## 2 Community Detection across Space and Its Variation across Time
### Community Detection across Space  
#### 2.2.1 Similarity of Nodes  
Besides the connectivity property, a transport system has a spatial heterogeneity in urban space. In other words, some land uses are more attractive to each other in terms of transportation, and in the function space they form aggregation patterns, i.e., community. Such community could imply a popular route at a specific time, or an agglomeration of living areas and work areas. Generally, the more intense the traffic flow interaction between land uses is, the higher probability that the land uses have to be grouped together. Since the transport system is modeled as a spatially embedded  
Figure 2: Variation of network with different cell sizes. (**a**,**c**) represent the variation of tessellation using different cell sizes, and (**b**,**d**) are the corresponding networks. network, we can then use the community detection method to extract the clustering patterns of land uses. It should be noted that, compared to the classic graph measures, the concept of community in our context has its own characteristics. Specifically, community detection in a network traffic flow should not only take into account the graph structure factor, but should also consider the traffic flow volumes among different regions. Therefore, we propose that the measures of attraction degree and structure similarity be integrated to group the nodes of network traffic flow. Before defining the attraction degree, we first introduce the related concept, as follows. **Definition 3**. : _(Attraction factor): In a network traffic flow, the traffic volume characteristics between a pair of directly connected nodes reveal their closeness relationship, and we term this connection the attracting factor. For the directly connected nodes \(N_{1}\) and \(N_{2}\), their degrees \(d_{N_{1}}\) and \(d_{N_{2}}\) are defined as the number of their connected edges, respectively. Assuming the edge between \(N_{1}\) and \(N_{2}\) is \(E_{(N_{1},N_{2})}\) with associated weight \(W_{(N_{1},N_{2})}\), the attracting factor \(f_{(N_{1},N_{2})}\) between \(N_{1}\) and \(N_{2}\) is calculated as follows:_  
\[f_{(N_{1},N_{2})}=\ln\left(1+\frac{d_{N_{1}}}{\sum_{j=1}^{d_{N_{1}}}W_{1j}}*W_{ (N_{1},N_{2})}\right) \tag{2}\]  
In Figure 3, there are six nodes, including \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\), \(N_{5}\) and \(N_{6}\), with their associated edges. In this community, \(d_{N_{1}}=3\), \(W_{(N_{1},N_{5})}=1\), \(\sum_{j=1}^{d_{N_{1}}}W_{1j}=3+2+1=6\), \(f_{(N_{1},N_{5})}=\ \ln\left(1+\frac{3}{6}*1\right)=0.4055\), \(d_{N_{5}}=3\), \(W_{(N_{5},N_{1})}=1\), \(\sum_{j=1}^{d_{N_{5}}}W_{1j}=3+1+1=5\), \(f_{(N_{5},N_{1})}=\ \ln\left(1+\frac{3}{6}*1\right)=0.4700\). **Definition 4**. : _(Attraction degree): Assuming nodes \(N_{1}\) and \(N_{2}\) are directly connected, their attracting degree can be measured as following._  
\[Attr_{(N_{1},N_{2})}=\frac{f_{(N_{1},N_{2})}+f_{(N_{2},N_{1})}}{2} \tag{3}\]  
In Figure 3, the attraction degree between \(N_{1}\) and \(N_{5}\) is \(Attr_{(N_{1},N_{5})}=\frac{f_{(N_{1},N_{5})}+f_{(N_{5},N_{1})}}{2}=\frac{0.40 55+0.4700}{2}=0.43775\). Definitions 3 and 4 model the force of attraction between any two directly connected nodes (i.e., cells). However, in reality, a node is attracted not only by its directly connected nodes, but also by its indirectly connected nodes. Therefore, we propose the extension of Definition 4 to take into  
Figure 3: A sample of a community. account the attraction degrees from both the directly connected nodes and the indirectly connected nodes. Assuming nodes \(N_{i}\) and \(N_{j}\) are indirectly connected by the path \(PT\) (\(N_{i}\), \(N_{i+1}\), \(N_{i+2}\),..., \(N_{j}\)), their attractive degree can be calculated as the product of the attractive degrees of all the pairs of directly connected nodes in \(PT\), i.e., \(f_{(N_{i}N_{i+1})}\) (\(i\leq k<j\)).