Abstract

Cities or human settlements are usually conceived of as examples of complex systems.  In this research, the question is examined empirically of whether cities self-organize through real and simulated data.  The results show that cities tend to self-organize themselves into a critical state, reminiscent of the sand pile model.

1  Introduction

Cities are the outcomes of interacting spatial decisions, in particular land development decisions.  To understand how cities evolve, it is therefore necessary to understand how land development behavior occurs and how these decisions interact with each other. This bottom up approach to describing urban spatial evolution has been done through computer simulations, namely cellular automata (e.g., White and Engelen, 1993; Benati, 1997; Couclelis, 1997; Phipps and Langlois, 1997; Casti, 1999; Lai and Han, 2014).  The findings derived from these simulations have provided intriguing insights into how and why cities grow and how we should plan for urban complexity (e.g., Byrne, 2003; Sengupta, Rauws, and de Roo, 2016).  For example, the spatial patterns of cities measured in terms of some appropriate indicators, say population densities, are fractal in nature, showing strong evidence of self-organization (e.g., Batty, 1995; 2005). In this research, I set aside the understanding of individual land development behaviors in depth.  But I do try to model these behaviors based on bounded rationality in that developers learn through development processes and change their development strategies accordingly.  The backing for this approach is provided indirectly by Simon's (1998) argument that the study of human thinking (a kind of artifices) need not be reduced to the knowledge of basic sciences, because it suffices to uncover characteristics common to human thinking in behavioral studies.  Similarly, in studying spatial evolution, if we could uncover behavioral characteristics common to land development processes, we could then model spatial evolution based on that knowledge without lose much generalizability.

Therefore, my focus here is on whether these interacting, individual development decisions together come up with spatial patterns which somehow result in some order.  I conducted two empirical studies, one through examining existing land use patterns in two settlements in Taiwan and the other through designing a computer experiment based on a simplified land development behavioral model to observe how order emerges from the complex system.  The results from the two empirical studies indicated strong evidence that complex spatial systems tend to self-organize themselves into a critical state in terms of power law where frequencies and scales of spatial units have a linear, logarithmic relationship.

The research is therefore grounded on the hypothesis that the evolution of complex spatial systems tends to self-organize themselves into a critical state, or self-organized criticality.  I conducted two empirical studies to test this hypothesis: the examination of land use mixture patterns of Min Jen Township in Nan Tou County and San Tsong City in Taipei metropolitan area, and a computer experiment on spatial patterns resulting from interacting land development decisions.  The results indicated that the spatial patterns derived from both the empirical data and the computer simulation followed a simple power law showing an evidence of self-organization.  Section 2 introduces basic ideas about self-organization and its relationship to spatial structures.  Section 3 describes the research design and Section 4 depicts the results.  Section 5 provides a conclusion.

2  Self-organization in Spatial Structures

Self-organization is defined as some order of patterns in complex systems that emerges from many agents interacting in these systems.  The most well-known example is a sand pile (Bak and Chen, 1991).  Consider dropping a set of grains of sand onto a plane, one at a time.  The sand on the plane gradually forms a growing pile.  When the pile reaches a certain height, avalanches start to occur so that the slope of the pile remains constant.  The curious phenomenon is that these avalanches caused by the chain effect of the interacting grains through gravity tend to self-organize into a critical state, or a power law.  That is, if we plot the occurrences of these avalanches by scales and frequencies, the data show a linear, logarithmic relationship.  This power law relationship occurs in many natural phenomena, such as earth quakes, stock markets, and spatial distribution of cities (Krugman, 1996).  In particular, Krugman (1996) found, from population data, that the distribution of cities in the U. S. depicts strong evidence of the power law relationship in that the numbers of cities with populations exceeding certain amounts are linearly related to the population scales of these cities in logarithmic scale.  Though this finding is concerned about inter-city comparison, it has been shown that similar observations were found in intra-city spatial structures (e. g., Batty and Longley, 1995).  Though there would be a close relationship between inter-city and intra-city spatial evolutions, the research focuses here on the latter, setting aside the inter-city interaction effects on urban growth.  In addition, the research concentrates mainly on spatial patterns resulting from interacting development decisions, leaving the question of why self-organization occurs in these patterns for future studies.

3  Research Design

Self-organized criticality has been the criterion of judging whether a complex system possesses the characteristics of self-organization.  In short, self-organized criticality explains how small events in large systems occur and how they aggregately maintain these systems in a critical state that is stable over a long period of time.  The most well-known example of explaining self-organized criticality is the dropping of a set of grains of sand, one at a time, to a pile on a plane (Bak and Chen, 1991).  When the pile reaches a certain height, the dropping of a grain of sand causes a chain of avalanches of different scales, while the slope of the pile remains constant, i. e., a steady state.  I argue that complex spatial systems, e. g., human settlements, tend to evolve toward such self-organized criticality in that in this critical state, a change of land use in a particular location causes a chain effect of land use patterns propagating the whole systems, while some relationship among the traits of the complex spatial systems, such as different land use patterns, holds over time.

3.1 Hypotheses

Though it would be debatable to apply directly the self-organization hypothesis found in natural sciences to studying social phenomena, I argue however that under the general framework of complexity and if addressing the systems at an appropriate level, the resulting patterns among these systems would be comparable independent of the detailed nature of these systems, such as organizations and built environments.  The differences among these systems result mainly from individual actions of the agents constituting the systems and how these agents interact with each other.  If we can focus on the interaction of these actions at a higher level without addressing the underlying processes of these actions, a general framework based on complexity describing the seemingly different complex systems could be developed.  In the context of spatial structures, I hypothesize therefore that land use patterns in cities tend to self-organize spatially into a critical state evidenced by the power law as described in the previous section.

3.2 Empirical Validation

In order to test this hypothesis, I conducted two studies based on real and simulated spatial data respectively.  In the first study, I examined the spatial data derived from current land use maps of two settlements in Taiwan, Min Jen Township and San Tsong City, by first computing fractal dimensions for spatial units in grid systems and then determining whether these dimensions depicted the evidence of self-organization.  In the second study, I designed a computer simulation based on Arthur's (1994; 2015) experiment on interacting individual behavior of bounded rationality to model land development decision making and the interaction among these decisions.  The purpose of this study was to discover through the simulation whether the seemingly uncoordinated actions came up with spatial patterns of self-organization.

3.2.1 The Case Study

In the first case study, I first collected the digital maps for the two areas under consideration (Min Jen Township and San Tsong City) (see Figure 1).  

Figure 1 The land use pattern for Min Jen Township

For each of the two areas, I computed the fractal dimensions for the cells in grid systems.  The grid systems were designed in order to test whether the sizes of cells would affect the distribution of fractal dimensions.  There were six such systems ranging from cell sizes of 50x50, 100x100, 150x150, 200x200, 250x250, to 300x300 squared meters.  The fractal dimensions or semi-fractal dimensions were calculated based on the following formula (Batty, 2005):

D_i = log(n_i+1n_i)log(d_id_i+1), where

d_i : the measurement scale for the ith iteration,

d_i+1 : the measurement scale for the i+1th iteration,

ni : the total number of steps for the ith iteration, and

n_i+1 : the total number of steps for the i+1th iteration.

There were totally six pairs of measurement scales (iterations) which were randomly chosen from the interval between 0 through 2.5.  They were 2.2450 and 1.8625; 2.1925 and 0.3075; 2.2050 and 0.0025; 1.0650 and 0.6050; 2.4550 and 0.5300; and 1.5775 and 0.7200.

3.2.2 The Computer Experiment

Based on the metaphor of sand pile, I conducted a simulation of two dimensional cellular automata and observed the resulting spatial patterns of land use distribution.  In the computer experimental design, each cell in the grid system was allowed for two types of development decisions made by a developer, commercial and residential.  The transition rules were organized through a questionnaire survey on real land developers so that the characteristics of bounded rationality of making land development decisions was considered.  In particular, the development rule for a certain cell changed over time depending on whether that rule had succeeded in the previously same situation encountered.  More specifically, at each time step, the developer's decision on the development of a particular cell depended on its eight neighboring cells according to the surveyed transition rules.  However, if the same local configuration was encountered by the developer in a previous time step, that decision would also depend on whether the previous decision succeeded.  Therefore, the transition rules evolved, or were learned by the developer, according to the developer's previous experience, making the simulation distinct from the regular CA models and more realistic.  

More specifically, the computer experiment was designed on a grid system of 100x100 cells.  In the initial time step, the two types of land uses, i. e., commercial and residential, were randomly assigned to the cells.  Starting from the second time step, the land uses or development decisions for the cells changed according to a set of adaptive transition rules.  The transition rules were designed based on a questionnaire survey from real developers.  For example, if a cell was currently in commercial use, the developer was asked to determine how he or she would have the cell developed at the next time step, given its spatial configuration of the eight neighboring cells.  Different from the traditional cellular automata model (e.g., Wolfram, 2002), the transition rules were not fixed in that each cell “learned” or “adapted” through time by changing its decision rule according to the effectiveness of that rule in previous time steps.  

That is, for each cell, the developer first determined whether the neighboring spatial configuration currently faced had been encountered before.  If it was new to the developer, she or he would select randomly from the rule database a development strategy associated with that configuration; otherwise, the developer would decide whether the associated rule was successful in the previous step based on an interactive payoff table which was also derived from the questionnaire survey.  Only if the associated rule had been a successful development strategy would the developer adopt it in the current situation; otherwise she or he would choose randomly another rule associated with the current spatial configuration.  

There were totally three such simulations: adaptive learning (as depicted above), random walk, and fixation.  In the random walk simulation, the decision rules were randomly chosen from the rule database derived from the questionnaire survey.  In the fixation simulation, a particular set of rules were used throughout the simulation.

4  Results and Discussion

Both the studies showed strong evidence of spatial self-organization.  In the first case study, the resulting scatterplots between r (the radius) and N(r) (the number of cells whose fractal dimensions exceed the mean within the radius r) showed a linear, logarithmic relationship, meaning that the spatial distribution of fractal dimensions was itself fractal in nature.  In the computer experiment, there was no strong evidence of self-organization in the random walk and fixation simulations.  The adaptive learning simulation showed, however, self-organized pattern in that the frequencies of clusters of cells in commercial use and the scales of these clusters were linearly related in logarithmic scales (see Figure 2).

Figure 2 Spatial patterns and cluster patterns from the computer simulation (Time Step 100)

The results implied that self-organized criticality was the attractor of the evolution of such complex spatial systems in that the spatial distribution of land uses depicted a frequency-cluster power law relationship reminiscent of the rank-size relationship observed in spatial distribution of human settlements.  The simulation is arguably simple in its assumptions and design, but it captures the basic behavioral constructs of the land development process.  With a generalization to some extent of idiosyncratic geographic attributes of real spatial settings, the assumption of the homogeneous spatial configuration in the simulation should provide some explanation of the evolution of real complex spatial systems.

The approach to urban spatial modeling introduced here is relative new and still in the experimenting stage.  It needs further research effort, both in methodology and philosophy, before any result could be claimed as practically useful.  The approach is, however, promising in that it deals directly with individual differences and possible links between micro-motives and macro-behaviors, which are exactly what have been missing in the traditional economic approaches.  

5  Conclusions

I conducted two studies based on real and simulated spatial data.  The results of the two studies support our hypothesis that spatial configurations of human settlements tend to self-organize themselves into a critical state as indicated by a power law relationship. The implication derived from both studies is that the self-organization characteristics observed from real spatial data might be caused by the adaptive, learning processes of land development decisions, rather than fixed rules that most cellular automata model seem to adopt.  Though the design of the research was far from complete in that the selection of case areas and the construction of behavioral models of land development decisions were either unstratified or simple minded, it provides intriguing insight into how urban spatial evolution occurs.  Further development could be done by incorporating more rigorous land development behavioral models into the simulation.

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This article is a working paper and has not been officially published.