# 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

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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