## 2 Methodology
### Graph Isomorphism Network (GIN)  
The graph isomorphism network (GIN) is formulated to accept networks as its input. [PERSON] et al. introduced GIN as a specific instance of spatial GNN well-suited for tasks involving graph classification [22]. The network operationalizes the aggregate and combine functions by summing the node features, as depicted in Equation (1) below.  
\[h_{v}^{(k)}=\ MLP^{(k)}((1+\in^{(k)})\cdot\ h_{v}^{(k-1)}+\sum_{ueN(v)}h_{u}^{( k-1)}) \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Adolescent Stage** & **Age Range** & **Total Subjects** & **Sex Distribution** \\ \hline Pre & 8–12 & 129 & 55 male/74 female \\ \hline Early & 12–14 & 113 & 58 male/55 female \\ \hline Middle & 14–16 & 109 & 54 male/55 female \\ \hline Late & 16–18 & 130 & 59 male/71 female \\ \hline Post & 18–22 & 141 & 53 male/88 female \\ \hline \hline \end{tabular}
\end{table}
Table 2: Distribution of subjects in adolescence.  
Figure 1: The distribution of sexes in the various stages of adolescence.  
In Equation (1), \(\epsilon^{(k)}\) represents a parameter that is adaptable at the \(k\)th layer, while MLP denotes a multi-layer perceptron incorporating non-linearities. The notation \(h_{v}^{(k)}\) signifies the feature vector at the \(k\)th layer for the \(v\)th node. The feature vector corresponds to layer \(k\), where it integrates the previous node feature \(h_{v}^{(k-1)}\) with aggregated node features to output the node feature of the current \(k\)th layer, denoted as \(h_{v}^{(k)}\). For a graph-level readout, the embedded node features from each layer are summed and subsequently concatenated to derive the final graph feature \(h_{G}\), as illustrated in Equations (2) and (3) [38].  
\[h_{G}^{(k)}=\text{sum}(h_{0}^{(k)},\;h_{1}^{(k)},\dots,h_{N}^{(k)}) \tag{2}\]  
\[h_{G}=\text{concatenate}(\text{h}_{G}^{(\mathbf{k})}\;|\;k=0,\;1,\dots,K) \tag{3}\]  
[PERSON] et al. asserted that the proposed network architecture has the capacity to acquire an injective mapping of the function \(g\), potentially endowing the model with comparable effectiveness to the WL test for tasks involving graph classification [22; 38].  
\[h_{v}^{(k)}=\;\emptyset\left(h_{v}^{(k-1)},\;f\left(\left\{h_{u}^{(k-1)}:u \in N(v)\right\}\right)\right) \tag{4}\]  
In Equation (4), \(h_{v}^{(k)}\) is the feature vector of node \(v\) at the \(k\)th layer or iteration. \(N(v)\) is the set of nodes adjacent to \(v\), and the equation aggregates and updates node features, where the function \(f\) operates on a multiset of node features and \(\phi\) is the injective function. Furthermore, the WL test employs a pre-defined injective hash function \(g\) to update the node labels \(l_{v}\):  
\[l_{v}^{(k)}=g\left(l_{v}^{(k-1)},\left\{l_{u}^{(k-1)}\:u\in N\;(v)\right\}\right) \tag{5}\]  
Figure 2: Overview of the proposed framework. The fMRI time-series data were preprocessed and divided into 264 regions of interest (ROIs) based on the Power coordinates template. Then, the mean BOLD time course data were extracted, and the Pearson correlation coefficient between ROIs was calculated and represented as a functional connectivity matrix. Then, the functional network graph was input to our MGIN model to classify sex. Lastly, the GNNExplainer method was applied to identify critical subnetwork structures and to identify important ROIs.