A Livable City is a Complex but Small Network under the Topological Perspective
A livable city is not just a place where basic needs are met, but a dynamic environment that fosters well-being, accessibility, and sustainability for its inhabitants. Such urban spaces are not linear constructs but complex systems in their essence. More specifically, urban spaces can be depicted as networks comprised of various nodes and their corresponding edges, such as the urban streets ranging from vast highways to modest pedestrian paths and their junctions within the streets (Jiang and Claramunt 2004). To construct such spaces, it is crucial to enhance network connectivity through the reduction of distances or, in other words, make the network smaller—not just in terms of geometric Euclidean distances but also topological distances. The topology refers to the properties and spatial relations unaffected by the continuous shape changes of geometric figures. For example, consider a city where a large park and a residential area are separated by a river. In terms of Euclidean distance, these two places might appear to be far from each other, especially if the river is wide. However, if a well-designed bridge or a series of interconnected pedestrian paths are introduced, the topological distance between the park and the residential area significantly decreases. Therefore, a livable city should be a small network with close topological connections or small topological distance.
The concept of a small network, which is summarized as the small world network (Watts and Strogatz 1998), is characterized not just by its minimal separation from distant nodes but also by a high degree of clustering among neighboring ones. The advantages of small-world networks lie in their ability to maintain tight-knit community connections while ensuring rapid dissemination of information or resources, thus striking a balance between efficiency and connectivity. The architecture of such networks is quantitatively described by two principal metrics: the average path length (L) and the clustering coefficient (C). A diminished average path length signifies the ease with which any two nodes can connect through a limited series of steps, whereas a heightened clustering coefficient reflects the network’s propensity for nodes to create tightly-knit clusters. Consequently, an optimally designed network would exhibit a high C value alongside a low L value, embodying the quintessential characteristics of small-world networks that facilitate both robust community engagement and topologically efficient communication pathways.
The Watts-Strogatz (WS) small-world model provides a mathematical framework for understanding the balance between global connectivity and local clustering within complex networks. Introduced as a means to mathematically model the small world phenomenon, the WS model bridges the gap between two traditionally distinct types of networks. It delineates three key representations of networks: regular, random, and the small-world itself. In a regular network, nodes are connected to all their nearest neighbors in a lattice-like structure, exhibiting high clustering (C) but long path lengths (L) since it only connects to the nearest neighbors. Conversely, a random network is characterized by nodes connected in a haphazard manner, resulting in low clustering (C) but short path lengths (L). The small-world network ingeniously combines elements of both: it maintains the high degree of clustering inherent in regular networks while also achieving the short path lengths typical of random networks. This hybrid structure is achieved by introducing a small number of random links to a regular lattice, thereby drastically reducing the average path length between nodes without significantly disrupting the network’s local clustering properties.
Small-world networks demonstrate the overall connectivity of complex networks through a topological approach, while the connectivity among individual nodes statistically exhibits a long-tail distribution where there are far more less-connected nodes than well-connected ones. This long-tail distribution describing the relationship between the number of nodes (y) and their corresponding degree of connectivity (x) can be modelled by the power law: y=x-α, with α representing the power-law exponent. This power law is scale-free, meaning it maintains the same form regardless of the scale of observation (Barabási and Albert 1999). The scale-free nature of this relationship indicates that the network’s structure is invariant to size; as the network grows, the proportion of less-connected to well-connected nodes remains constant in the power law distribution. Taking urban street networks as an example, this principle can significantly impact urban planning and traffic management. In such networks, most roads (less-connected nodes) serve local neighborhoods, while a few major roads and highways (well-connected nodes) handle the bulk of the traffic, connecting different parts of the city efficiently. These different classes of roads are organized as a scaling hierarchy, and to ensure the smooth operation of the traffic system, it often suffices to ensure that the major roads remain unobstructed. Conversely, if the road network were uniformly distributed, then minor changes could lead to significant disruptions throughout the network. The robustness of scale-free networks lies in their ability to localize disruptions, thereby maintaining overall system efficiency and reducing the likelihood of widespread traffic congestion.
PageRank (Page and Brin 1998) is an algorithm designed to compute the most significant nodes or hubs within networks. Unlike simpler metrics such as connectivity, which merely tallies the number of direct links a node possesses, PageRank assesses both the quantity and the quality of these connections. It is predicated on the understanding that links from highly ranked nodes bolster the significance of a node more than links from lower-ranked ones. In essence, PageRank sees a link from Node A to Node B as a vote of confidence from A to B, with votes from more pivotal nodes having greater impact. However, the similarity between node significance and connectivity lies in their scale-free nature, manifesting a long-tail distribution where far more nodes are either less-connected or less significant than a small number of well-connected or crucial nodes. By employing head/tail breaks (Jiang 2013), it is possible to identify these priority nodes and uncover the underlying hierarchical structure of the network, thereby highlighting the most significant nodes crucial for maintaining the stability of the complex network.
A city can be regarded as a complex network. More importantly, a livable city is a small world network characterized by small topological distances. The WS small-world network model exhibits networks with high clustering coefficients and short average path lengths, signifying not only tightly interconnected locales but also a high degree of global integration. Statistically, all nodes follow a long-tail distribution that fits a power law, indicating a prevalence of far more less-connected nodes than well-connected ones. This distribution is scale-free, displaying similar statistical distributions across different levels of nodes, thus contributing to the system’s stability, as the overall distribution often remains unchanged even when the state of individual nodes changes. Additionally, the importance of nodes can also be calculated using the PageRank algorithm. The principles of complex networks provide profound insights into urban systems, offering not just an understanding of the city’s inherent connections and dynamics, but also introducing innovative perspectives and strategies for urban planning and development, making cities more resilient and livable spaces.
References:
- Barabási A. L. and Albert R. (1999), Emergence of scaling in random networks, Science, 286(5439), 509–512.
- Jiang B. and Claramunt C. (2004), Topological analysis of urban street networks, Environment and Planning B: Planning and design, 31(1), 151–162.
- Jiang B. (2013), Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution, The Professional Geographer, 65(3), 482–494.
- Page L. and Brin S. (1998), The anatomy of a large-scale hypertextual Web search engine, Proceedings of the seventh international conference on World Wide Web, 7, 107–117.
- Watts D. J. and Strogatz S. H. (1998), Collective dynamics of ‘small-world’ networks, Nature, 393(6684), 440–442.