## 3 Methodology  
Inspired by HGSM [28] and the spreading ability of the node in GGC [26], we propose Hybrid Community-based Gravity Centrality (HCGC). Our method considers both the local and global information of each node in a network. Additionally, the method also includes the local modularity calculated by the Louvain algorithm due to its popularity for community detection [30]. This local modularity is used to improve the local influence scores \((LI)\). HCGC is then computed by using \(LI\) as masses in the gravity model with a truncation radius \(R\) to capture the global information of the network. This approach has the potential to identify the vital hubs for transportation networks. The HCGC is defined by the following equation:  
\[LI(v)=e^{\frac{LM(v)}{N}}\times DC^{+}(v)\times Ks(v) \tag{9}\]  
where \(N\) is the number of nodes in a network. \(LM(v)\) is the local modularity calculated by the community detection algorithm, which measures how well each individual node fits within its assigned community. Equation (10) defines the local modularity.  
\[LM(v)=\frac{E_{internal}(v)}{E_{total}(v)}=\frac{\sum_{u\in N(v)}\delta(c_{v}, c_{u})}{k_{v}} \tag{10}\]  
where \(E_{internal}(v)\) counts edges connecting node n to nodes in the same community. \(E_{total}(v)\) is the degree of node v. \(N(v)\) is the set of neighbors of node v. \(\delta(c_{v},c_{u})\) is 1 if nodes v and u are in the same community, 0 otherwise. \(k_{v}\) is the degree of node n.  
The community assignment \(\delta(c_{v},c_{u})\) is calculated by the Louvain algorithm which is widely used for community detection in networks [30]. The Louvain algorithm maximizes global modularity \(Q\) to determine communities. This global modularity \(Q\) measures the quality of the entire community partition, while local modularity measures how well each individual node fits within its assigned community. The global modularity \(Q\) can be expressed as:  
\[Q =\frac{1}{2m}\sum_{v,u}\biggl{[}A_{uv}-\frac{k_{v}k_{u}}{2m} \biggr{]}\delta(c_{v},c_{u}) \tag{11}\] \[=\frac{1}{2m}\biggl{[}\sum_{v,u}-\frac{\sum_{v}k_{v}\sum_{u}k_{u} }{2m}\biggr{]}\delta(c_{v},c_{u})\]  
\[\delta(c_{v},c_{u})=\begin{cases}0,\,c_{v},c_{u}\,\text{belong to different community}\\ 1,\,c_{v},c_{u}\,\text{belong to same community}\end{cases}\]  
where \(A_{uv}\) is the weight of the edge between nodes \(v\) and \(u\), whose value is 1 for unweighted networks. \(k_{v}\) and \(k_{u}\) are the sums of weights of edges attached to nodes \(v\) and \(u\). \(m\) is the sum of all edge weights in the graph. \(c_{v}\) is the community to which node \(v\) belongs.  
HCGC represents the impact of neighbor nodes within a truncation radius \(R\) following gravity-based centrality measures [26; 29].  
\[HCGC(v)=\sum_{d_{uv}\leq R,v\
eq u}\frac{LI(v)\times LI(u)}{d_{uv}^{2}} \tag{12}\]  
Algorithm 1 shows the pseudocode of the HCGC.  
```
0: Network \(G(V,E)\) with nodes labels \(0,1,\ldots,N\), \(R\)
0:\(HCGC\) values for all nodes
1: Calculate \(LM\) using method in Equation (10) for all nodes
2: Calculate degree and average neighbor degree \(DC^{+}\) for all nodes
3: Calculate \(Ks\) using K-shell decomposition method for all nodes
4:for each node \(v\) in \(V\)do
5:\(LI(v)\gets e^{\frac{LM(v)}{N}}\times DC^{+}(v)\times Ks(v)\) // Calculate \(LI\) for node \(v\) according Equatio (9)
6:endfor
7:for each node \(v\) in \(V\)do
8:\(HCGC\gets 0\)
9:\(R_{v}\leftarrow\) a set of \(R\)-neighborhood nodes of node \(v\)
10:for each node \(u\) in \(R_{v}\)do
11:\(HCGC+=\frac{LI(v)\times LI(u)}{d\sigma_{v}}\) // Calculate the \(HCGC\) according to Equation (12)
12:endfor
13:endfor
14:return\(HCGC\)
```  
**Algorithm 1** The algorithm of HCGC measure