Systemic Resilience of Complex Urban Systems
Abstract
Two key paradigms emerge out of the variety of urban forms: certain cities resemble trees, others leaves. The structural difference between a tree and a leaf is huge: one is open, the other closed. Trees are entirely disconnected on a given scale: even if two twigs are spatially close, if they do not belong to the same branch, to go from one to the other implies moving down and then up all the hierarchy of branches. Leaves on the contrary are entirely connected on intermediary scales. The veins of a leaf are disconnected on the two larger scales but entirely connected on the two or three following intermediary scales before presenting tiny tree-like structures on the finest capillary scales. Deltas are leaves not trees. Neither galaxies nor whirlpools are trees.
We will see in this paper that historical cities, like leaves, deltas, galaxies, lungs, brains and vein systems are all fractal structures, multiply connected and complex on all scales. These structures display the same degree of complexity and connectivity, regardless of the magnification scale on which we observe them. We say that these structures are scale free. Mathematical fractal forms are often generated recursively by applying again and again the same generator to an initiator. The iteration creates an arborescence. But scale free structure is not synonymous with a recursive tree-like structure. The fractal structure of the leaf is much more complex than that of the tree by its multiconnectivity on three or more intermediary levels. In contrast, trees in the virgin forest, even when they seem to be entangled, horizontal, and rhizomic, have branches that are not interconnected to form a lattice.
As we will see, the history of urban planning has evolved from leaf-like to tree-like patterns, with a consequent loss of efficiency and resilience. Indeed, in a closed foliar path structure, the formation of cycles enables internal complexification and flow fluctuations due to the possibility of flow transfers, as is the case of historical cities.
One of the central demonstrations in this paper is that an urban system’s structural resilience is highest when it is configured according to a scale free structure for its parts and for its connections. The spatial distribution and the intensity of connections in such a structure obeys a Pareto distribution – that is, an inverse power law found throughout the organization of living organisms and economic systems. The scale relationships between the different hierarchic levels of an arborescence, a leaf, and the blood and oxygen circulation systems in our bodies obey such a mathematical law. It states the frequency of an element’s appearance and the span of a connection based on its hierarchic level: the smaller an element is, the more often it will be encountered in the system; the bigger an element is the rarer it will be. This fundamental law defines in itself the manner in which living organisms and things should be organized to optimize their access to energy, the use that they make of it, and their resilience
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