[Editor's Note]
In traditional city networks, the way cities interact usually appears in the form of city pairs, where an edge representing interaction connects two cities and forms the basic unit of the city network. Even the cluster characteristics and community structure between multiple cities are still studied from the basic unit of each city pair. In this issue, we will interpret the complex interaction between cities from the perspective of high-order networks, and jointly explore the future development direction of urban networks.
1. From city to city "group"
If one of the world's cities were to map out a network of interactions, it would be easy to imagine that cities within the same country would have similar properties and characteristics. For example, in the global city network, China's cities generally show the characteristics of aggregate advantage and per capita disadvantage, and the urban network constructed by some aggregate indicators (such as total population flow, foreign direct investment, etc.), and the domestic cities represented by Shanghai, Beijing and Shenzhen are generally located in the center of the network, but some per capita indicators, such as the number of per capita producer service institutions and per capita foreign investment, are difficult for these cities to enter the core layer of the network. Cities such as New York, Los Angeles, San Francisco, and Chicago in the United States generally have nominal per capita GDP and will also show an advantage in per capita indicators in the global city network.
In the same way, cities in the same cultural circle and cities with similar historical contexts will also be reflected in similar clusters in the big picture of the global city network, so although the city network constructed by a single element stream clearly clusters these cities, it cannot depict other interactions between these clusters. These obvious clusters have done little to express the interactive characteristics of clusters and capture the effects of other unknown variables.
Some network scholars are beginning to wonder how to consider various types of interactions in a network, such as whether cities such as Shanghai, Beijing, and Shenzhen can appear in the world's network of cities as a cluster of cities, after all, they all exhibit some degree of similarity. In addition, the ability to represent multiple forms of interaction between different cities in a single diagram (in addition to the multi-layer network approach). High-order city networks with hypergraph and simplex as the main tools provide a great way to do this.
2. Hypergraph and simplex: two ways to express higher-order networks
The so-called hypermap is the promotion form of the general network. In the traditional network, we generally build a network based on a single element traffic, the nodes of the network are also a single city, and the edges of the network represent a single form of interaction. Supermap provides a form of promotion, where multiple cities can work together as a node. As a simple example, if we compare a city to a person, the traditional city network is an individual in our WeChat address book, which can only have a point-to-point conversation. And the high-level city network is the groups in our WeChat address book. Different groups will form an associated "edge" because of one or more people, which is called hyperedge.
In Figure 1, the author plots a simple network of cities, with Beijing, Shanghai, and Hong Kong all being metropolises in China with similar cultural characteristics. New York, Chicago, and Los Angeles are all metropolises in the United States with more cultural interaction; Hong Kong and Singapore are both cities with complete economic characteristics and similar economic attributes; Singapore and New York have more cooperation and interaction in finance. Based on these interactive characteristics, a high-level network of these seven cities is constructed, and thus a super-edge representing various attributes such as national characteristics, city characteristics, and financial cooperation is formed.
Another form of higher-order network is simplex, which is a topological structure in which several vertices are connected to each other. To put it bluntly, it is the generalized form of line segments and triangles. For example, a two-dimensional simplex is a triangle, in a triangle, the three vertices are all connected to each other (like a group of three people), and a three-dimensional simplex is a tetrahedron, and the four vertices are connected to each other. Therefore, a k-dimensional simple complex is a group of k+1 cities.
Also taking the above-mentioned high-order city network as an example, Beijing, Shanghai and Hong Kong form a group, forming a two-dimensional simplex, New York, Chicago and Los Angeles are another simplex, and Hong Kong and Singapore, Singapore and New York are all one-dimensional simplexes. Such a higher-order network is essentially a network complex composed of four simplexes, sometimes called a simplicial complex.
Fig.1 Urban network expressed in hypergraphs and simplex Source: The author
3. Future directions and challenges
On the basis of this concept, an important question is, what is the difference in nature between such urban networks and traditional urban networks? At present, high-order networks are still in their infancy in complex networks, and their application and practice in urban networks have not yet been carried out on a large scale. Therefore, these differences are not yet known, but the special properties of some higher-order network dynamics provide some inspiration for exploration.
For example, in a higher-order network, a node is not only related to its neighbors, but also affected by the cluster in which it is located, and the impact of higher-order interactions on nodes is intertwined and complex. In addition, the conditions required for critical changes in some traditional networks will also change in higher-order networks, such as the critical phase transition of the network and the steady state of the network. But there are also some similar aspects, such as some basic characteristics such as the centrality of the network, which will still be there, and so on.
At the same time, there are some practical problems in interpreting urban networks in the form of higher-order networks, among which the most important problem is how different interaction modes and factor flow modes between cities affect the characteristics and attributes of higher-order urban networks. How to construct effective high-order network indicators to measure and interpret cities is also in its infancy. Nevertheless, with the improvement of the functions of the large platform of the city, it is still a bright direction to use high-level networks to depict the interaction of the complex platform economy and society.
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Hosted by Dr. Dai Yuehua of the Shanghai Institute of Development Strategy, the column focuses on the cutting-edge trends of urban science development, explains the general characteristics and laws of urban science and human behavior dynamics in cities, and explores the paths and methods of using cutting-edge urban science theories to optimize urban governance.