## 2 Methodology
### Intercity Efficiency Model  
[PERSON] [26] and [PERSON] [36] introduced a methodology to assess the effectiveness of transportation networks by integrating population density with circuity. Incorporating the actual population distribution as a factor in the circuitry weight leads to varied transportation efficiencies in regions with similar network structures, but differing population distributions. Therefore, this indicates that these networks possess varying degrees of importance in enhancing the overall transportation efficiency within the city, aligning with typical trends. The calculation of the new model relies on the traffic flow between various city pairs. However, obtaining population data with high precision presents a challenge and serves as a bottleneck. Due to the expansive geographical coverage of city clusters, obtaining accurate statistics on passenger flows between city pairs is challenging, costly, and time consuming and might not be available in less-developed areas. Numerous studies utilize digital traffic data mining techniques to estimate passenger travel flow. However, the application of this method to extensive and intricate regions is presently hindered by limitations in data availability and computational capabilities. This study references the works of [PERSON] [40] and [PERSON] [41] and posits that the likelihood of traveling from city \(i\) to city \(j\) is directly related to the appeal of city \(j\) to city \(i\). Variations in resource allocation and industrial distribution among cities within the urban agglomeration result in distinct interrelationships between each pair of cities. The intensity of these interrelationships directly influences the distribution pattern of intercity passenger flows [42]. The gravity model [43] combines the spatial distribution of different regions with their attributes, to measure the level of connectivity between these regions. It represents the prevailing model utilized in research concerning the extent of spatial ties and predicting travel distribution. The basic principle states that the strength of spatial connection is assumed to be directly proportional to the product of the masses of two cities and inversely proportional to the square of the distance between them. This model is characterized by its simplicity and broad adaptability. Consequently, this study employs the gravity model to forecast intercity travel distribution within urban agglomerations. Its calculation formula is as follows:  
\[A_{ij}=\frac{\sqrt{P_{i}\cdot GDP_{i}}\times\sqrt{P_{j}\cdot GDP_{j}}}{d_{ij}} \tag{4}\]  
where \(A_{ij}\) is the attractiveness of city \(i\) to city \(j\); \(P_{i}\) and \(P_{j}\) are regional demographic indicators for the two cities, usually expressed in terms of total population; and \(GDP_{i}\) and \(GDP_{j}\) denote the gross product of the two cities. This study collects pertinent data at the prefecture-level city level, which is then aggregated to the city cluster scale. The primary data utilized comprise economic and social indicators, such as population and GDP. The data are sourced from the China Urban Statistical Yearbook, China Regional Economic Statistical Yearbook, as well as provincial and municipal statistical yearbooks. Therefore, the movement of traffic between cities can be expressed as follows:  
\[q_{ij}^{a}=q_{i}^{a}\cdot\frac{A_{ij}^{a}}{\sum\limits_{j=1,j\
eq i}^{\frac{m _{a}}{a}}A_{ij}^{a}} \tag{5}\]  
where \(q_{ij}^{a}\) is the passenger traffic flow from city \(i\) to city \(j\) in city cluster \(a\); \(n_{a}\) is the number of cities in city cluster \(a\); \(A_{ij}^{a}\) is the attractiveness of city \(i\) to city \(j\) and it can be calculated usingthe gravity model; and \(q_{i}^{a}\) is the total passenger traffic flow in city \(i\), which represents the total number of passengers departing from that city in a given period of time. To simplify the calculation, this study uses the total population of the city as a proxy [26].