## 3 Application in Age-Development fMRI Data
### Extracted Smoothness Harmonics and Their Merits  
The analysis of the smoothness harmonics, both in terms of spatial smoothness brain maps (\(\mathbb{E}_{\text{spat}}(f)\)) and temporal smoothness index (\(\mathbb{E}_{\text{temp}}(f)\)), reveals distinct cortical dynamics between children and adults. Figure 3A illustrates the results, showing that children exhibit overall more stable brain activity throughout the movie clip compared to adults. This finding aligns with previous research suggesting that children may demonstrate greater attentiveness during cartoon-watching situations compared to adults [14]. Of particular interest is the significant difference observed in the spatial smoothness of brain maps between adults and children (Figure 3B). In cortical regions associated with higher cognitive functions and the development of the theory of mind (caring about others' thoughts), such as the superior frontal sulcus (SFS), right temporoparietal junction (Right TPJ) [26], and the inferior frontal gyrus (IFG) [27], notable distinctions between children and adults are observed. These findings are consistent with previous studies investigating the neural basis of developmental differences between children and adults in developing the theory of mind [28; 29; 30]. The use of smoothness harmonics as discriminative features in inter-subject similarity and between-group classification analyses yielded interesting results. Figure 3C and the first row of Table 1 demonstrate that highly consistent inter-subject brain patterns were identified when using smoothness harmonics for both adults and children. In contrast, the utilization of the static functional connectomes (sFCs) as an alternative only achieved compatible inter-subject similarity for the children's group, as indicated by the overall score of 0.628. It is noteworthy that the overall score for smoothness harmonics was much lower (0.446) compared to that of the sFC approach. This lower score can be interpreted as a failure of the sFC approach to effectively capture the group-wise differences between adults and children. These results highlight the superior performance of smoothness harmonics as discriminative features in capturing meaningful variations and distinctions in cortical dynamics between different age groups. \begin{table}
\begin{tabular}{c c c} \hline  & **sFCs** & **Smoothness Harmonics** \\ \hline \multirow{3}{*}{Inter-subject similarity} & Overall: 0.628 & Overall: 0.446 \\ \cline{2-3}  & Between-adults: 0.318 & Between-adults: 0.654 \\ \cline{1-1}  & Between-children: 0.624 & Between-children: 0.632 \\ \hline \multirow{2}{*}{Between-group classification accuracy (\%)} & \multirow{2}{*}{\(64.6\pm 2.12\)} & \multirow{2}{*}{\(80.7\pm 1.13\)} \\ \cline{1-1}  & & \\ \end{tabular}
\end{table} 
Table 1: Usages of sFCs and smoothness harmonics in inter-subject similarity and between-group analyses. Figure 3: Extracted smoothness harmonics in the age-development fMRI dataset [14]. (**A**) Attained temporal smoothness indexes for the group-wise adult and child fMRI data. The raw indexes were normalized to the range of \([0,1]\) for ease of comparison. (**B**) Differences in the extracted spatial smoothness harmonics between adults and children are represented in a pairwise spatial format, with the color bar indicating the harmonic value difference between the two groups. The three most-differentiated brain regions, highlighted in light red colors, are the superior frontal sulcus (SFS), the right temporoporarietal junction (Right TPJ), and the inferior frontal gyrus (IFG). The produced brain maps are displayed for the hemispheres from coronal, sagittal, and horizontal views from left to right. (**C**) The inter-subject similarity was assessed based on either the static functional connectome (sFC) on the left or the estimated smoothness patterns on the right. In both correlation matrices, the first 33 subjects represent adults, while the remaining subjects represent children. The color bar indicates the cosine similarity.