2.4.4. Agglomeration

Agglomeration can characterize the nature of grouping urban network nodes. Generally, the stronger the connection among urban nodes, the larger the value of the clustering coefficient of urban nodes. The local clustering coefficient ignores the importance of urban nodes in the weighted network; thus, we consider using the local weighted clustering coefficient to measure the structure agglomeration resilience of the urban network. The formula for this is [45]:

$$\mathbb{C}\_{i}^{w} = \frac{1}{k\_{i}(k\_{i}-1)} \sum\_{j,k} \left( w \hat{v}\_{ij} w \hat{v}\_{ik} w \hat{j}\_{k} \right)^{\frac{1}{3}} \tag{13}$$

where *Cwi* is the local weighted clustering coefficient, *ki* is the number of neighbors of node *i*, and *wij*, *wik*, and *wjk* are the weights of the edges among nodes, which are processed using the network maximum weight standardization method. The more urban transmission among nodes and the stronger the interaction capabilities, the higher the dependence among nodes, the less "robust" the network is, the less the city's ability to resist interference from the outside world is, and the more network connections will be disrupted in the event of any local outage; as such, the local weighted clustering coefficient's numerical size and the network structure are inversely proportional to the level of resilience [46].

#### **3. Evaluation of Urban Network Structure Resilience in the Three Provinces of Northeast China**

#### *3.1. Spatial Pattern of the Urban Network Structure*

Based on the information of the connection matrix, transportation connection matrix, innovation connection matrix, and economic connection matrix, the multiple connection networks were classified according to the natural breakpoint method. Further, ArcGIS was used to realize spatial visualization and draw the multiple network connection distribution maps (Figure 3). Through the Quadratic Assignment Procedure (QAP) correlation analysis in UNICET (Table 3), we found that the correlation coefficients among the information network, transportation network, innovation network, and economic network are significantly correlated at the 1% level, indicating that the multiple urban networks in the three provinces of Northeast China exhibit strong correlation characteristics. Among them, the correlation between the transportation network and the economic network is the highest (0.610), and the correlation between the innovation network and the economic network is the lowest (0.307); this indicates that there are certain similarities among the multiple urban networks in the three provinces, but, at the same time, high differences also exist. Hence, it is necessary to further explore the structure resilience characteristics of each network.

**Figure 3.** Distribution of the network connection strength of multiple cities in the three provinces of Northeast China.

**Table 3.** Correlation coefficients of multiple connection networks in 34 cities in the three provinces of Northeast China.


Note: \*\*\* indicates passing the 1% significance test.

In order to reflect the urban network spatial pattern of the three provinces in Northeast China more clearly, the natural breakpoint method is used to classify the element connections between cities, and ArcGIS is used to visualize them [10]. The urban networks in the three provinces demonstrate an overall spatial pattern of "dense in the north and sparse in the south", but each network also has different characteristics (Figure 3). The urban information network has a multi-center network structure with Shenyang, Dalian, Harbin, and Changchun as the core (Figure 3a), whereby the first level constitutes a "cross" spatial pattern, showing a more complex network pattern than other networks. The difference of the urban transportation network is more evident (Figure 3b), influenced by spatial proximity, such that the first level of the transportation network is mainly the connection among Shenyang-Fushun, Anshan-Liaoyang, and Shenyang-Benxi, and the fifth level mainly constitutes the connection among urban nodes. The overall connection of the urban innovation network is looser than that of other networks, and the connections among city nodes are relatively weak (Figure 3c). The intra-provincial linkage of the innovation

network is closer, and the cross-provincial linkage is mainly between provincial capital cities and sub-provincial cities. The first level of the urban economic network in Liaoning Province presents an "N"-shaped structure, with closer intra-provincial ties and closer inter-provincial ties than the transportation network and innovation network (Figure 3d). The connections among urban nodes in the integrated network are more complicated than those in other networks, and the connections among cities are closer, and the urban network structure is more robust (Figure 3e).

#### *3.2. Urban Network Structure Resilience*

#### 3.2.1. Network Hierarchy

Urban nodes with strong radiation and dispersal capabilities in the information network and transportation network are mainly Harbin, Changchun, Shenyang, and Dalian, indicating that the provincial capital cities and sub-provincial cities are leading in terms of socio-economic development, and the rest of the cities are dependent on them owing to their spatial proximity and radiation-driven effects, resulting in a high hierarchy of city nodes around them (Figure 4a,b). The cities with a high hierarchy in the innovation network are Changchun and Jilin, which form a single core pattern in space with evident polarization characteristics (Figure 4c). The spatial distribution of economic network hierarchy shows a "ridge-type" trend with "Harbin–Changchun–Shenyang–Dalian" as the axis, decreasing from the middle to both ends (Figure 4d). In the integrated network, the urban nodes located at the fifth level are mainly Harbin, Changchun, Shenyang, Dalian, and Anshan, and a spatial axial development trend is evident (Figure 4e).

**Figure 4.** Spatial distribution of weighted degree and weighted average nearest-neighbor degree of the multi-city linkage network in the three provinces of Northeast China.

We drew the weighted degree distribution fitting curve according to the weighted degree calculation results of each urban node (Figure 5). From the slope of each curve (0.883 < |a| < 3.235), we can determine that all kinds of networks have strong hierarchical characteristics. Among them, the curve slope of the innovation network has the largest value (|a| = 3.235), which indicates that the innovation network has the highest hierarchical level, the core position of the urban node is more prominent, and the three-dimensional development trend of the innovation network is evident. The curve slope of the economic network has the second highest value (|a| = 1.603), indicating that the network has a more evident hierarchical structure of urban nodes. The transportation network (|a| = 1.431) and the information network are less hierarchical (|a| = 0.883), and the high-value areas are mainly provincial capitals and sub-provincial cities, which show spatial homogeneity. The slope of the integrated network curve (|a| = 1.117) is relatively smooth compared with that of the innovation, economic, and transportation networks, indicating that the degree of external connection of urban nodes under the integrated network is relatively reasonable, and the difference in the hierarchical level among urban nodes is not significant; relatively speaking, the integrated network shows a flat development.

**Figure 5.** Distribution of the weighted degree of the urban network in the three provinces of Northeast China.

#### 3.2.2. Network Matching

The NWAD values of urban nodes with higher hierarchy in each urban network are smaller (Figure 4). In the integrated network, the number of urban nodes with NWAD in the first group is the largest, and they are mainly the urban nodes with higher values of weighting degree, which, to some extent, indicates that there are more communication and contact paths between the core urban nodes and other urban nodes, which is more conducive to the flow of elements among nodes.

The weighted degree correlation of all five types of network coefficients is negative (−0.041 < b < −0.008), indicating that the information, transportation, innovation, economic, and integrated networks all have heterogeneous characteristics. Among them, the information network has the most evident heterogeneity (b = −0.041), and the urban nodes with higher weighted values can maintain good interaction with the nodes at the same level and can also communicate and cooperate with the urban nodes at different levels. The transportation network has strong heterogeneity (b = −0.027), and the city nodes with evident transportation advantages have a radiating and driving effect on their neighboring cities. In addition, the well-connected transportation network also contributes to the development of regional linkages, and the path connections among city nodes tend to be heterogeneous. The heterogeneity characteristics of the economic network are weak (b = −0.015), with strong mobility of economic factors among core cities but weak mobility of economic factors among peripheral cities, with significant spatial differences in the intensity of the flow of economic factors and the low structural resilience of the economic network. The heterogeneity of the innovation network is not evident compared with other networks (b = −0.008), and the phenomenon of homogeneous grouping exists, the connection among nodes in core cities and nodes in peripheral cities is weak, and the cross-regional exchange and cooperation regarding the innovation factor flow are restricted. The combined network heterogeneity is lower than that of the information network and higher than that of the transportation, economic, and innovation networks (b = −0.025), indicating that there may be a certain degree of bias in measuring the structural resilience of urban networks based on a single factor flow. The combined effect of multiple factor flows can enhance the "robustness" of the connection paths among urban nodes to some degree and jointly improve the level of the urban network.

#### 3.2.3. Network Transmission and Agglomeration

The spatial difference of the economic network transmission is the most evident, followed by that of information, integrated, and transportation networks. Meanwhile, the innovation network transmission has the smallest spatial variability (Figure 6), in which the information network transmission shows an overall hierarchical structure that increases from south to north. Moreover, the spatial pattern of the transportation, economic, and integrated network transmission has a certain similarity, and the innovation network transmission has a multi-core distribution pattern. Shenyang exhibits the most significant influence on the transmission of the information and innovation networks. In the information network, there are three groups that have the same degree of influence on the network transmission after the failure of city nodes: Songyuan and Baicheng, Qiqihar and Jixi, and Mudanjiang and Heihe; the degrees of influence are 0.3808, 0.3818, and 0.3836, respectively. Two groups of cities in the innovation network have the same impact on the innovation network transmission: Anshan and Jinzhou, Baishan and Jixi; the network efficiency after the failure of the city node is 0.4363 and 0.4484, respectively, and the network efficiency after the failure of city nodes in Changchun contributes the most to the transport network transmission, while Anshan, Suihua, Fushun, Tieling, Yingkou, Benxi, Tonghua, and Mudanjiang have the same degree of influence on the network efficiency, at 0.1467, 0.1485, 0.1488, and 0.1491, respectively. The urban node that bears the main transmission function in the economic network is Harbin, in which, the cities of Tonghua-Dandong and Jilin-Jixi-Hegang have equivalent influence on the transmission of the economic network, and the network efficiency after the failure of the city node is 0.6139 and 0.6169, respectively.

In the integrated network, Shenyang is the city node that undertakes the main transmission function, and Qitaihe and Baishan have the lowest transmission function.

**Figure 6.** Spatial distribution of the transmission efficiency of the multi-city connection network in the three provinces of Northeast China.

The average weighted clustering coefficients of the information, transportation, innovation, economic, and comprehensive networks are all approximately 0.2, indicating that the clustering characteristics of each type of network are weak, and with the expansion of the network scale, the core urban nodes have a wider radiation range, and other city nodes rely on the core city nodes to achieve cross-regional cooperation. The local weighted clustering coefficients of the rest of the networks except the transportation network show that the local weighted clustering coefficients of provincial capital cities and sub-provincial cities are in the first group (Figure 7). This indicates that the relationship between provincial capital cities and sub-provincial cities and other cities is not very close, but rather, there is a one-way connection between other city nodes and core city nodes, whereby there is less interactive cooperation among other city nodes, such that the node is not ye<sup>t</sup> evident. The interaction and cooperation among other city nodes are lower, and no evident network has been formed. From the perspective of network structural resilience, the weak connection between core city nodes and other city nodes facilitates the penetration of external information, thereby enhancing the "robustness" of the city network in response to external information interference.

**Figure 7.** Spatial distribution of the local weighted clustering coefficients in the multi-city connection network of the three provinces of Northeast China.

#### 3.2.4. Urban Node Type Identification

The city nodes whose transmission and agglomeration are located in the first and second groups are regarded as dominant city nodes. There are five city nodes, which are provincial capital cities (Harbin, Shenyang, and Changchun), sub-provincial cities (Dalian), and resources-based cities (Daqing and Anshan). The cities in this category are in prominent positions in diverse networks, with strong comprehensive strength, leading the construction and development of the neighboring cities. The city nodes with both transmission and agglomeration in the fourth and fifth groups are considered as vulnerable city nodes, such that there are nine city nodes in total, namely, Songyuan, Tonghua, Baicheng, Jixi, Yichun, Heihe, Hegang, Shuangyashan, and Baishan, all of which are located in Jilin Province and Heilongjiang Province; most of them are peripheral cities with a low level of socio-economic development and imperfect public service facilities (Figure 8). Under the effect of administrative barriers, vulnerable city nodes are distant from provincial capital cities. As such, in considering future construction and development, it is necessary to heed and support the development of such cities and enhance their capability to cope with unexpected risks.

**Figure 8.** Spatial distribution of the types of urban nodes in the three provinces of Northeast China.

#### **4. Influencing Factors of Urban Network Structure Resilience in the Three Provinces of Northeast China**

#### *4.1. Variable Selection*

Exploring the factors influencing urban network structure resilience can provide relevant references for optimizing the resilience of the urban network structure. Presently, research on urban network structure resilience is still in the exploration stage, and there are few studies on the factors influencing urban network structure resilience. The resilience of the urban network structure is the result of multiple factors interacting and working together, combined with existing research results on the urban network structure and urban resilience [47–51]. The four properties of comprehensive urban network structure resilience are used as dependent variables. Economic scale, knowledge thickness, political status, geographic conditions, urban vitality, governmen<sup>t</sup> capacity, openness, labor wages, and science and education level are used as the drivers affecting the resilience of the urban network structure (Table 4).


**Table 4.** Influencing factors of urban network structure resilience.

#### *4.2. Regression Results*

The least-squares method was used to analyze the influencing factors of urban network structure resilience, and the regression results are shown in Table 5. These results show that the R2 is between 0.455 and 0.793, which can explain 45.50% to 79.30% of urban network structure resilience. Overall, the fitting effect of the least-squares method is more suitable.

**Table 5.** Regression results of influencing factors.


Note: \*, \*\*, \*\*\* indicate passing the 10%, 5%, and 1% significance test, respectively.

The best fit for the hierarchical nature of the city network structure, by controlling the other variables, denotes that political status, urban vitality, and labor wages can have a significant positive effect on enhancing its hierarchical characteristics, and all pass the 1% significance level test. The improvement of governmen<sup>t</sup> capacity also has an effect, albeit to a lesser extent, but passes the 5% significance level test, consistent with the results for provincial capital cities and sub-provincial cities as core city nodes. Under the interaction of political status, urban vitality, labor wages, and governmen<sup>t</sup> capacity, the non-heterogeneous pattern of the urban network structure is evident.

In terms of urban network structure matching, the regression results of its influencing factors are similar to those of hierarchy, in which governmen<sup>t</sup> capacity, political status, and urban vitality significantly impact it negatively. Hence, it is necessary to focus on the interaction and cooperation between provincial capital cities and sub-provincial cities through the macro-regulatory role of the government, enhance the radiation capacity of core city nodes, and promote the development of other urban nodes through core urban nodes.

The transmission of the urban network structure is significantly affected by political status, governmen<sup>t</sup> capacity, and knowledge thickness, and the transport function of the provincial capital cities and sub-provincial cities occupies a significant position in the entire network. After the failure of such city nodes, the transmission of the urban network structure will be significantly affected, and the network efficiency will decrease. Therefore, it is necessary to enhance the transmission efficiency of other urban nodes to ensure that the urban network can maintain regular operation in unexpected situations.

The agglomeration of the urban network structure is mainly related to urban vitality, governmen<sup>t</sup> capacity, and knowledge thickness. The local weighted clustering coefficient of urban nodes with a large number of urban residents and a large total number of patent applications is lower, which is consistent with the results of the previous analysis. Therefore, we need to emphasize improving the level of science and technology within the region and strengthening the construction of transportation infrastructure to promote the flow of information, transportation, innovation, economy, and other factors among city nodes to realize regional interaction and cooperation.
