## 2 Materials and Methods
### Node Importance  
The adjacency matrix resulting from the WGCNA analysis was preprocessed to contain only the top 10% of the scores extracted from the TOM. Then, it was loaded using the _igraph_[27] Python package to make a weighted graph of co-expressed genes. We checked the connectivity of the genes in the module, i.e., the sum of the weights of all the edges connected to a given node. A higher total connectivity implies stronger and more frequent interactions between the nodes and the presence of multiple pathways in order to propagate information through the network. Conversely, a lower total connectivity suggests that the nodes are less connected, which may result in fewer interactions and fewer avenues for information to flow. Local topological metrics were computed for nodes in each network module using Syntacle [32]: (i) _degree_ centrality, i.e., the number of edges incident to a node. One way to understand degree centrality is to consider that every edge in a network can be thought of as a path of length 1. Therefore, degree centrality can be seen as a measure that counts the number of these paths that originate from a particular node; (ii) _betweenness_ centrality, i.e., the number of shortest paths between all pairs of nodes in the graph that pass through that node. In other words, it measures how often a node appears on the shortest path between any two other nodes in the graph. Nodes with high betweenness centrality are important because they act as \"bridges\" between different parts of the graph, allowing for efficient communication and the flow of information; (iii) _closeness_ centrality, i.e., the average shortest distance from each node to each other node. This is an inverse measure because larger values indicate lower or reduced centrality, as opposed to higher values indicating greater centrality; and (iv) _clustering coefficient_, i.e., the extent to which the neighbors of a node are close to being a clique. If a node has a high clustering coefficient, it suggests that its neighboring nodes are closely connected to each other. On the other hand, if the clustering coefficient of a node is low, it indicates that its neighboring nodes are less connected to each other. The overall clustering coefficient of a network is calculated by taking the average of the clustering coefficients of all the nodes in the network. We also measured the density of the modules by resorting to the _completeness_ index [33; 34]. A key-player-based analysis was also performed by computing the (i) \"Key Player Problem/Negative\" (KPP-Neg) index, which gives insights on the contribution of a given node to the cohesiveness of a network (breaking down a network into smaller components, or in cases where fragmentation is not possible, increasing the path lengths between nodes to a degree that they become almost disconnected), and the (ii) \"Key Player Problem/Positive\" (KPP-Pos) index, which instead focuses on the level of connectivity/embedding of nodes within a network [16], in other words, identifying nodes that can establish direct links or short paths to the maximum number of nodes. Therefore, we computed a KPP-Neg-based measure of network fragmentation, the \({}^{D}F\) index, in order to check if the removal of key players leads to a significant increase in the number of disconnected components, and two KPP-Pos-based indices: \({}^{D}R\), which is the weighted proportion of all nodes reached by a set of nodes, which is also known as the \"kp-set\", and \(m\)-reach, which is the count of the number of unique nodes reached by any member of the kp-set in \(m\) links or less. In this study, we considered only \"fast paths\" by setting \(m=3\). The key-player analysis was performed on groups of nodes of size 1, namely on individual nodes. Finally, genes composing the module of interest were subjected to gene set enrichment analysis (GSEA) using PANTHER [35] against the Gene Ontology (GO), which was run with default parameters.