Geospatial Analysis Requires a Different Way of Thinking: The Problem of Spatial Heterogeneity

Geospatial analysis is very much dominated by a Gaussian way of thinking, which assumes that things in the world can be characterized by a well-defined mean, i.e., things are more or less similar in size. However, this assumption is not always valid. In fact, many things in the world lack a well-defined mean, and therefore there are far more small things than large ones. This paper attempts to argue that geospatial analysis requires a different way of thinking - a Paretian way of thinking that underlies skewed distribution such as power laws, Pareto and lognormal distributions. I review two properties of spatial dependence and spatial heterogeneity, and point out that the notion of spatial heterogeneity in current spatial statistics is only used to characterize local variance of spatial dependence. I subsequently argue for a broad perspective on spatial heterogeneity, and suggest it be formulated as a scaling law. I further discuss the implications of Paretian thinking and the scaling law for better understanding of geographic forms and processes, in particular while facing massive amounts of social media data. In the spirit of Paretian thinking, geospatial analysis should seek to simulate geographic events and phenomena from the bottom up rather than correlations as guided by Gaussian thinking.Comment: 13 pages, 4 figures, and 3 table

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 that 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 beautiful built and natural environments or with a high degree of wholeness.Comment: 14 pages, 7 figures, 2 table

Street Hierarchies: A Minority of Streets Account for a Majority of Traffic Flow

Urban streets are hierarchically organized in the sense that a majority of streets are trivial, while a minority of streets is vital. This hierarchy can be simply, but elegantly, characterized by the 80/20 principle, i.e. 80 percent of streets are less connected (below the average), while 20 percent of streets are well connected (above the average); out of the 20 percent, there is 1 percent of streets that are extremely well connected. This paper, using a European city as an example, examined, at a much more detailed level, such street hierarchies from the perspective of geometric and topological properties. Based on an empirical study, we further proved a previous conjecture that a minority of streets accounts for a majority of traffic flow; more accurately, the 20 percent of top streets accommodate 80 percent of traffic flow (20/80), and the 1 percent of top streets account for more than 20 percent of traffic flow (1/20). Our study provides new evidence as to how a city is (self-)organized, contributing to the understanding of cities and their evolution using increasingly available mobility geographic information.Comment: 15 pages, 10 figures, 4 tables, submitted to Int. J. of Geographic Information Scienc

Different Ways of Thinking about Street Networks and Spatial Analysis

Street networks, as one of the oldest infrastructures of transport in the world, play a significant role in modernization, sustainable development, and human daily activities in both ancient and modern times. Although street networks have been well studied in a variety of engineering and scientific disciplines, including for instance transport, geography, urban planning, economics, and even physics, our understanding of street networks in terms of their structure and dynamics remains limited, especially when dealing with such real-world problems as traffic jams, pollution, and human evacuations for disaster management. One goal of this special issue is to promote different ways of thinking about understanding street networks, and of conducting spatial analysis.Comment: 3 page

Defining Least Community as a Homogeneous Group in Complex Networks

This paper introduces a new concept of least community that is as homogeneous as a random graph, and develops a new community detection algorithm from the perspective of homogeneity or heterogeneity. Based on this concept, we adopt head/tail breaks - a newly developed classification scheme for data with a heavy-tailed distribution - and rely on edge betweenness given its heavy-tailed distribution to iteratively partition a network into many heterogeneous and homogeneous communities. Surprisingly, the derived communities for any self-organized and/or self-evolved large networks demonstrate very striking power laws, implying that there are far more small communities than large ones. This notion of far more small things than large ones constitutes a new fundamental way of thinking for community detection. Keywords: head/tail breaks, ht-index, scaling, k-means, natural breaks, and classificationComment: 9 pages, 3 figures, 3 tables; Physica A, 2015, xx(x), xx-x

Strong Subadditivity and Emergent Surface

In this paper, we introduce two bounds which we call the Upper Differential Entropy and the Lower Differential Entropy for an infinite family of intervals(strips) in quantum field theory. The two bounds are equal provided that the theory is translational invariant and the entanglement entropy varies smoothly with respect to the interval. When the theory has a holographic dual, strong subadditivity of entanglement entropy indicates that there is always an emergent surface whose gravitational entropy is exactly given by the bound.Comment: 18 pages, 8 figures, replace "residual entropy" to "differential entropy

Hidden Conformal Symmetry and Quasi-normal Modes

We provide an algebraic way to calculate the quasi-normal modes of a black hole, which possesses a hidden conformal symmetry. We construct an infinite tower of quasi-normal modes from the highest-weight mode, in a simple and elegant way. For the scalar, the hidden conformal symmetry manifest itself in the fact that the scalar Laplacian could be rewritten in terms of the $SL(2,R)$ quadratic Casimir. For the vector and the tensor, the hidden conformal symmetry acts on them through Lie derivatives. We show that for three-dimensional black holes, with appropriate combination of the components the radial equations of the vector and the tensor could be written in terms of the Lie-induced quadratic Casimir. This allows the algebraic construction of the quasi-normal modes feasible. Our results are in good agreement with the previous study.Comment: 23 pages; references added; typos corrected, more clarifications, published versio