Understanding Urban Complexity through the Perspective of Living Structure
Urban spaces are complicated, comprising a wide array of subsystems such as transportation networks, infrastructure facilities, energy supply systems and so on. But more crucially, urban spaces exhibit complexity, denoting that these subsystems are not only complicated in terms of size but also well-connected among each other like a network or semilattice. This complexity of such urban spaces lies in its inherent living structure that evokes people’s sense of wellbeing to the urban living environment (Alexander 2002–2005). The concept of living structure views the city as a coherent whole with each of its subspaces hierarchically organized. For example, a stable transportation system is constructed extending from highways to urban main roads, and further down to neighborhood streets and sidewalks, rather than being made up solely of isolated lanes. Similarly, larger cities often support smaller cities by providing infrastructure and economic opportunities that enhance the overall functionality of the region. Therefore, the purpose of understanding urban complexity is not only about creating additional habitable spaces but transforming the city into a more stable system capable of addressing complex challenges.
Living structure is a mathematical concept that underlies the notion of urban space complexity, composed of inner substructures or subspaces. Thus, the degree of urban complexity is determined by both the number of substructures and their inherent hierarchy. In other words, the more substructures the city has, the more complex the city is, and the higher hierarchy of the substructures, the more complex the city is. The hierarchy ubiquitously emerges with the phenomenon of there are far more small substructures than large ones as mentioned in the examples from last paragraph. In particular, it is not about more smalls than larges, but “far more” indicates the disproportion between smalls and larges, and their occurring numbers. Furthermore, the notion of far more smalls than larges recurs across scales or on a global scale rather just once.
Statistically, the hierarchy in the concept of living structure exhibits a long-tail distribution in the relationship between scale and the number of substructures, displaying an “L” shape on the rank-size plot. This distribution deviates from the Gaussian, embracing nonlinearity, where small causes can lead to disproportionately large effects. An illustrative example of this is the butterfly effect, where the mere flap of a butterfly’s wings in Brazil could trigger a massive tornado in Texas, or how a single driver’s abrupt stop can lead to a major traffic jam. This nonlinearity creates complexity and heterogeneity across scales, meaning that in this distribution a very small number of large events or structures often control the main behavioral characteristics of the system. This nonlinearity also indicates that urban spaces cannot be predicted linearly like simple systems, but complex interactions and dynamic changes need to be considered, so their behavior patterns and development trends are usually unpredictable and changeable.
The scaling relationship or scaling law between scale (x) and number of details (y) of far more smalls than larges can be represented by the power law: y=x-α, with α representing the power-law exponent and larger α indicates a longer tail in the distribution. This power law is scale-free, maintaining the same form at any observation scale, hence considered average-independent. For instance, in wealth distribution, a small portion of the population controls the majority of wealth, a pattern that remains constant across different average wealth levels. In contrast, height distribution is scale-bounded, with most people’s heights clustering around the mean, and shifts in the average height lead to significant variations in individual heights. The scale-free nature signifies that the long-tail distribution remains stable and unaffected by minor scale variations. Moreover, scale-free properties are recursive, meaning the predominance of smaller details over larger ones recurs at various scales. However, representing the long-tail distribution solely with power laws is not comprehensive. Nonlinear phenomena and scaling laws of “far more smalls than larges” can also be described using functions like logarithms and exponents. Therefore, the nonlinearity in urban complexity is often statistical rather than precise that can be depicted in limited mathematical representations, but it can be simply derived from head/tail breaks (Jiang 2013) to detect the scaling law if the notion of far more smalls than larges recurs at least twice.
The hierarchical organization of urban space is characterized by two distinct attributes: spatial heterogeneity and spatial homogeneity. On the one hand, cities are a coherent whole consist of far more small subspaces than large ones across all scales, so called the scaling law. Different scales of subspaces form a hierarchy, and the characteristics of features within each level are not the same. For example: The preferred destinations for affluent and low-income individuals are likely to differ significantly. Wealthier individuals often opt for upscale shopping centers, while those with lower incomes tend to gravitate towards free public parks for recreation. On the other hand, there are more or less similar elements available at each scale, and these elements are interrelated, with closer elements being more closely related, so called the Tobler’s law. For example, two affluent individuals are more likely to visit similar types of tourist destinations rather than opting for locations similar to those frequented by low-income individuals. Therefore, living structure is said to be governed by these two fundamental laws (Jiang 2019). Among the two laws, the scaling law is the first, or dominant law, as it is universal, global, and across scales, while Tobler’s law is available locally or on each of the scales.
Living structure provides a new perspective for not just understanding the urban complexity but also creates organized complexity for making and remaking the urban environments more livable or living (Alexander 2003). The key aspect of understanding complexity is to perceive the urban spaces as a living structure or a whole that consists of far more small substructures than large ones across all scales, and more or less similar sized substructures on each of the scales. The notion of living structure is defined organically in a bottom-up manner from the substructures and derived hierarchy, which means the whole is more than the sum of its parts instead of the whole is equal to the sum of its parts. In terms of urban design and planning, the concept of living structure implies that subspaces must be differentiated at different scales yet to be adapted to each other at similar levels of scale. This distinguishes it from traditional disciplines like biology and physics, which do not focus on creating complexity, whereas urban science can create more vibrant and efficient societal structures through the view of living structure.
References:
- Alexander C. (2002–2005), The Nature of Order: An essay on the art of building and the nature of the universe, Center for Environmental Structure: Berkeley, CA.
- Alexander C. (2003), New Concepts in Complexity Theory: Arising from studies in the field of architecture, an overview of the four books of The Nature of Order with emphasis on the 15 scientific problems which are raised, http://natureoforder.com/library/scientific-introduction.pdf.
- Jiang B. (2013), Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution, The Professional Geographer, 65(3), 482–494.
- Jiang B. (2019), Living structure down to earth and up to heaven: Christopher Alexander, Urban Science, 3(3), 96.