Wholeness as a hierarchical graph to capture the nature of space

According to Christopher Alexander’s theory of centers, a whole comprises numerous, recursively defined centers for things or spaces surrounding us. Wholeness is a type of global structure or life-giving order emerging from the whole as a field of the centers. The wholeness is an essential part of any complex system and exists, to some degree or other, in spaces. This paper defines wholeness as a hierarchical graph, in which individual centers are represented as the nodes and their relationships as the directed links. The hierarchical graph gets its name from the inherent scaling hierarchy revealed by the head/tail breaks, which is a classification scheme and visualization tool for data with a heavy-tailed distribution. We suggest that (1) the degrees of wholeness for individual centers should be measured by PageRank (PR) scores based on the notion that high-degree-of-life centers are those to which many high-degree-of-life centers point, and (2) that the hierarchical levels, or the ht-index of the PR scores induced by the head/tail breaks, can characterize the degree of wholeness for the whole: the higher the ht-index, the more life or wholeness in the whole. Three case studies applied to the Alhambra building complex and the street networks of Manhattan and Sweden illustrate that the defined wholeness captures fairly well human intuitions on the degree of life for the geographic spaces. We further suggest that the mathematical model of wholeness be an important model of geographic representation, because it is topological oriented, which enables us to see the underlying scaling structure. The model can guide geodesign, which should be considered as the wholeness-extending transformations that are essentially like the unfolding processes of seeds or embryos, for creating built and natural environments of beauty or with a high degree of wholeness.     

A power-law-based approach to mapping COVID-19 cases in the United States

This paper examines the spatial and temporal distribution of all COVID-19 cases from January to June 2020 against the underlying distribution of population in the United States. It is found that, as time passes, COVID-19 cases become a power law with cutoff, resembling the underlying spatial distribution of populations. The power law implies that many states and counties have a low number of cases, while only a few highly populated states and counties have a high number of cases. To further differentiate patterns between the underlying populations and COVID-19 cases, we derived their inherent hierarchy or spatial heterogeneity characterized by the ht-index. We found that the ht-index of COVID-19 cases persistently approaches that of the populations; that is, 5 and 7 at the state and county levels, respectively. Mapping the ht-index of COVID-19 cases against that of populations shows that the pandemic is largely shaped by the underlying population with the R-square value between infection and population up to 0.82.     

CRG Index: A More Sensitive Ht-Index for Enabling Dynamic Views of Geographic Features

The ht-index is a recently proposed tool for capturing dynamic views of the evolution process of geographic features, but the ht-index is not sensitive to small changes during this process. To address this problem, an improved ht-index, a cumulative rate of growth (CRG) index, was proposed. Compared to the ht-index, the CRG index is more sensitive to changes during the evolution process of geographic features, and the process is better captured by the CRG index as a result. In addition, the head/tail breaks method (used to derive the ht-index) was adjusted, making the CRG index a universal tool for capturing the dynamic views of both increasingly heterogeneous and increasingly homogeneous geographic features.