Zipf's Law in Hierarchically Ordered Open System

Tadeusz Zipser, Magdalena Mlek and Wawrzyniec Zipser ()

ERSA conference papers from European Regional Science Association

Abstract: 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 differentation, resulted from geographical determinants or accidental historical events can constitute a set of preliminary priorities affecting the development of 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 remaining part of the system. When it is deleted, there is no evidence, which would indicate that the system was changed in any way. Same thing pertains when a city is dropped from the middle of the hierarchy. Similarly, adding a city does not require change of the existing rules. Cosidering 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 (certain number of people) for only 'flow-through' contacts. Special attention was devoted to an assesment of stability and endurance towards disruptions in ideal hierachical order, so as to examine at which point the disturbance of hierarchical structure disrupts the characteristics consistent with Zipf's law. Both arbitrarily inserted concentration of origin and destination loads and hierarchical ordering caused the rank-sized curves to became similar or nearly identical to the ideal Zipf's law graph. Quantitive growth of old subsystems does not destroy the characteristic of the entire system. The comparison of these modellings with recently accomplished selftrainig modelling of settlement system reveals important relations between the parameter values emerged in that experiment and the hierarchical network. An other hypothetical factor is the detriment of percolation effect among local nodal areas.

Date: 2011-09
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