Q-analysis is a mathematical framework to describe and analyze
set system
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
s, or equivalently
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es. This idea was first introduced by Ronald Atkin in the early 1970s. Atkin was a British mathematician teaching at the University of Essex. Crediting the inspiration of his idea to Clifford Dowker’s paper (Homology Groups of Relations, Annals of Mathematics, 1952), he became interested in the algebra of relations in social structures. He tried to explain his idea in both mathematical and also accessible forms to both technical and general audience. His main ideas are reflected in ''The Mathematical Structure of Human Affairs'' (1974). That book covers the key ideas in q-analysis and its application to a wide range of examples, like analyzing game of chess, urban structures, politics at university, people and complexes, works of abstract art, and to physics. He contended that q-analysis can be considered as a powerful generalized method wherever we are dealing with relationships among sets.
Description
A
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
of vertices can be represented as a polyhedron in dimensions, so that for example a triangle of three vertices can be drawn on a plane of two dimensions and is accordingly called a 2-simplex. When simplices share vertices, the intersections of their vertex sets are themselves simplices of equal or lower dimension. For example, two triangles with two vertices in common share not only the two 0-simplex vertices but the 1-simplex line between them. The triangles are said to be both 1- and 0-connected because they share 1- and 0-dimensional faces.
Q-analysis of a simplicial complex consists in stepping through all up to the dimension of the largest simplex and constructing for each a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the simplices that are -connected at each level, and in particular, determining how many
connected components are present for each .
[* Atkin, R. (1974). Mathematical Structure in Human Affairs. London, Heinemann.]
Q-analysis can thus provide a rich summary of (literally) multi-faceted relationships between entities.
Applications
* Analysis of large-scale systems structure
* Analysis of
social network
A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for an ...
*
Decision making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rati ...
See also
*
Systems theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
*
Living systems theory
Living systems are open self-organizing life forms that interact with their environment. These systems are maintained by flows of information, energy and matter.
In the last few decades, some scientists have proposed that a general living systems ...
Notes
References
* Atkin, R. (1972). From cohomology in physics to q-connectivity in social science. International Journal of Man-Machines Studies vol. 4, 139–167.
* Atkin, R. (1974). Mathematical Structure in Human Affairs. London, Heinemann.
* Atkin, R. (1976). An algebra for patterns on a complex II. International Journal of Man-Machines Studies vol. 8, 483–498.
* Atkin, R. (1977). Combinatorial Connectivities in Social Systems. Basel, Birkhäuser Verlag.
Systems theory
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