Hierarchy: Structure, Terminology, and Applications across various fields

A hierarchy organizes objects as “above,” “below,” or “at the same level” in various disciplines. Hierarchical structures are fundamental in fields like organizational theory, computer science, and biology.

By Brad Nakase, Attorney

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Table of Contents

1. Introduction

2. Nomenclature

3. Etymology

4. Putting hierarchies into perspective

5. Visual hierarchy

6. Informal representation

7. Mathematical representation

8. Subcategories

9. Applications and Contexts

10. Criticism

Introduction

A grouping of objects (names, objects, categories, values, categories) that are portrayed as being “at the identical level”, “below”, or “above” other objects is called a hierarchy (Greek: ἱεραρχία, from hierarkhes, hierarkhia, ‘control of the superior priest’, ‘president of holy rituals’). Numerous disciplines, including architecture, philosophy, mathematics, design, computer science, systems theory, organizational theory, social sciences (particularly political science), and systematic biology consider hierarchy to be fundamental.

Entities can be connected across a hierarchy in both vertical and diagonal directions, indirectly or directly. As far as hierarchical relationships are concerned, the sole direct relationships in a hierarchy are with one’s immediate boss or any of the subordinates, albeit a structure that is primarily hierarchical may also have other hierarchies.

Using several links pointing in a single direction, hierarchical linkages can stretch “vertically” upward or downward. If a portion of the hierarchy is not connected vertically, it can still be connected “horizontally” by going upwards in the hierarchy to locate a shared indirect or direct superior and afterward back down. This can be compared to two colleagues who share the same comparative level of power but answer to different superiors. There are organizational structures that are complementary to and different from the hierarchy. Among these is heterarchy.

Nomenclature

There is a specific terminology for hierarchies. Diagramming a hierarchy (see below) makes these terms the easiest to comprehend.

The following words are frequently used in relation to hierarchies in an institutional context:

* Object: A single entity such as an individual, division, idea, component of a setup, or a participant in a group

* System: The full set of items (like an administration) that are organized hierarchically

* Dimension: An additional term for “system” coming from analytical processing on the internet (cubes, for example)

* Member: A (dimension, class system, or taxonomy) that contains an (object or element) at any (rank or level)

* Terminologies related to positioning

Rank: Relative worth, intricacy, authority, level, power, and relevance of the object among other factors
Tier or Level: A collection of items that are equal in importance or rank
Ordering: The way the (levels or ranks) are organized
Hierarchy: A specific group of people arranged into (levels or ranks). It is possible to create multiple hierarchies for each (classification system or dimension taxonomy), flattening the structure by omitting certain sections of the dimensions.

* Terminologies Regarding Placement

Hierarch: At the highest level, a realm that is made up of a solitary orphan (member or object) at the top of the hierarchy. The foundation of an inverse tree
Member: A node or member related to both superior and inferior members at any tier of the hierarchy within a dimension
Orphan: The member lacking a parent who is in any tier of the dimension, is frequently the tip of an unconnected branch. By developing a connection (interaction) with the parent at the next superior level, orphans may be transplanted to the hierarchy.
Leaf: Any member of the dimension at any level without any hierarchy-based subordinates
Neighbor: A participant next to another participant at the same (rank or level). Constantly a peer/colleague.
Superior: A greater rank or something that is rated higher (an ancestor or a parent)
Subordinate: A lesser rank or an item that is rated lower (a descendant or a kid)
Collection: every item at a single level (e.g., peers)
Peer: A similar-ranking object, hence at the identical level
Interaction: An object’s relationship to its immediate superior or inferior (an inferior/superior pair)

When an object is exactly one level higher or lower than another, there is an immediate interaction (for example, when two objects are separated by a line on the tree).

Distance: The least no. of connections—that is, one fewer than the total number of items that must be “crossed” in order to form a path—between two things.
Span: The qualitative explanation of a level’s width in a graphic, such as how many subordinates the object possesses

* Terminologies Related to Nature

Attribute: Genetic trait shared by (members and subordinates) at a level (for example, hair color)
Attribute-value: Particular worth of an inherited trait (Auburn, for example)

The general nomenclature used differs in the context of a mathematics setting (graph theory).

Although most hierarchies have more specialized terminology related to their topic, they all operate on the same principle. In data structures, for instance, objects are referred to as nodes, subordinates as children, and superiors as parents. In the workplace, a peer represents a coworker, whereas a boss/superior is generally a manager or supervisor.

The extent of branching

The total number of immediate children or subordinates an object possesses is referred to as its branching degree. In a graph with direction, this number is equal to the no. of additional vertices that the node in question has linked to it by outgoing arcs. The highest degree or the maximum degree found within the framework in its entirety, can be used to classify hierarchies. Two broad categories result from this kind of categorization: linear and branched.

One degree is the maximum in the linear hierarchy. Put otherwise, every object (except the bottom and top objects) may be seen as a lineup, with exactly one immediate superior and one immediate subordinate for every object. Each hierarchy contains this feature with regard to levels, although often every level may include an endless number of items. This refers only to the objects instead of the levels.

A minimal degree of two or greater is achieved when any number of items in the branching hierarchy have a degree of two or more. Many people naturally associate the term “hierarchy” with a hierarchical structure that branches out. There are several types of systems that have branching hierarchies, such as corporations and categorization schemes. The degree allows for more division within the general grouping of a branching hierarchy.

The flat hierarchy, sometimes referred to as a flat organization in the context of businesses, is a branch hierarchy where the highest degree is close to infinity, meaning it spans a large area. Most typically, systems that are intuitively thought to be hierarchical only have a limited span. As a result, the structure of the hierarchy is sometimes seen as lacking any hierarchy whatsoever. Graphite and Diamond, for instance, are like flat hierarchies composed of many carbon atoms which are further subdivided into subatomic components.

When an object has 2 parent objects, it is said to have an overlapping hierarchy, which corresponds to a branch hierarchy. A graduate pupil may, for instance, have 2 co-supervisors who receive direct and equal reports from the student and who possess the same degree of authority in the university administration (they hold the same level or tenure standing).

Etymology

The Oxford Dictionary of English lists 1881 as the possible first instance of the word hierarchy, which was then applied to Pseudo-Dionysius the Areopagite’s depiction of the 3 levels of angels (fifth and sixth centuries). Pseudo-Dionysius made mention of the two separate ecclesiastical and celestial hierarchies using the associated Greek term (ἱεραρχία, hierarchia). The word hierarchia comes from the Greek hierarches (ἱεράρχης, “president of the holy rites, great-priest”) which derives from hiereus (ἱερεύς, “priest”) and arche (ἀρχή, “the first position or power, authority”). It signifies the “rule of an eminent priest.” It is said that Dionysius used it as a conceptual word for the first time.

As a result of the “hierarchical” tables of the organization that Eastern Orthodox and Roman Catholic churches (refer to Catholic Church order) had—customarily with God at the top of the peak of the hierarchy—the term “hierarchical” eventually began to be used for comparable organizational techniques in non-religious contexts.

Putting hierarchies into perspective

The common visual representation of a hierarchy is a pyramid, in which the width of each level indicates the number of things at the given level in relation to the total and the elevation of the level indicates the rank of that particular level. For instance, multitudes of people without direct subordinates could make up the backbone of a corporation, while a handful of Directors might sit at the top.

Although not all pyramid/triangle diagrams are hierarchical—take the 1992 United States Department of Agriculture food advisory pyramid, for instance—such pyramids are sometimes diagrammed with triangles to highlight the size disparities between the tiers. On the right is an illustration of the triangle-shaped diagram.

An additional popular way to depict a hierarchical plan is with a tree layout. This method is frequently used to portray phylogenetic trees, organizational charts, and sports playoff brackets.

Since computers have made it possible to store and navigate ever-larger sets of information, a number of techniques are being developed to portray hierarchy in a way that maximizes the amount of information on the screen of a computer. Treemaps, radial trees, and fractal maps are a few examples.

Visual hierarchy

Effective layout and content formatting in documents are highly reliant on the principles of visual hierarchy in the creative industry, particularly in graphic design. The correct arrangement of documents on desktops also requires a visual hierarchy.

Nested clusters are one way to graphically depict hierarchy. Layers of data are used by nested clusters to depict hierarchical relationships. The Venn diagram illustrates an example of a child element contained within a parent element. Basic hierarchical relationships are best represented by this structure. To access something on a desktop computer, for instance, one might point the user to the primary folder first, followed by the subfolders inside the primary folder. Before they find the intended file, they will continue to open files inside the folders.

Visual stacking is used by the stair design to depict hierarchical links in increasingly complex hierarchies. Imagine in your mind’s eye the highest point of the staircase that descends to the right and starts from the left. On the stairwell, parent components sit at the highest point and child components remain at the bottom. By using visual stacking, this structure illustrates hierarchical linkages.

Informal representation

To put it simply, a hierarchy is a collection of objects where:

1) There is no element better than itself, and

2) There is only one element in the collection that is greater than the others, the (hierarch or apex).

It is also understood that this initial condition states that there cannot be any circular links within a hierarchy; there can only ever be a transitive relationship between two things. According to the 2nd criterion, every object in the hierarchy must share a leader/root.

Mathematical representation

In mathematical terms, a partially arranged set, called poset, is referred to as a hierarchy. In this instance, the complete poset, made up of elements, is the system. Each component in the system possesses a specific, clear property. A class is any level that results from grouping entities with an identical attribute value together.

“Hierarchy” is specifically used to describe a poset where classes are arranged according to increasing levels of complexity. A specific order or sequence is frequently followed when doing operations like addition, subtraction, division, and multiplication. Multiplication & division are typically utilized to solve a problem before addition & subtraction are carried out. Using brackets to indicate which action should be completed before the next illustrates hierarchy as well. Consider this: (2 + 7) × (5 – 4). In general, one will multiply 7 by 5 in this situation first according to the mathematical hierarchy rules. However, after the brackets are inserted, one will understand how to finish the procedures inside them before moving forward to the next step of the problem.

The majority of algebraic problems, or those requiring several stages to solve, follow these guidelines. Mathematics uses hierarchy to solve problems fast and effectively, saving time by avoiding the need to carefully analyze each step of the equation. These days, the majority of these guidelines are recognized as the correct methods for resolving specific mathematical problems.

Subcategories

Nested hierarchy

Hierarchically arranged nested sets are called nested hierarchies or inclusion hierarchies. Matryoshka dolls (Russian) are typical examples of nesting. From the inner doll to the outer doll, every doll is encircled by another. The outside doll contains every inner doll; each of the inside dolls that are left is contained by the following outer doll, & so on. In a generalized nested hierarchy, several objects can exist within levels, however each object can only have a single parent at a time. This is in contrast to matryoshkas, which show a nested hierarchy having only 1 item per level, or one of every kind of doll. The next example provides a mathematical formulation and demonstration of the overall idea:

There are always other terms for squares, such as quadrilaterals, polygons, or shapes. It is hierarchical in this sense. Still, think about the collection of polygons with this arrangement. A square is not a triangle, hexagon, or any other shape; it is always going to be a quadrilateral.

The organizational structures underlying taxonomies & systematic categorization are called nested hierarchies. For instance, a human could be defined as follows utilizing the initial Linnaean taxonomy, which he established in the Systema Naturae (tenth edition):

The idea of nested hierarchy remains constant, even though taxonomies may undergo periodic changes, as shown by biological taxonomy.

Nested hierarchies, such as Russian dolls, also serve to show the characteristics of recursion and self-similarity in numerous programming taxonomies & syntax models (and also in mathematical fractals). Hierarchical programming includes recursion as one subset, & recursive thinking is comparable to a type of hierarchical logic and thought.

Containment hierarchy

A straightforward extension of the idea of a nested hierarchy is the containment hierarchy. Though each set has to be “strict”—none of the pair of sets may be the same—all sorted sets are nonetheless nested. It is possible to alter the previous shapes illustration to show this:

Programming that is object-oriented uses class inheritance as a general illustration of the containment hierarchy.

Subsumptive and compositional containment are the two different forms of containment hierarchies. Just as compositional hierarchies are “composed” of their children, subsumptive hierarchies “subsume” their offspring. It’s also possible for a hierarchy to be compositional and subsumptive. Additionally, a hierarchy could be compositional as well as subsumptive.

Subsumptive containment hierarchies

Object classes are arranged from the broad to the particular in subsumptive containment hierarchies. This kind of hierarchy is also known as an “IS-A hierarchy” and a “taxonomic hierarchy.” A lower-rank item “is a” component of the greater category, as the final phrase explains the link between every level. The above-described taxonomy structure is a hierarchy of subsumptive containment. Relying on the Linnaean taxonomy example, we can see that something belonging to the order Mammalia “is a” part of the class Animalia; that is, a human being “is a” primate, a primate “is a” mammal, & furthermore. A system of “concepts” is another abstract definition of the subsumptive hierarchy. Using the above-described Linnaean hierarchy, for instance, a being name such as Animalia can be used to group every species that corresponds to the concept of animals.

Compositional containment hierarchies

Systems are “composed” of their constituent pieces, which are arranged in an order known as compositional containment hierarchies. This is a general breakdown of most engineering structures, artificial or natural.

The Compositional hierarchy of life is what each and every individual experiences on a daily basis. There is only one kind of human, and that is an organ system, which is made up of organs, tissues, cells, molecules, and atoms. The final two stages, at least on a macroscopic level, really pertain to all the matter. Furthermore, every one of these tiers inherited every one of their offspring’s attributes.

Along with scalar qualities (molecules greater than atoms & cells larger than molecules, etc.), emergent characteristics in this example refer to functions not observed at lower levels (for example, cognition is a trait of a brain, not of neurons). While both of the ideas are frequently seen in a compositional hierarchy, they aren’t necessarily general properties. Bi-directional causality is the defining feature of such level hierarchies. Higher-level entities can have properties caused by lesser-level entities through an interaction process known as upward causation. Parents are made up, at least in part, of their offspring.

The impact that an entity’s interactions and properties may have when it is included in an entity at a higher position is known as “downward causation.” In addition, every level has autonomous entities.

Applications and Contexts

According to Kulish (2002), practically all human-implemented organizational systems are hierarchical. The concepts of “nation” & “government” have common definitions that imply all nations have governments and all governments are hierarchical. The social division of societies is a concept that sociologists can use to study socioeconomic systems. All organized categorization schemes, or taxonomies, follow the hierarchy. Regardless of internal governance arrangements, the majority of organized religions function as hierarchies under priesthoods and deities. The ecclesiastical hierarchy in many Christian faiths is autocephalous.

When it comes to ancestry (as seen in the family tree), inheritance (heirship and succession), and cousinship (first cousin after removed, 2nd cousin, etc.), families are often thought of as structures that are hierarchical. In accordance with Maslow’s pyramid describing human needs, all the requirements of a balanced existence and living can be arranged. A hierarchical structure is common in learning steps. In order to comprehend differential equations, you must first study calculus, and in order to master calculus, you first need to understand elementary algebra, & so forth. As many systems, including biomass pyramids, the structure of life, & Linnaean taxonomy, strive to describe, nature provides hierarchical patterns.

Although the examples above are classic and frequently show a clear hierarchical form, hierarchies are present in numerous environments where this separation/branching pattern is not readily apparent. For instance, hierarchical postal code structures are the norm. The top-level binding notion, the “postal region,” is made up of eighteen items, or letters, taking the postal code structure of Canada as a case. The “zone” represents the subsequent level below, wherein the elements are the numbers 0 through 9. Given that each one of these ten items has eighteen parents, this is an illustration of the overlapping hierarchy.

The hierarchy descends further to provide 7,200,000 distinct codes in the form of A0A 0A0 (twenty objects per letter for the 2nd and 3rd positions). Another common feature of library classification schemes is hierarchy. Since the no. of digits that can be utilized beyond the point of the decimal is limitless, the system of the Dewey Decimal is indefinitely hierarchical.

Organizations

A hierarchy of dominance can be used to structure an organization. A single individual or group holds the greatest authority or power in an institutional hierarchy, with each level below it denoting a smaller degree of authority. This is the structure of most organizations, including corporations, military forces, militias, governments, and structured religions. An organizational diagram can show the individuals or units within a company in a hierarchical format.

The metaphorical pyramid of power is inverted in a reversed hierarchy, with the base at the very top and the peak at the lowest. The notion that individuals in higher positions bear some responsibility for those in lower rankings is embodied by this style.

Biology

Empirically, we may observe that a significant fraction of (intricate) ecosystems in nature have hierarchical structures. Theoretically, one may anticipate that complex systems would be hierarchical in a universe where complexity needed to arise from simplicity. The empirical groundwork for the field which eventually evolved into hierarchical ecology in the 1980s was built by system hierarchies research carried out during the 1950s.

Thermodynamics provides a summary of the theoretical underpinnings. In the broadest sense, biological processes that are structured as physical phenomena are thermodynamic systems that are open with self-organizing behavior, along with the subset/set linkages between dissipative systems that can be described in a hierarchical fashion.

Other hierarchical biological representations include taxonomic hierarchies, which include phylogenetic trees and the Linnean classification scheme, and ecological pyramids, which show trophic levels or energy flow in ecosystems. Phylogenetic trees also reflect inferred trends in evolutionary relationships among extinct and living species.

Imaging using computers

Model hierarchies are commonly used in computer animation and CGI applications. An illustration of this would be a 3D human model, the chest is the parent of the left arm (upper), which is the parent of the left arm (lower), which is the parent of the hand. The majority of objects created as 3D digital models employ this pattern in modeling and animation.

Linguistics

There is a hierarchy in a lot of grammatical theories, like phrase structure.

Mapudungun and Cree are examples of direct-inverse linguistics that use a hierarchy of people rather than distinct object and subject markers to identify objects and subjects on verbs.

The three people in this system—or four if Algonquian languages are used—occur in the hierarchy of importance. Inverse markers have been used when an object surpasses the subject’s position in order to identify what is the object versus what is the subject.

Conversely, a range of non-hierarchical phenomena are included in languages. For instance, a pronoun’s relationship to the preceding noun phrase it refers to frequently transcends grammatical bounds free of hierarchy manner.

Music

Fred Lerdahl (Composer) and Ray Jackendoff’s (Linguist) 1985 work Generative Theory of Tonal Music is one example of how the arrangement of a piece of music is often interpreted hierarchically. Other examples include Heinrich Schenker’s (1768–1835) analysis. It is believed that the total pitches in a composition constitute an all-encompassing surface that can be broken down into progressively more basic and sparse forms of motion.

The three levels of arrangement that function according to Schenker’s concept are the background, also known as Ursatz, which serves as a fundamental “long-range contrast” structure that is common to all harmonic music literature; the foreground, which is evident in every aspect of a musical score; and the center ground, which is essentially a synopsis of a crucial contrapuntal advancement as well as voice-leading.

Tonal music is structured using pitches arranged in a hierarchical manner. Each pitch has significance based on its connection to a harmonic key, and subordinate themes in different keys are summarized by returning to the harmonic as the main theme.

Criticism

Significant criticisms of hierarchy epistemology are made in the works of a variety of thinkers, including Michel Foucault (1926–1984), Hayden White (1928–2018), and William James (1842–19010). In Radical Empiricism, James famously states that the purpose of scientific thinking is always to discover unambiguous distinctions between types and categories; this is an unwritten objective that is celebrated upon discovery. Scientific issues, however, are frequently seen as unanswered if there are areas of the globe that are structured differently and involve inherent and persistent ambiguity.

Many ideologies, including critical theory, anarchism, communism, feminist theory, and Marxism, critique the hierarchies that are frequently present in society as a whole, particularly in interpersonal relationships. In organizations, educational institutions, households, and other societal contexts, hierarchies exist. It’s common to see these connections as essential. Animals, people, plants, and other entities are arranged hierarchically.

Identity theories, behavioral psychology, and ethics

According to different types of theories of virtue, different qualities are listed and occasionally arranged hierarchically in ethics.

Certain arbitrary instances exhibit an imbalance in “compositional” importance among tiers of the organization, indicating that the importance of minor segments of the overall hierarchical array relies on their inclusion in more significant segments. Human existence is characterized by an order of actions: productive endeavors support or are directed by moral life; moral life is governed by rational thought; rational thought (used in ethical & political activity) supports contemplative reasoning (in which we meditate God). Contemplative reason is given resources and time when it comes to practical reason.

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