## 4 Methodological Approach
### Network Centrality Analysis  
In this step, the diagonals of the correlation matrix given in Table 3 were replaced with 0 s, and this matrix was then used as the input to the network centrality to identify and evaluate the relative significance of challenges within the network, providing a deeper understanding of their structural importance and interconnectivity. Two of the most common metrics for assessing the significance of nodes in a network that were used were betweenness centrality and eigenvector centrality [93; 94]. These measures offer distinct but complementary perspectives on node importance. Betweenness centrality measures the degree to which a node acts as a bridge positioned along the shortest paths between other nodes in a network, highlighting nodes that act as bridges or intermediaries. Such nodes are critical for controlling and influencing communication or resource flow within the network. Eigenvector centrality provides a more detailed view of centrality by taking into account not just the quantity of a node's connections but also the quality of those connections. A node with fewer direct links can still achieve high eigenvector centrality if its connections are to other highly influential nodes. The formulas for computing these two metrics are defined in Equations (1) and (2).  
\begin{table}
\begin{tabular}{c c} \hline
**Code** & **Challenges** \\ \hline C11 & “Lack of motivation and empowerment for employees” \\ C12 & “Lack of incentives and rewards” \\ C13 & “An absence of a customer-centric mindset to gauge and meet current and future needs” \\ C14 & “The service-oriented-thinking gap between managers and employees” \\ C15 & “Overlapping functions between departments” \\ C16 & “Absence of innovation strategy” \\ C17 & “Unclear value proposition for the new or improved service” \\ C18 & “Lack of coordination and poor internal communication between government entity departments” \\ C19 & “A gap in marketing internally for new or improved services” \\ C20 & “Lack of integration between government entities for the shared services” \\ C21 & “Absence of clear and formalized innovation process” \\ C22 & “Regulations and legislation that limit service innovation” \\ \hline \end{tabular}
\end{table}
Table 2: Cont.  
Using these two formulas, the computed betweenness centrality and eigenvector centrality values for the 22 challenges are presented in Table 4. These metrics collectively capture the different dimensions of a node's importance, enabling a comprehensive understanding of its role within the network.  
\[BC_{(n_{k})}=\sum_{i<j}\frac{g_{ij}(n_{k})}{g_{ij}} \tag{1}\]  
where \(BC_{(n_{k})}\) is the betweenness centrality of node \(n_{k}\), \(g_{ij}\) is the number of shortest paths connecting node \(n_{i}\) and node \(n_{j}\), and \(g_{ij}(n_{k})\) = the number of shortest paths connecting node \(n_{i}\) and node \(n_{j}\) passing through node \(n_{k}\).  
\[X_{i}=\frac{1}{\lambda}\sum_{j}A_{ij}X_{j} \tag{2}\]  
\(X_{i}\) is the eigenvector centrality of a node \(i\), \(\lambda\) is a constant representing the largest eigenvalue in the adjacency matrix \(A\), \(A_{ij}\) is the adjacency matrix indicating the connection between nodes \(i\), and \(j\), \(X_{j}\) is the eigenvector centrality of the neighboring node \(j\).