# The Relative Concentration of Interaction—A Proposal for an Integrated Understanding of Centrality and Central Places

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*Longue Durée*)

## Abstract

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## 1. Introduction

## 2. Central Place Theory

- market principle: maximizing the number of centers for the best supply ([5], pp. 77–79),
- transportation principle: reducing the transport distance of centers ([5], pp. 79–81),
- administration principle: no competition between centers by including all lower order centers in the market area of the higher order center ([5], pp. 82, 83).

## 3. A Generalized Definition of Centrality

Centrality is the relative concentration of interaction

## 4. Christaller Centrality and Network Centrality

## 5. Centrality Potential and Actual Centrality

## 6. Centrality Vector

- The intensity of centrality that can be understood as the sum of interactions with other places. The number of central functions supplied by a center can be used as a simple estimator for the intensity of centrality. The use of actual flows offers a higher accuracy but requires more data. The degree of centrality can be used to measure the intensity of centrality in networks when the edges are weighted with flows.
- Considering two places, one with a higher intensity but a limited range and the other one with a lower intensity but very remote connections, it is not easy to say which place is more central. Hence, the range or sphere of influence of a central place, is another dimension of centrality. It is a measure of the longest distance of an activity in which a central place is involved. For networks, we can use the closeness as an appropriate measure to assess the sphere of influence of a place.
- A place with many subordinated places but just two levels of hierarchy may have the same centrality as a place with few subordinated places but many levels of hierarchy. While Christaller’s approach allows us to estimate the hierarchical level of a place, this is not necessarily possible for networks which often are intended to be non-hierarchical. Hence, there is no appropriate network centrality index for this purpose.
- Is a place passively receiving interactions or actively controlling its own and other interactions? How many connections have to pass a certain node in a network? Betweenness is the index to answer such a question of control of interaction, i.e., a measure of the network location of a place in terms of assumed flows of information. The original concept of Christaller [5] does not cover this aspect, but it is possible to ask how exclusive a Christaller-center is as an interaction partner for the other places of a region—especially when we consider its network integration.

## 7. A Semi-Quantitative Method to Analyze Centrality

#### 7.1. Central Functions as a Tool to Assess Centrality

#### 7.2. Central Functions as Part of the Generalized Definition of Centrality

- Class 1 (dark gray): Extraordinary occurrence of centrality indicators; the centrality intensity is assumed to be very high.
- Class 2 (gray): Centrality indicators are well observable and indicate a high level of centrality intensity.
- Class 3 (light gray): Only few centrality indicators occur. The centrality intensity is medium to low.
- Class 4 (white): None or only marginal traces of centrality indicators are observable. This indicates a very low level of centrality intensity.

## 8. Case Studies

#### 8.1. Felix Romuliana

“The place chosen for the resting place of a mighty ruler, Diocletian’s adopted son and, consequently, member of Jove’s family, must have had quite a special architectural character. Galerius, glorified as a new Romulus and Alexander throughout the Empire after his triumph over King Narseus of Persia, was certainly not likely to consent that the edifice dedicated to him in the place where he was born and which was to bear the name of his mother should look like a provincial civil or military settlement”.([49], p. 124)

#### 8.2. Heuneburg

## 9. Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Christaller, W. Die Zentralen Orte in Süddeutschland—Eine ökonomisch-Geographische Untersuchung über die Gesetzmäßigkeiten der Verbreitung und Entwicklung der Siedlungen mit Städtischer Funktion; Gustav Fischer: Jena, Germany, 1933. [Google Scholar]
- Nakoinz, O. Zentralität—Theorie, Methoden und Fallbeispiele zur Analyse Zentraler Orte; Edition Topoi: Berlin, Germany, in print.
- Bebermeier, W.; Beck, D.; Gerlach, I.; Klein, T.; Knitter, D.; Kohlmeyer, K.; Krause, J.; Marzoli, D.; Meister, J.; Müller-Neuhof, B.; et al. Ancient Colonization of Marginal Habitats. A Comparative Analysis of Case Studies from the Old World. eTopoi. J. Anc. Stud.
**2016**, 6, 1–44. [Google Scholar] - Schmid, S.G.; Mouton, M. (Eds.) Men on the Rocks: The Formation of Nabataean Petra: proceedings of a Conference Held in Berlin 2–4 December 2011; Number 1 in Supplement to the Bulletin of Nabataean Studies; Logos: Berlin, Germany, 2013. [Google Scholar]
- Christaller, W. Die Zentralen Orte in Süddeutschland—Eine ökonomisch-Geographische Untersuchung über die Gesetzmäßigkeiten der Verbreitung und Entwicklung der Siedlungen mit Städtischer Funktion; Reprint of edition 1933; Wissenschaftliche Buchgesellschaft: Darmstadt, Germany; Jena, Germany, 1968. [Google Scholar]
- Kohl, J.G. Der Verkehr und die Ansiedelungen der Menschen in ihrer Abhängigkeit von der Gestaltung der Erdoberfläche; Arnoldische Buchhandlung: Dresden, Germany, 1841. [Google Scholar]
- Reynaud, J. Villes. In Encyclopédie Nouvelle; Gosselin: Paris, France, 1841; Volume 8, pp. 670–687. [Google Scholar]
- Lösch, A. Die Räumliche Ordnung der Wirtschaft: Eine Untersuchung über Standort, Wirtschaftsgebiete und Internationalen Handel; Fischer: Jena, Germany, 1940. [Google Scholar]
- Lösch, A. Economics of Location; Yale University Press: New Haven, CT, USA; London, UK, 1954. [Google Scholar]
- Hudson, J.C. A Location Theory for Rural Settlement. Ann. Assn. Am. Geogr.
**1969**, 59, 365–381. [Google Scholar] [CrossRef] - Von Böventer, E. Walter Christaller’s Central Places and Peripheral Areas: The Central Place Theory in Retrospect. J. Reg. Sci.
**1969**, 9, 117–124. [Google Scholar] [CrossRef] - Parr, J.B. Models of the Central Place System: A More General Approach. Urban Stud.
**1978**, 15, 35–49. [Google Scholar] [CrossRef] - Parr, J.B. Frequency Distributions of Central Places in Southern Germany: A Further Analysis. Econ. Geogr.
**1980**, 56, 141–154. [Google Scholar] [CrossRef] - Parr, J.B. Alternative Approaches to Market-area Structure in the Urban System. Urban Stud.
**1995**, 32, 1317–1329. [Google Scholar] [CrossRef] - Arlinghaus, S.L. Fractals Take a Central Place. Geogr. Ann. Ser. B Hum. Geogr.
**1985**, 67, 83–88. [Google Scholar] [CrossRef] - Gradmann, R. Schwäbische Städte. Zeitschrift der Gesellschaft für Erdkunde zu Berlin
**1916**, 7, 425–457. [Google Scholar] - Pacione, M. Urban Geography: A Global Perspective; Routledge: London, UK; New York, UY, USA, 2009. [Google Scholar]
- Pumain, D. Alternative explanations of hierarchical differentiation in urban systems. In Hierarchy in Natural and Social Sciences; Pumain, D., Ed.; Springer: Dordrecht, The Netherlands, 2006; pp. 169–222. [Google Scholar]
- Bernbeck, R. Theorien in der Archäologie; Francke: Tübingen, Germany; Basel, Switzerland, 1997. [Google Scholar]
- Nakoinz, O. Models of Centrality. eTopoi. J. Anc. Stud.
**2012**, 3, 217–223. [Google Scholar] - Knitter, D.; Nakoinz, O.; Del Fabbro, R.; Kohlmeyer, K.; Meyer, M.; Schütt, B. The Centrality of Aleppo and its environs. eTopoi. J. Anc. Stud.
**2014**, 3, 107–127. [Google Scholar] - Nakoinz, O. Zentralortforschung und zentralörtliche Theorie. Archäol. Korresp.
**2009**, 39, 361–380. [Google Scholar] - Nakoinz, O. Zentralorte in parallelen Raumstrukturen. In Parallele Raumkonzepte, Workshop des Exzellenzclusters Topoi vom 15.–17. März 2010; Hansen, S., Meyer, M., Eds.; Topoi. Berlin Studies of the Ancient World; De Gruyter: Berlin, Germany, 2013; pp. 83–104. [Google Scholar]
- Nakoinz, O. Concepts of Central Place Research in Archaeology. In Landscapes and Human Development: The Contribution of European Archaeology, Proceedings of the International Workshop ‘Socio-Environmental Dynamics over the Last 12,000 Years: The Creation of Landscapes, 1–4 April 2009’; Kiel Graduate School ‘Human Development in Landscapes’, Ed.; Habelt: Bonn, Germany, 2010; pp. 251–264. [Google Scholar]
- Nakoinz, O. Räumliche Interaktionsmodelle. Praehist. Z.
**2013**, 88, 226–257. [Google Scholar] [CrossRef] - Freeman, L.C. The Development of Social Network Analysis: A Study in the Sociology of Science; Empirical Press: Vancouver, BC, Canada; BookSurge: North Charleston, SC, USA, 2004. [Google Scholar]
- Scott, P.J.P. Social Network Analysis: A Handbook, 2nd ed.; Sage Publications Ltd.: Thousand Oaks, CA, USA, 2000. [Google Scholar]
- Freeman, L.C. Centrality in social networks conceptual clarification. Soc. Netw.
**1978**, 1, 215–239. [Google Scholar] [CrossRef][Green Version] - Freeman, L.C.; Roeder, D.; Mulholland, R.R. Centrality in social networks: II. Experimental results. Soc. Netw.
**1979**, 2, 119–141. [Google Scholar] [CrossRef] - Kadushin, C. Understanding Social Networks: Theories, Concepts, and Findings; Oxford University Press: New York, NY, USA, 2012. [Google Scholar]
- Aggarwal, C.C. (Ed.) Social Network Data Analytics; Springer: Boston, MA, USA, 2011. [Google Scholar]
- Sun, J.; Tang, J. A Survey of Models and Algorithms for Social Influence Analysis. In Social Network Data Analytics; Aggarwal, C.C., Ed.; Springer: Boston, MA, USA, 2011; pp. 177–214. [Google Scholar]
- Taylor, P.J. World City Network: A Global Urban Analysis; Routledge: London, UK, 2004. [Google Scholar]
- Taylor, P.J.; Hoyler, M.; Verbruggen, R. External Urban Relational Process: Introducing Central Flow Theory to Complement Central Place Theory. Urban Stud.
**2010**, 47, 2803–2818. [Google Scholar] [CrossRef][Green Version] - Knitter, D.; Blum, H.; Horejs, B.; Nakoinz, O.; Schütt, B.; Meyer, M. Integrated centrality analysis: A diachronic comparison of selected Western Anatolian locations. Quat. Int.
**2013**, 312, 45–56. [Google Scholar] [CrossRef] - Yan, X.; Zhai, L.; Fan, W. C-index: A weighted network node centrality measure for collaboration competence. J. Informetr.
**2013**, 7, 223–239. [Google Scholar] [CrossRef] - Li, H.; Zhang, Z. Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms. arXiv, 2017; arXiv:1708.05959. [Google Scholar]
- Wang, H.; Hernandez, J.M.; Van Mieghem, P. Betweenness centrality in a weighted network. Phys. Rev. E
**2008**, 77. [Google Scholar] [CrossRef] [PubMed][Green Version] - Opsahl, T.; Agneessens, F.; Skvoretz, J. Node centrality in weighted networks: Generalizing degree and shortest paths. Soc. Netw.
**2010**, 32, 245–251. [Google Scholar] [CrossRef] - Zipf, G.K. Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology; Addison-Wesley Press: Cambridge, UK, 1949. [Google Scholar]
- Johnson, G.A. Aspects of Regional Analysis in Archaeology. Annu. Rev. Anthropol.
**1977**, 6, 479–508. [Google Scholar] [CrossRef] - Wilkinson, T.J.; Bintliff, J.; Curvers, H.H.; Halstead, P.; Kohl, P.L.; Liverani, M.; McCorriston, J.; Oates, J.; Schwartz, G.M.; Thuesen, I.; et al. The Structure and Dynamics of Dry-Farming States in Upper Mesopotamia [and Comments and Reply]. Curr. Anthropol.
**1994**, 35, 483–520. [Google Scholar] [CrossRef] - Blotevogel, H.H. Zentrale Orte. In Handwörterbuch der Raumordnung; Ritter, E.H., Ed.; Akademie für Raumforschung und Landesplanung: Hannover, Germany, 2005; pp. 1307–1315. [Google Scholar]
- Denecke, D. Der geographische Stadtbegriff und die räumlich-funktionale Betrachtungsweise bei Siedlungstypen mit zentraler Bedeutung in Anwendung auf historische Siedlungsepochen. In Vor- und Frühformen der Europäischen Stadt im Mittelalter: Bericht über ein Symposium in Reinhausen bei Göttingen in der Zeit vom 18. bis 24. April 1972; Jankuhn, H., Schlesinger, W., Steuer, H., Eds.; Abhandlungen der Akademie der Wissenschaften in Göttingen 3; Vandenhoeck & Ruprecht: Göttingen, Germany, 1972; pp. 33–55. [Google Scholar]
- Gringmuth-Dallmer, E. Kulturlandschaftsmuster und Siedlungssysteme. Siedlungsforschung
**1996**, 14, 7–31. [Google Scholar] - Gringmuth-Dallmer, E. Zentren unterschiedlichen Ranges im nordwestslawischen Gebiet. In Frühgeschichtliche Zentralorte in Mitteleuropa; Internationale Konferenz und Kolleg der Alexander von Humboldt-Stiftung zum 50. Jahrestag des Beginns archäologischer Ausgrabungen in Pohansko bei Břeclav, Břeclav, Tschechische Republik, 5–9 October 2009; Macháček, J., Ungerman, S., Henning, J., Leube, A., Biermann, F., Eds.; Number 14 in Studien zur Archhäologie Europas; Habelt: Bonn, Germany, 2011; pp. 431–440. [Google Scholar]
- Crumley, C.L. Heterarchy and the Analysis of Complex Societies. Archeol. Pap. Am. Anthropol. Assoc.
**1995**, 6, 1–5. [Google Scholar] [CrossRef] - Crumley, C.L. Remember How to Organize: Heterarchy Across Disciplines. In Nonlinear Models for Archaeology and Anthropology: Continuing the Revolution; Beekman, C.S., Baden, W.W., Eds.; Ashgate Publishing: Aldershot, UK, 2005; pp. 35–50. [Google Scholar]
- Srejović, D.; Čedomir, V. Emperor’s Galerius’s buildings in Romuliana (Gamzigrad, eastern Serbia). AnTard
**1994**, 2, 123–141. [Google Scholar] [CrossRef] - Christodoulou, D.N. Galerius, Gamzigrad, and the Fifth Macedonian legion. J. Rom. Archaeol.
**2002**, 15, 275–281. [Google Scholar] [CrossRef] - Von Bülow, G. The two Defensive Systems of the Late Roman Imperial Palace of Romuliana-Gamzigrad (Dacia Ripensis). In Focus on Fortifications; Frederiksen, R., Schnelle, M., Muth, S., Schneider, P., Eds.; Oxbow Books: Oxford, UK, 2016; pp. 314–324. [Google Scholar]
- Živić, M. Felix Romuliana—50 Years of Solving; National Museum Zaječar: Belgrade, Serbia, 2003. [Google Scholar]
- Protic, N.; Martinovic, L.; Milicic, B.; Stevanovic, D.; Mojasevic, M. The Status of Soil Surveys in Serbia and Montenegro. In Soil Resources of Europe; Jones, R.J.A., Houšková, B., Bullock, P., Montanarella, L., Eds.; European Soil Bureau Joint Research Center, Office for Official Publications of the European Communities: Luxembourg, 2005; pp. 297–315. [Google Scholar]
- Jović, V. Geologic and geomorphologic features of Eastern Serbia. In Archaeology of Eastern Serbia—Symposium; Lazić, M., Ed.; Center for Archaeological Research: Belgrade, Serbia, 1997; pp. 31–32. [Google Scholar]
- Srejović, D.; Lazić, M. The Bronze Age settlements ad cementeries in Timoćka Krajina. In Archaeology of Eastern Serbia—Symposium; Lazić, M., Ed.; Center for Archaeological Research: Belgrade, Serbia, 1997; pp. 245–247. [Google Scholar]
- Tasić, N. Eneolithic and Early Bronze Ages in the North-East Serbia. In Archaeology of Eastern Serbia—Symposium; Lazić, M., Ed.; Center for Archaeological Research: Belgrade, Serbia, 1997. [Google Scholar]
- Petrović, P. Romans in the Timok Valley. In Archaeology of Eastern Serbia—Symposium; Lazić, M., Ed.; Center for Archaeological Research: Belgrade, Serbia, 1997; pp. 129–131. [Google Scholar]
- Bowman, A.K.; Garnsey, P.; Cameron, A. (Eds.) The Cambridge Ancient History: The Crisis of Empire, a.d. 193–337, 2nd ed.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2005; Volume XII. [Google Scholar]
- Leadbetter, B. Galerius and the Will of Diocletian; Routledge: Milton Park, Abingdon, Oxon, UK; New York, NY, USA, 2009. [Google Scholar]
- Petković, S.; Živić, M. Traces of metallurgical activities in late Roman romuliana: Results of recent investigations. Metal. J. Metall.
**2006**, 12, 111–128. [Google Scholar] - Škundrić, J.; Wulf-Rheidt, U. Gamzigrad, Serbien: Der spätantike Kaiserpalast Felix Romuliana und sein Umfeld. e-Forschungsberichte
**2016**, 1, 115–123. [Google Scholar] - Fernández-Götz, M.; Krausse, D. Rethinking Early Iron Age urbanisation in Central Europe: the Heuneburg site and its archaeological environment. Antiquity
**2013**, 87, 473–487. [Google Scholar] [CrossRef][Green Version] - Stegmaier, G. Zur Entstehung, Datierung und Besiedlung der Heuneburg-Vorburg. Ergebnisse der Grabungen 2000–2003. In “Fürstensitze” und Zentralorte der frühen Kelten, Abschlusskolloquium des DFG-Schwerpunktprogramms 1171 in Stuttgart, 12–15 October 2009; Krausse, D., Ed.; Forsch. u. Ber. Vor- u. Frühgesch. Baden-Württemberg; Theiss: Stuttgart, Germany, 2010; Volume 120, pp. 257–268. [Google Scholar]
- Kurz, S. Zur Entstehung, Datierung und Besiedlung der Heuneburg-Vorburg. Ergebnisse der Grabungen 2000-2003Zur Genese und Entwicklung der Heuneburg in der späten Hallstattzeit. In “Fürstensitze” und Zentralorte der frühen Kelten. Abschlusskolloquium des DFG-Schwerpunktprogramms 1171 in Stuttgart, 12–15 October 2009; Krausse, D., Ed.; Forsch. u. Ber. Vor- u. Frühgesch. Baden-Württemberg; Theiss: Stuttgart, Germany, 2010; Volume 120, pp. 239–256. [Google Scholar]
- Nakoinz, O. Quantifying Iron Age urbanism. In Delicate Urbanism in Context: Pre-Roman German Urbanism; Stoddart, S., Ed.; McDonald Institute for Archaeological Research: Cambridge, UK, 2017; pp. 87–95. [Google Scholar]
- Nakoinz, O. Archäologische Kulturgeographie der Ältereisenzeitlichen Zentralorte Südwestdeutschlands; Universitätsforsch. prähist. Arch.; Habelt: Bonn, Germany, 2013; Volume 224. [Google Scholar]
- Stegmaier, G. Monumentalität und Anspruch auf Macht: Differenzierte Betrachtungen zum Großgrabhügel‚ Hohmichele’ und weiteren Bestattungen im Umfeld der Heuneburg. In Interpretierte Eisenzeiten. Fallstudien, Methoden, Theorie. Tagungsbeiträge der 7. Linzer Gespräche zur Interpretativen Eisenzeitarchäologie; R. Karl, J.L., Ed.; Studien zur Kulturgeschichte von Oberösterreich; Oberösterreichisches Landesmuseum Linz: Linz, Austria, 2017; Volume 47, pp. 27–36. [Google Scholar]
- Schumann, R. Status und Prestige in der Hallstattkultur; Münchner Archäologische Forschungen; Verlag Marie Leidorf GmbH: Rahden, Germany, 2015; Volume 3. [Google Scholar]
- Kimmig, W. Die Heuneburg an der Oberen Donau; Führer zu archäologischen Denkmälern in Baden-Württemberg; Theiss: Stuttgart, Germany, 1983; Volume 1. [Google Scholar]
- Modarressi-Tehrani, D. Untersuchungen zum Früheisenzeitlichen Metallhandwerk im Westlichen Hallstatt- und Frühlatènegebiet.; Bochumer Forsch. Ur- u. Frühgesch. Arch.; Verlag Marie Leidorf GmbH: Rahden, Germany, 2009; Volume 2. [Google Scholar]
- Nakoinz, O. Fingerprinting Iron Age Communities in South-West-Germany and an Integrative Theory of Culture. In Fingerprintig the Iron Age; Popa, C., Stoddart, S., Eds.; Oxbow: Oxford, UK, 2014; pp. 187–199. [Google Scholar]

1 | If required, or supported by the available data, more classes—shades of grey in the figure—can be used to express a wider range of intensity values. |

2 | The original concept included further subdivisions which are not presented in this paper due to the better readability of the reduced concept; interested readers are referred to Nakoinz [2]. |

**Figure 1.**Each offered central function has an upper limit (determined by the maximum distance people will travel to access it) and lower limit (determined by the minimum population required to sustain a central function). Based on this, each central place has a circular complementary region. In this configuration, some areas are served by more than one central place while other areas are not served at all. A hexagonal configuration of complementary regions is the most efficient way to ensure that everybody has access to the central function (after Ref. [5], pp. 65–72).

**Figure 2.**There are various measures of centrality in social network theory that describe different characteristics of a social network and its interacting individuals. The most common node centrality measures are: (

**a**) degree, i.e., the number of adjacent nodes; (

**b**) closeness that measures the centrality by calculating the average shortest distances to all other nodes; (

**c**) betweenness that highlights critical nodes in the network structure by measuring to what degree a certain node lies in the shortest paths of other nodes ([32], pp. 180–182). Equations and a more detailed characterization is given in Freeman [28].

**Figure 3.**Sketch of the different centrality concepts. Not every settlement is a central place. For a place to be central, it has to offer central functions to its complementary region. This region can be defined based on Christaller or network centrality. The network edges in the figure show that only some sites interact, i.e., exchanging central functions. The differences between actual and potential centrality result from the combination of network integration of central places and their ability to serve their complementary region with central functions. Large deviations between potential and actual centrality point to the importance of historical contingency or intervening opportunities that influence the flows of interaction.

**Figure 4.**Centrality Graph as a tool to visualize and compare the different aspects of settlement characteristics, Christaller and network centrality (see text for a detailed explanation of the different elements).

**Figure 6.**Centrality Graph of Felix Romuliana. (

**a**) actual centrality of the place and its environs; top: before and after the palace phase; bottom: during the palace phase; (

**b**) potential centrality of the site (see text for a detailed description).

**Figure 7.**Centrality Graph of Heuneburg (see text for a detailed explanation of the different elements).

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Knitter, D.; Nakoinz, O.
The Relative Concentration of Interaction—A Proposal for an Integrated Understanding of Centrality and Central Places. *Land* **2018**, *7*, 86.
https://doi.org/10.3390/land7030086

**AMA Style**

Knitter D, Nakoinz O.
The Relative Concentration of Interaction—A Proposal for an Integrated Understanding of Centrality and Central Places. *Land*. 2018; 7(3):86.
https://doi.org/10.3390/land7030086

**Chicago/Turabian Style**

Knitter, Daniel, and Oliver Nakoinz.
2018. "The Relative Concentration of Interaction—A Proposal for an Integrated Understanding of Centrality and Central Places" *Land* 7, no. 3: 86.
https://doi.org/10.3390/land7030086