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This Topic and my PowerPoint Show are based on my PhD dissertation: "Hierarchy Theory - Some Common Properties of Competitively-Selected Systems", System Science Department, Binghamton University, NY, 1996. If you wish to pursue further research in this area please contact me at firstname.lastname@example.org. A few copies of my dissertation are available.
The material that follows contains more detail than the PowerPoint Show.
Most complex structures are compositional or control hierarchies. An example of a compositional hierarchy is written language. A word is composed of characters. A simple sentence is composed of words. A paragraph is composed of simple sentences, and so on. An example of a control hierarchy is a management structure, where a manager controls a number of foremen or team leaders, and they, in turn, control a number of workers.
Optimal Span Hypothesis:
Optimal Span is about the same, between five and nine, for virtually all complex structures that have been competitively selected.
That includes the products of Natural Selection (Darwinian evolution) and the products of Artificial Selection (Human inventions that competed for acceptance by human society).
The hypothesis is supported by empirical data from varied domains and a derivation from Shannon’s Information Theory and Smith and Morowitz’s concept of intricacy.
What is a Hierarchy?
A more recent book, Complexity – The Emerging Science at the Edge of Order and Chaos, observes that the “hierarchical, building-block structure of things is as commonplace as air.” [ Waldrop, 1992 ]. Indeed, a bit of contemplation will reveal that nearly all complex structures are hierarchies.
There are two kinds of hierarchy. A few well-known examples will set the stage for more detailed examination of modern Hierarchy Theory:
1 -Management Structure (Control Hierarchy)
2 -Software Package (Control Hierarchy)
3 – Written Language (Containment Hierarchy)
4 – “Chinese boxes” (Containment Hierarchy)
Traversing a Hierarchy
Folding” a “String”
The above “parade” was described with the Chief Exec at the head of it, but you could just as well turn it around and have the lowest-level employees lead and the Chief Exec at the rear. When military hierarchies go to war, the lowest-level soldiers are usually at the front and the highest-level Generals well behind.
A more practical example is the text you are reading right now! It was transmitted over the Internet as a string of “bits” – “1″ and “0″ symbols. Each group of eight bits denotes a particular character. Some of the characters are the familiar numbers and upper and lower-case letters of our alphabet and others are special characters, such as the space that demarks a word (and is counted as a character that belongs to the word), punctuation characters such as a period or comma or question mark, and special control characters that denote things like new paragraph and so on.
You could say the string of 1′s and 0′s is folded every eight bits to form a Character. The string is folded again at each Space Character to form Words. Each group of Words is folded yet again at each comma or period symbol that denotes a Simple Sentence. Each group of Simple Sentences is again folded to make Paragraphs, and so on.
You could lay out a written document as a tree structure, similar to a Management hierarchy. The Characters would be at the bottom, the Words at the next level up, the Simple Sentences next, the Paragraphs next, and so on up to the whole Section, Chapter, and Book.
What is Optimal Span?
According to System Science research and Information Theory, there is a single equation that may be used to determine the most beneficial Span. Thatoptimum value maximizes the effectiveness of the resources. A Management Structure should have the Span of Control that makes the best use of the number of employees available. A Written Language Structure should have the Span of Containment that makes the best use of the number of characters (or bits in the case of the Internet) available, and so on.
The simple equation for Optimal Span derived by [ Glickstein, 1996 ] is:
So= 1 + De
(Where D is the degree of the nodes and e is the Natural Number 2.71828459)
In the examples above, where the hierarchical structure may be described as a single-dimensional folded string where each node has two closest neighbors, the degree of the nodes is, D = 2, so the equation reduces to:
So= 1 + De = 1 + 2 x 2.71828459 = 6.43659
“Take home message”: OPTIMAL SPAN, So = ~ 6.4
Also see Quantifying Brooks Mythical Man-Month (Knol) , [Glickstein, 2003 ] and [ Meijer, 2006 ] for the applicability of Optimal Span to Management Structures.
[Added 4 April 2013: The Meijer, 2006 link no longer works. His .pdf document is available at http://repository.tudelft.nl/assets/uuid:843020de-2248-468a-bf19-15b4447b5bce/dep_meijer_20061114.pdf ]
Examples of Competitively-Selected Optimal Span
Management Span of ControlManagement experts have long recommended that Management Span of Control be in the range of five or six for employees whose work requires considerable interaction. Depending upon the level of interaction, experts recommend up to nine employees per department.This recommendation comes from experience with organizations with different Spans of Control. The most successful tend to have Spans in the recommended range, five to nine,an example of competitive-selection.
When the lowest level consists of service-type employees, whose interaction with each other is less complex, there may be a dozen or two or more in a department, but there will usually be one or more foremen or team leaders to reduce the effective Management Span of Control to the range five to nine.Corporate hierarchies usually haveabout the same range of first-level departments reporting to the next level up and so on.
Say you had a budget for 49 employees and had to organize them to make most effective use of your human resources. Which of the following seems most reasonable?
(A) you have ONE manager and 48 workers, which is a BROAD hierarchy. Management experts would say a Management Span of Control of 48 is way too much for anyone to handle!
(B) you have a third-level chief executive, three executive-level managers, each with three department managers, totaling THIRTEEN managers in a three-level management hierarchy and only 36 workers, which is a TALL hierarchy with an average Management Span of Control of only 3.3. Management experts would say this is way too inefficient with too many managers!
(C) you have a second-level manager and six department managers, totaling SEVEN managers and 42 workers in a MODERATE hierarchy with an average Management Span of Control of about 6.5. Management experts would say this is about right for most organizations where the workers have to interact with each other. Optimal Span theory supports this common-sense belief!
Human Span of Absolute JudgementEvolution and Natural Selection have produced the human brain and nervous system and our senses of vision, hearing, and taste. It turns out that these senses are generally limited to five to nine gradations that can be reliably distinguished. It is also the case that we can remember about five to nine chunks of information at any one time. This is another example of competitive-selection, where, over the eons of evolutionary development, biological organisms competed and those that best fit the environment were selected to survive and reproduce.
George A Miller wrote a classic paper titled The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information [ Miller, 1956 ]. He showed that human senses of sight, hearing, and taste were generally limited to five to nine gradations that could be reliably distinguished. Miller’s paper begins as follows:
"My problem is that I have been persecuted by an integer [7 +/- 2]. For seven years this number has followed me around, has intruded in my most private data, and has assaulted me from the pages of our most public journals. This number assumes a variety of disguises, being sometimes a little larger and sometimes a little smaller than usual, but never changing so much as to be unrecognizable. The persistence with which this number plagues me is far more than a random accident. There is, to quote a famous senator, a design behind it, some pattern governing its appearances. Either there really is something unusual about the number or else I am suffering from delusions of persecution.Miller’s paper is well worth reading and is available on the Internet at this link [Miller, 1956]"
Glickstein’s Theory of Optimal Span
Hierarchy and Complexity
Weak Statement of Hypothesis
Strong Statement of Hypothesis