarchicalepistemology are advanced. James famously asserts in his workRadical Empiricism that clear distinctions of type and category are a constant but unwritten goal of scientific reasoning, so that when they are discovered, success is declared. But if aspects of the world are organized differently, involving inherent and intractable ambiguities, then scientific questions are often considered unresolved.
Feminists,Marxists,anarchists,communists,critical theorists and others, all of whom have multiple interpretations, criticize the hierarchies commonly found within human society, especially in social relationships. Hierarchies are present in all parts of society: in businesses, schools, families, etc. These relationships are often viewed as necessary. Entities that stand in hierarchical arrangements are animals, humans, plants, etc.
Ethics, behavioral psychology, philosophies of identity
Career-oriented purposes can be diagrammed using a hierarchy describing how less important actions support a larger goal.
Inethics, variousvirtues are enumerated and sometimes organized hierarchically according to certain brands ofvirtue theory.
In some of these random examples, there is an asymmetry of 'compositional' significance between levels of structure, so that small parts of the whole hierarchical array depend, for their meaning, on their membership in larger parts. There is a hierarchy of activities in human life: productive activity serves or is guided by the moral life; the moral life is guided by practical reason; practical reason (used in moral and political life) serves contemplative reason (whereby we contemplate God). Practical reason sets aside time and resources for contemplative reason.
^abcdefDawkins, Richard (1976). Bateson, Paul Patrick Gordon; Hinde, Robert A. (eds.).Hierarchical organization: a candidate principle for ethology. Growing points in ethology: based on a conference sponsored by St. John's College and King's College, Cambridge. Cambridge, England: Cambridge University Press. pp. 7–54.ISBN0-521-29086-4.
^Douglas Lemke (2002).Regions of War and Peace. Cambridge: University of Cambridge. p. 49.
^abLehmann, Fritz (1996). Eklund, Peter G.; Ellis, Gerard; Mann, Graham (eds.).Big Posets of Participatings and Thematic Roles. Conceptual structures: knowledge representation as interlingua—4th International Conference on Conceptual Structures, ICCS '96, Sydney, Australia, August 19–22, 1996—proceedings. Lecture Notes in Artificial Intelligence 115. Germany: Springer. pp. 50–74.ISBN3-540-61534-2.
^abLane, David (2006). "Hierarchy, Complexity, Society". In Pumain, Denise (ed.).Hierarchy in Natural and Social Sciences. New York, New York:Springer-Verlag. pp. 81–120.ISBN978-1-4020-4126-6.
^Corballis, Michael (2011).The Recursive Mind. Princeton University Press.ISBN978-0691145471.
^Kopisch, Manfred; Günther, Andreas (1992). "Configuration of a passenger aircraft cabin based on conceptual hierarchy, constraints and flexible control". In Belli, Fevzi (ed.).Industrial and Engineering Applications of Artificial Intelligence and Expert Systems.Industrial and engineering applications of artificial intelligence and expert systems: 5th international conference, IEA/AIE-92, Paderborn, Germany, June 9–12, 1992 : proceedings. Lecture Notes in Computer Science Series. Vol. 602.Springer. pp. 424–427.doi:10.1007/BFb0024994.ISBN3-540-55601-X.ISSN0302-9743.
^Parsons, David (2002).Object Oriented Programming in C++. Cengage Learning. pp. 110–185.ISBN0-8264-5428-3.
^Kulish, V. V. (2002).Hierarchical Methods: Hierarchy and hierarchical asymptotic methods in electrodynamics. Vol. 1.Springer. pp. xvii–xx,49–71.ISBN1-4020-0757-4.
^Kulish, Viktor Vasylevych (11 April 2006) [2002].Hierarchical Methods: Hierarchy and Hierarchical Asymptotic Methods in Electrodynamics. Volume 123 of Fundamental Theories of Physics. Vol. 1. New York: Springer Science & Business Media (published 2006). p. xviii.ISBN9780306480614. Retrieved26 November 2021.At the same time, it is a trivial phenomenon of everyday life, too. Indeed, one can be convinced that there is a hierarchy in everyday life everywhere. We can affirm once more that a person lives in a completely hierarchical world.
^Tipton, Frank (January 2003). "Politics, Governance, and Diplomacy between Neoclassicism and Modernism".A History of Modern Germany Since 1815. London: A&C Black (published 2003). p. 239.ISBN9780826449108. Retrieved26 November 2021.Kühne believes the Prussian franchise lasted for over sixty years because it accurately reflected the social realities of the relatively closed, hierarchical, and conformist communities of rural Prussia.
^Compare:Lee, Stephen J. (12 February 2016) [1987]. "Dictatorship in Germany".European Dictatorships 1918-1945 (4, revised ed.). London: Routledge (published 2016). p. 217.ISBN9781317294221. Retrieved26 November 2021.[...] Nazi administration is now regarded as an overlapping and confusing 'polycracy'. The basis of the criticism is that the multi-layering of agencies and the lack of proper delineation between them generated conflict and hampered efficiency. There was, for example, extensive rivalry between the Four-Year plan Office, the Ministry of Economics, the War ministry and the Plenipotentiary for Economics.
Wicken, J. S.; Ulanowicz, R. E. (1988), "On quantifying hierarchical connections in ecology",Journal of Social and Biological Systems,11 (3):369–377,doi:10.1016/0140-1750(88)90066-8
Galindo, Cipriano; Fernández-Madrigal, Juan-Antonio (2007). Kacprzyk, Janusz (ed.).Multiple Abstraction Hierarchies for Mobile Robot Operation in Large Environments. Studies in Computational Intelligence. Berlin: Springer Berlin Heidelberg.ISBN978-3-540-72688-3.
