Abstract
We discuss two complexity indicators reported in the literature for partially ordered sets (posets), the first one based on linear extensions and the second one on incomparabilities. Later, we introduce a novel indicator that combines comparabilities and incomparabilities with a Shannon’s entropy approach. The possible values the novel complexity indicator can take are related to the partitions of the number of order relationships through Young diagrams. Upper and lower bounds of the novel indicator are determined and analysed to yield a normalised complexity indicator. As an example of application, the complexity is calculated for the ordering of countries based on their performance in chemical research. Finally, another complexity indicator is outlined, which is based on comparabilities, incomparabilities, and equivalences.
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- 1.
We use ≼ instead of ≺ to allow the relation between one object and itself.
- 2.
The fact of having entangled diagrams does imply that the order information of the poset is hidden. In fact, there are different approaches to extract information from posets, as discussed by Brüggemann and Patil (see Brüggemann and Patil 2011, pp. 36–38).
- 3.
Given the posets (X, ≼ ′) and (X, ≼ ′ ′), their intersection yields the poset (X,≼) where ≼ = {(x,y)|x ≼ ′ y ∧ x ≼ ′ ′ y; x, y ∈ X}.
- 4.
Here, P = (X,≼).
- 5.
Yannakakis found that for certain posets it is NP-hard to decide if their dimension is equal or lower than a particular natural number (see Yannakakis 1982).
- 6.
We take unlabelled posets since what is important in the kind of complexity we are considering is the connectivity among the objects on the poset rather than their identity.
- 7.
Note that ⌊x⌋ is the floor function that maps a real number to the largest previous following integer.
- 8.
Counting the number of unlabelled posets for a given N is a matter of current research in order theory. McKay and Brinkmann, in 2002, developed an algorithm to count this number up to N = 16 (see McKay and Brinkmann 2002). Some of the results they found are that for N = 4 , there are 16 posets; 16,999 for N = 8; 1,104,891,746 for N = 12 ; and 4,483,130,665,195,080 for N = 16.
- 9.
⌈x⌉ is the ceiling function that maps a real number to the smallest following integer.
- 10.
SCImago takes a citation window of 4 years less than the observation window. That is why even if the query was performed in 2012, the information corresponds to the period 1996–2007.
- 11.
Tenth International Workshop on Partial Order, Theory and Application, Berlin, 27–28 September 2012.
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Restrepo, G. (2014). Quantifying Complexity of Partially Ordered Sets. In: Brüggemann, R., Carlsen, L., Wittmann, J. (eds) Multi-indicator Systems and Modelling in Partial Order. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8223-9_5
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