Zipf’s law in hierarchically ordered open system

• Published in: Jahrbuch für Regionalwissenschaft 2011, Vol.31 (2), p.93-112
• Main Authors: Zipser, Tadeusz, Mlek, Magdalena, Zipser, Wawrzyniec
• Format: Article
• Language: English
• Subjects: Distribution; Economics; Economics and Finance; Environmental Economics; Geography; Hierarchical scales; Hierarchy; Landscape/Regional and Urban Planning; Methodology; Original Paper; Population Economics; Probability theory; Regional economics; Regional/Spatial Science; Simulation; Statistical analysis; Systems analysis; Zipf, George Kingsley
• ISSN: 0173-7600; 1613-9836
• DOI: 10.1007/s10037-011-0056-8

It was observed that inserting the arbitrary concentration into the simulation process makes the rank–size curves more similar to Zipf’s law graphs than simulations based on initially equal size of each node. The hierarchical network’s chain hypothesis based on the assumption that the prior cities’ size differentiation, resulted from geographical determinants or historical events can constitute a set of priorities affecting the contacts. “Hierarchical chain” with different influence in the “up” direction as opposed to the “down” direction in hierarchy can be compared to the structure of sentences in a language, as well as to functioning of economic initiatives. This setup has one important property: it allows for a part of the system to be cut off without affecting organization of the system. When an element is deleted or added, there is no evidence, which would indicate that the system was changed. Considering the easy ‘truncation’ or ‘expansion’ properties gives the proposed system high flexibility. Systems linked by such hierarchy lead to a clear distribution following the Zipf’s law, if they are “open” i.e. directing most of the contacts outside of the system, requiring handling for only ‘flow-through’ contacts. Special attention was devoted to an assessment of stability and endurance towards disruptions in ideal hierarchical order, so as to examine at which point the disturbance disrupts the Zipf’s law. The comparison of these modelings with recently accomplished selftraining modeling of settlement system reveals important relations between the parameter values emerged in that experiment and the hierarchical network.