## 3 Results on Philadelphia Neuro Developmental Cohort (PNC) Data
### Dynamic Functional Network Connectivity Analysis  
After the PNC data was reduced to a \(264\times 124\) matrix for each subject, the FC can be obtained by calculating the Pearson correlations coefficient (PCC) for each pair of ROIs. As we have discussed in the Introduction, the traditional method of calculating FC is based on the assumption that the functional network of the brain is static over the entire scanning session. However, many recent studies have shown that the functional network changes dynamically over time, and more meaningful information of brain activation patterns can be mined by considering the presence of temporal variability [46; 47]. \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Children** & **Young Adults** \\ \hline Number & 193 & 204 \\ Gender (male/female) & 91/102 & 81/123 \\ Age (Mean \(\pm\) SD, months) & 124.06 \(\pm\) 11.33 & 231.50 \(\pm\) 12.14 \\ \hline Ethnicity & & \\ ASIAN & 3 (1.5\%) & 0 (0\%) \\ AFRICAN & 77 (39.9\%) & 74 (36.3\%) \\ AMERICAN & 0 (0\%) & 2 (1\%) \\ OTHER/MIXED & 20 (10.4\%) & 17 (8.3\%) \\ CAUCASIAN/WHITE & 92 (47.7\%) & 111 (54.4\%) \\ HAWAIIAN/PACIFIC & 1 (0.5\%) & 0 (0\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Demographic characteristics of the subjects. Figure 2: Flowchart for investigating the developmental differences. Resting-state functional magnetic resonance (rs-fMRI), blood oxygen level dependent (BOLD), functional connectivity (FC), intrinsic connectivity networks (ICNs). In this work, the dFC is estimated by using the sliding window approach along with time series. For this method, a window is moved along the resting-state fMRI time series and the correlation between two ROIs is calculated for each window. Assume that the time series has \(T\) time points, thus there are \(N=[(T-l)/s]+1\) sub-sequences generated by the sliding time window with window length \(l\) and step size \(s\). It is critical to choose the appropriate length of the windows, since longer windows are closer to the whole signal while shorter windows are more sensitive to transient changes and may introduce more noise to the correlation analysis [48]. To effectively capture dynamic information, a window length between 30 s and 60 s is suggested, which can reflect the cognition of an individual [48; 49]. For the PNC data, we choose \(l=20\) and \(s=1\). In this way, each time window has 20 time points with the window length of 60 s and the step of 1 s. Therefore, for each subject, as the sliding window moves, the 124 time points of the whole BOLD signal will be divided into \([(124-20)/1]+1=105\) sub-sequences. Based on this, we calculate the PCC between every two ROI regions within every sliding window. For each subject, if we denote the time series in the \(n\)-th time window of ROI \(i\) (\(i\in\{1,2,\cdots,264\}\)) as \(\{x_{ik}^{n}\}_{k=1}^{l}\), then the PCC between the \(s\)-th ROI and the \(t\)-th ROI in the \(n\)-th time window is calculated as follows  
\[r_{st}^{n}=\frac{\sum_{k=1}^{l}(x_{sk}^{n}-\bar{x}_{sn})(x_{tk}^{n}-\bar{x}_{tn })}{\sqrt{\sum_{k=1}^{l}(x_{sk}^{n}-\bar{x}_{sn})^{2}}\sqrt{\sum_{k=1}^{l}(x_ {tk}^{n}-\bar{x}_{sn})^{2}}} \tag{7}\]  
where  
\[\bar{x}_{sn}=\frac{1}{l}\sum_{k=1}^{l}x_{sk}^{n},\quad\bar{x}_{sn}=\frac{1}{l }\sum_{k=1}^{l}x_{tk}^{n}.\]  
Thus, considering the symmetry of PCC for two ROIs, for each subject, the dFC feature matrix is \(D\in\mathbb{R}^{N\times C_{264}^{2}}\) with \(N=105\) being the number of sliding windows, where \(C_{264}^{2}=264\times(264-1)/2=34,716\). For each subject, the average connectivity information is further extracted from matrix \(D\) by calculating the mean of each column. The obtained feature vector \(V\in\mathbb{R}^{34716}\) for each subject is a description of the average dynamic functional connectivity between each of two ROIs. The calculation approach of the feature vector for each subject is shown in Figure 3.  
Figure 3: The flowchart for calculating the feature vector of each subject. Pearson correlations coefficient (PCC).