## 2 Community Detection across Space and Its Variation across Time
### Community Detection across Space
#### 2.2.2 Algorithm
For example, if \(S_{1}=S_{2}\), \(Ssim_{(S_{1},S_{2})}=1\); if \(S_{1}\cap\ S_{2}=\emptyset\), \(Ssim_{(S_{1},S_{2})}=\ 0\)._

Equation (10) only measures the similarity between two node sets (i.e., two communities), each of which plays a different role in the corresponding graph structure. Specifically, for a graph, some communities are more important than others, and, in order to measure the global similarity between two graphs (i.e., two sets of communities), we propose calculation of the sum of the weighted similarity between two sets of communities. In this process, we define the weight of a community as its contribution rate in the corresponding graph. In general, the more nodes the community has, the largercontribution rate it has for the whole system. Assuming the graph \(C\) consists of the communities {\(C_{1}\), \(C_{2}\),..., \(C_{k}\)} and \(S_{1}\), \(S_{2}\),..., \(S_{k}\) are the node sets of \(C_{1}\), \(C_{2}\),..., \(C_{k}\), respectively, the contribution rate of \(C_{i}\) (\(i\) = 1, 2,..., \(k\)) is defined as follows:

\[comR_{C_{i}}=\frac{S_{i}}{S} \tag{11}\]

Based on the contribution rate of community, we then calculate the weighted similarity between two network traffic flows as follows. **Definition 7**. : _(Similarity of spatially embedded networks): Let \(C_{p}\) and \(C_{q}\) be two spatially embedded networks with k-size community structures {\(C_{1}^{p}\), \(C_{2}^{p}\),..., \(C_{k}^{p}\)} and \(C_{1}^{q}\), \(C_{2}^{q}\),..., \(C_{k}^{q}\)}, respectively. \(comR_{1}^{p}\), \(comR_{2}^{p}\),..., \(comR_{k}^{p}\) are the contribution rates of the communities \(C_{1}^{p}\), \(C_{2}^{p}\),..., \(C_{k}^{p}\), respectively, and \(comR_{1}^{q}\), \(comR_{2}^{q}\),..., \(comR_{k}^{q}\) are the contribution rates of the communities \(C_{1}^{q}\), \(C_{2}^{q}\),..., \(C_{k}^{q}\), respectively. \(S_{1}^{p}\), \(S_{2}^{p}\),..., \(S_{k}^{p}\) are the node sets of the communities \(C_{1}^{p}\), \(C_{2}^{p}\),..., \(C_{k}^{p}\), respectively, and \(S_{1}^{q}\), \(S_{2}^{q}\),..., \(S_{k}^{q}\) are the node sets of the communities \(C_{1}^{q}\), \(C_{2}^{q}\),..., \(C_{k}^{q}\), respectively. Then, the similarity between the two graphs \(C_{p}\) and \(C_{q}\) is defined as:_

\[FsimC=\max_{i_{1},i_{2}-i_{k}}\frac{2*PsimC_{(i_{1},i_{2}-i_{k})}}{\sum_{j=1}^ {k}\left(comR_{j}^{p}+comR_{j}^{q}\right)} \tag{12}\]

_where \(PsimC_{(i_{1},i_{2},\ldots,i_{k})}\) is calculated as follows:_

\[ \begin{array}{c}PsimC_{(i_{1},i_{2}-i_{k})}=\frac{1}{2}*\left(comR_{1}^{p}+ comR_{i1}^{q}\right)*Sim_{(S_{1}^{p},S_{1}^{q})}+\frac{1}{2}*\left(comR_{2}^{p}+ comR_{i2}^{q}\right)*Sim_{(S_{2}^{p},S_{1}^{q})}\\ +\ldots+\frac{1}{2}*\left(comR_{k}^{p}+comR_{ik}^{q}\right)*Sim_{(S_{1}^{p},S _{k}^{q})}\end{array}  \tag{13}\]

_where \((i_{1},i_{2}\ldots i_{k})\) is a full permutation of the set \(I=\{1,2\ldots k\}\)._

The community structure is a partition of all the land uses (i.e., nodes) of the network traffic flow for a specific time slice. Hence, the variation of networks across time can be analyzed by measuring the similarity between the corresponding community structures. The better the matching between two community structures, the more similar the corresponding traffic conditions in different time slices. ## 3 Data Sets and Settings

We conducted a series of experiments to explore the transport system of Beijing city using taxi trajectory points. As the capital of China, Beijing is the national political, economic, and administrative center. By the end of 2016, the number of taxis in Beijing had reached 71,600, with a permanent population of 21.729 million. More than 55% of residents take taxis every week [32]. In this paper, we chose the central zone of the city (i.e., the range of latitude 39\({}^{\circ}\) 49\({}^{\circ}\) 41\({}^{\circ}\)-39\({}^{\circ}\) 59\({}^{\circ}\) 17\({}^{\circ}\) N and longitude 116\({}^{\circ}\) 15\({}^{\circ}\) 47\({}^{\circ}\)-116\({}^{\circ}\) 29\({}^{\circ}\) 09\({}^{\circ}\) E) as the study region (Figure 4). The research area is within the third ring area of Beijing.