Spatial complexity

In mathematics, spatial complexity is defined ^[1] as the complexity of a spatial entity, whether it is a surface or a solid body, or even a spatial object of dimension two or higher. Whatever the spatial object, the description and assessment of its spatial complexity is eventually algorithmic, thus linking it to algorithmic complexity. Of particular interest is the spatial complexity of maps, satellite images, photographs, patterns, landscapes. According to spatial complexity theory, as introduced by Fivos Papadimitriou, spatial complexity is determined by three sets of factors: geometric, probabilistic and topological.

Assessment

For two dimensional surfaces, spatial complexity can be measured by two alternative metrics: one based on run-length encoding and another on edit distance.^[1] For three dimensions, another two metrics, that are based on Reeb graphs and on the Euler characteristic.^[1]

Properties

Spatial complexity is scale-dependent (it depends on the scale of observation), depends on the generalization that is made before its assessment (depends on the degree to which geometric or thematic entities are grouped together) and relates to entropy, topology and geometry, but its study should be interdisciplinary.

Applications

Besides mathematics, spatial complexity is significant for a number of disciplines, including the following:

Geomorphology^[2]
Mechanics^[3]
Materials science^[4]
Software engineering^[5]
Geoinformatics, Geography and Cartography^[6]
Ecology^[7]
Landscape Ecology^[8]
Wildlife conservation^[9]
Forestry^[10]
Neurology^[11]
MRI^[12]
Urban design and urban planning^[13]
Non-linear dynamics and optics^[14]
Geology^[15]
Zoology^[16]
Lasers^[17]
Cosmology^[18]
COVID-19^[19]

References

↑ ^1.0 ^1.1 ^1.2 Papadimitriou, Fivos (2020) (in en). Spatial Complexity: Theory, Mathematical Methods and Applications. Springer International Publishing. ISBN 978-3-030-59670-5. https://www.springer.com/gp/book/9783030596705.
↑ Danek P., Samonil P., Phillips D. (2016). Geomorphic controls of soil spatial complexity in a primeval mountain forest in the Czech Republic. Geomorphology 273, 180-291.
↑ Van der Heijden H.M., Champneys A.R., Thompson M.T. (1998). The spatial complexity of localized buckling inrods with noncircular cross section. SIAM Journal of Applied Mathematics, 59(1), 1980221.
↑ Liu S., Phillabaum B., Carlson E.W., Dahmen K.A., Dahmen N.S., Vidhyadhuraja M.M., Qazilbash M., Basov D.N. (2015). Random field driven spatial complexity at the Mott Transition in VO2. 18 Feb. 2015. https://arxiv.org/pdf/1502.05426.pdf
↑ Chhabra, Jitender Kumar; Aggarwal, K.K.; Singh, Yogesh (2004-08-01). "Measurement of object-oriented software spatial complexity" (in en). Information and Software Technology 46 (10): 689–699. doi:10.1016/j.infsof.2004.01.001. ISSN 0950-5849. https://www.sciencedirect.com/science/article/abs/pii/S0950584904000023.
↑ White R., & Engelen G. (1994). Cellular dynamics and GIS: Modelling spatial complexity. Geographical Systems, 1(3), 237–253.
↑ 7. Dieckmann U., Law R., Metz J.A., Metz J.A., Metz J. (2005). The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge: Cambridge University Press. https://www.cambridge.org/gr/academic/subjects/life-sciences/ecology-and-conservation/geometry-ecological-interactions-simplifying-spatial-complexity?format=PB
↑ Papadimitriou, Fivos (2009-01-01). "Modelling spatial landscape complexity using the Levenshtein algorithm" (in en). Ecological Informatics 4 (1): 48–55. doi:10.1016/j.ecoinf.2009.01.001. ISSN 1574-9541. http://www.sciencedirect.com/science/article/pii/S1574954109000028.
↑ Cushman, Samuel A., ed (2010) (in en). Spatial Complexity, Informatics, and Wildlife Conservation. Springer Japan. ISBN 978-4-431-87770-7. https://www.springer.com/gp/book/9784431877707.
↑ Andrieu E., Ladet S., Heintz W., Deconchat M. (2011). History and spatial complexity of deforestation and logging in small private forests. Landscape and Urban Planning, 103(2), 109–117.
↑ Jia H., Li Y., & Yu D. (2018). Normalized spatial complexity analysis of neural signals. Scientific Reports, 8, 7912. DOI:10.1038/s41598-018-26329-0. www.nature.com/scientificreports
↑ Schulz, Laura; Ischebeck, Anja; Wriessnegger, Selina C.; Steyrl, David; Müller-Putz, Gernot R. (2018-07-01). "Action affordances and visuo-spatial complexity in motor imagery: An fMRI study" (in en). Brain and Cognition 124: 37–46. doi:10.1016/j.bandc.2018.03.012. ISSN 0278-2626. PMID 29723681. http://www.sciencedirect.com/science/article/pii/S0278262618300563.
↑ o'Brien, Jamie (2019-06-10) (in en). Spatial Complexity in Urban Design Research : Graph Visualization Tools for Communities and their Contexts. Routledge. doi:10.4324/9781315624464. ISBN 978-1-315-62446-4. https://www.taylorfrancis.com/books/spatial-complexity-urban-design-research-jamie-brien/10.4324/9781315624464.
↑ Nonlinear dynamics and spatial complexity in optical systems : the Forty-first Scottish Universities' Summer School in Physics, Edinburgh, August 1992. Harrison, R. G. (Robert G.), 1944-, Uppal, J. S.. Boca Raton, FL. 4 May 2018. ISBN 978-1-351-08349-2. OCLC 1035845227. https://www.worldcat.org/oclc/1035845227.
↑ Mays, David C.; Faybishenko, Boris A.; Finsterle, Stefan (2002). "Information entropy to measure temporal and spatial complexity of unsaturated flow in heterogeneous media" (in en). Water Resources Research 38 (12): 49–1–49-11. doi:10.1029/2001WR001185. ISSN 1944-7973. Bibcode: 2002WRR....38.1313M. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2001WR001185.
↑ Liu X., Xu N., Jiang,A. (2015). Tortuosity entropy: A measure of spatial complexity of behavioral changes in animal movement. Journal of Theoretical Biology, 364, 197-205.
↑ Lugiato, L. A.; Oppo, G. L.; Tredicce, J. R.; Narducci, L. M.; Pernigo, M. A. (1990-06-01). "Instabilities and spatial complexity in a laser" (in EN). JOSA B 7 (6): 1019–1033. doi:10.1364/JOSAB.7.001019. ISSN 1520-8540. Bibcode: 1990JOSAB...7.1019L. https://www.osapublishing.org/josab/abstract.cfm?uri=josab-7-6-1019.
↑ Van de Weygaert R. (2016). Voids and the Cosmic Web: cosmic depression & spatial complexity. https://arxiv.org/pdf/1611.01222.pdf
↑ Geng X., Gerges F., Katul G.G., Bou-Zeid E., Nassif H., Boufadel M.C. (2020). Population agglomeration is a harbinger of the spatial complexity of COVID-19. Chemical Engineering Journal, available online 13 November 2020, 127702

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[:0-1] 1.0 ^1.1 ^1.2 Papadimitriou, Fivos (2020) (in en). Spatial Complexity: Theory, Mathematical Methods and Applications. Springer International Publishing. ISBN 978-3-030-59670-5. https://www.springer.com/gp/book/9783030596705.

[2] Danek P., Samonil P., Phillips D. (2016). Geomorphic controls of soil spatial complexity in a primeval mountain forest in the Czech Republic. Geomorphology 273, 180-291.

[3] Van der Heijden H.M., Champneys A.R., Thompson M.T. (1998). The spatial complexity of localized buckling inrods with noncircular cross section. SIAM Journal of Applied Mathematics, 59(1), 1980221.

[4] Liu S., Phillabaum B., Carlson E.W., Dahmen K.A., Dahmen N.S., Vidhyadhuraja M.M., Qazilbash M., Basov D.N. (2015). Random field driven spatial complexity at the Mott Transition in VO2. 18 Feb. 2015. https://arxiv.org/pdf/1502.05426.pdf

[5] Chhabra, Jitender Kumar; Aggarwal, K.K.; Singh, Yogesh (2004-08-01). "Measurement of object-oriented software spatial complexity" (in en). Information and Software Technology 46 (10): 689–699. doi:10.1016/j.infsof.2004.01.001. ISSN 0950-5849. https://www.sciencedirect.com/science/article/abs/pii/S0950584904000023.

[6] White R., & Engelen G. (1994). Cellular dynamics and GIS: Modelling spatial complexity. Geographical Systems, 1(3), 237–253.

[7] 7. Dieckmann U., Law R., Metz J.A., Metz J.A., Metz J. (2005). The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge: Cambridge University Press. https://www.cambridge.org/gr/academic/subjects/life-sciences/ecology-and-conservation/geometry-ecological-interactions-simplifying-spatial-complexity?format=PB

[8] Papadimitriou, Fivos (2009-01-01). "Modelling spatial landscape complexity using the Levenshtein algorithm" (in en). Ecological Informatics 4 (1): 48–55. doi:10.1016/j.ecoinf.2009.01.001. ISSN 1574-9541. http://www.sciencedirect.com/science/article/pii/S1574954109000028.

[9] Cushman, Samuel A., ed (2010) (in en). Spatial Complexity, Informatics, and Wildlife Conservation. Springer Japan. ISBN 978-4-431-87770-7. https://www.springer.com/gp/book/9784431877707.

[10] Andrieu E., Ladet S., Heintz W., Deconchat M. (2011). History and spatial complexity of deforestation and logging in small private forests. Landscape and Urban Planning, 103(2), 109–117.

[11] Jia H., Li Y., & Yu D. (2018). Normalized spatial complexity analysis of neural signals. Scientific Reports, 8, 7912. DOI:10.1038/s41598-018-26329-0. www.nature.com/scientificreports

[12] Schulz, Laura; Ischebeck, Anja; Wriessnegger, Selina C.; Steyrl, David; Müller-Putz, Gernot R. (2018-07-01). "Action affordances and visuo-spatial complexity in motor imagery: An fMRI study" (in en). Brain and Cognition 124: 37–46. doi:10.1016/j.bandc.2018.03.012. ISSN 0278-2626. PMID 29723681. http://www.sciencedirect.com/science/article/pii/S0278262618300563.

[13] 'Brien, Jamie (2019-06-10) (in en). Spatial Complexity in Urban Design Research : Graph Visualization Tools for Communities and their Contexts. Routledge. doi:10.4324/9781315624464. ISBN 978-1-315-62446-4. https://www.taylorfrancis.com/books/spatial-complexity-urban-design-research-jamie-brien/10.4324/9781315624464.

[14] Nonlinear dynamics and spatial complexity in optical systems : the Forty-first Scottish Universities' Summer School in Physics, Edinburgh, August 1992. Harrison, R. G. (Robert G.), 1944-, Uppal, J. S.. Boca Raton, FL. 4 May 2018. ISBN 978-1-351-08349-2. OCLC 1035845227. https://www.worldcat.org/oclc/1035845227.

[15] Mays, David C.; Faybishenko, Boris A.; Finsterle, Stefan (2002). "Information entropy to measure temporal and spatial complexity of unsaturated flow in heterogeneous media" (in en). Water Resources Research 38 (12): 49–1–49-11. doi:10.1029/2001WR001185. ISSN 1944-7973. Bibcode: 2002WRR....38.1313M. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2001WR001185.

[16] Liu X., Xu N., Jiang,A. (2015). Tortuosity entropy: A measure of spatial complexity of behavioral changes in animal movement. Journal of Theoretical Biology, 364, 197-205.

[17] Lugiato, L. A.; Oppo, G. L.; Tredicce, J. R.; Narducci, L. M.; Pernigo, M. A. (1990-06-01). "Instabilities and spatial complexity in a laser" (in EN). JOSA B 7 (6): 1019–1033. doi:10.1364/JOSAB.7.001019. ISSN 1520-8540. Bibcode: 1990JOSAB...7.1019L. https://www.osapublishing.org/josab/abstract.cfm?uri=josab-7-6-1019.

[18] Van de Weygaert R. (2016). Voids and the Cosmic Web: cosmic depression & spatial complexity. https://arxiv.org/pdf/1611.01222.pdf

[19] Geng X., Gerges F., Katul G.G., Bou-Zeid E., Nassif H., Boufadel M.C. (2020). Population agglomeration is a harbinger of the spatial complexity of COVID-19. Chemical Engineering Journal, available online 13 November 2020, 127702

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