Abstract
In this paper, we address the problem of measuring structural dissimilarity between two partial orders with n elements. We propose a structural dissimilarity measure, based on the distance between isomorphism classes of partial orders, and propose an interpretation in terms of graph theory. We give examples of structural dissimilarity computations, using a simulated annealing algorithm for numerical optimization.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For sake of simplicity, in the following, elements of X partially ordered by ≤ will be referred directly as elements of P.
- 2.
As stated above, we say that p and q are comparable if either p ≤ q or q ≤ p, thus P ⊆ Comp(P).
- 3.
The sequence is to be read from right to left.
- 4.
The dots “ ⋅ ⋅” stand for an unspecified pair of elements of the poset.
- 5.
For example, suppose P is a poset and suppose a, b, and c constitute an antichain in P. Then, A a b A b c P is not a poset, unless also A a c is applied to P.
- 6.
For the problem of determining the number of posets with n elements, see Schröder [2002].
- 7.
In the following, we write Π n to mean the partially ordered set \( ({\Pi }_{n},{\le }_{{\Pi }_{n}})\).
- 8.
The notion of height is well defined in Π n , since the Dedekind chain condition holds in meet semilattices (see Davey and Priestley 2002).
- 9.
In fact, one has to delete Δ comparabilities and add the corresponding reversed Δ comparabilities.
- 10.
In this example, the compared posets are defined over different label sets, so we can only focus on the distance between equivalence classes.
References
Annoni P, Brüggemann R, Saltelli A (2011) Partial Order investigation of multiple indicator systems using variance - based sensitivity analysis. Environ Model Softw 26:950–958
Axenovich M, Kézdy A, Martin R (2008) On the editing distance of graphs. J Graph Theory 58(2):123–138
Brualdi RA, Junga HC, Trotter WT Jr (1994) On the posets of all posets on n elements. Discrete Appl Math 50(2):111–123
Brüggemann R, Bartel HG (1999) A theoretical concept to rank environmentally significant chemicals. J Chem Inf Comput Sci 39:211–217
Brüggemann R, Patil GP (2011) Ranking and prioritization for multi-indicator systems - introduction to partial order applications. Springer, New York
Brüggemann R, Halfon E, Welzl G, Voigt K, Steinberg C (2001) Applying the concept of partially ordered sets on the ranking of near-shore sediments by a battery of tests. J Chem Inf Comput Sci 41:918–925
Davey BA, Priestley BH (2002) Introduction to lattices and order, Cambridge University Press, Cambridge
Fattore M, Grassi R (2012) Measuring dynamics and structural change of time-dependent socio-economic networks, Qual Quant May 2013
Gao X, Xiao B, Tao D, Li X (2010) A survey of graph edit distance. Pattern Anal Appl 13:113–129
Harary F (1969) Graph theory. Perseus Books, Cambridge
Kendall M (1938) A new measure of rank correlation. Biometrika 30(1–2):81–89
Klein DJ (1995) Similarity and dissimilarity in posets. J Math Chem 18(2):321–348
Monjardet B (1981) Metrics on partially ordered sets - a survey. Discrete Math 35:173–184
Patil GP, Taillie C (2004) Multiple indicators, partially ordered sets and linear extensions: multi-criterion ranking and prioritization. Environ Ecol Stat 11:199–228
Schröder BSW (2002) Ordered sets. Birkhäuser, Basel
Sœrensen PB, Brüggemann R, Thomsen M, Lerche D (2005) Applications of multidimensional rank-correlation. Match Comm Math Comput Chem 54:643–670
van Laarhoven PJM, Aart EHL (1987) Simulated annealing: theory and applications. Kluwer, Dordecht
Voigt K, Scherb H, Brüggemann R, Schramm KW (2011) Application of the PyHasse program features: sensitivity, similarity, and separability for environmental health data. Statistica & Applicazioni, Special Issue, 155–168
Zeng Z, Tung AKH, Wang J, Feng J, Zhou L (2009) Comparing stars: on approximating graph edit distance. Proc VLDB Endow 2(1):25–36
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fattore, M., Grassi, R., Arcagni, A. (2014). Measuring Structural Dissimilarity Between Finite Partial Orders. 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_4
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8223-9_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-8222-2
Online ISBN: 978-1-4614-8223-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)