## 2 Materials and Methods
### The ABIDE Dataset
\begin{table}
\begin{tabular}{l l c c} \hline \hline
**Site ID** & **Site Name** & **ABIDE** & **Total (ASD, TD)** \\ \hline CALTECH & California Institute of Technology & 1 & 38 (19, 19) \\ \hline CMU & Carnegie Mellon University & 1 & 27 (14, 13) \\ \hline KKI & Kennedy Krieger Institute & 1 & 55 (22, 33) \\ \hline LEUVEN\_1 & University of Leuven & 1 & 29 (14, 15) \\ \hline LEUVEN\_2 & University of Leuven & 1 & 35 (15, 20) \\ \hline MAX\_MUN & Ludwig Maximilians University Munich & 1 & 57 (24, 33) \\ \hline NYU & NYU Langone Medical Center & 1 & 184 (79, 105) \\ \hline OHSU & Oregon Health and Science University & 1 & 28 (13, 15) \\ \hline OLIN & Qin Neuropsychiatry Research Center & 1 & 36 (20, 16) \\ \hline PITT & University of Pittsburgh School of Medicine & 1 & 57 (30, 27) \\ \hline SBL & Social Brain Lab & 1 & 30 (15, 15) \\ \hline SDSU & San Diego State University & 1 & 36 (14, 22) \\ \hline STANFORD & Stanford University & 1 & 40 (20, 20) \\ \hline TRINITY & Trinity Centre for Health Sciences & 1 & 49 (24, 25) \\ \hline UCLA\_1 & University of California Los Angeles & 1 & 114 (65, 49) \\ \hline UCLA\_2 & University of California Los Angeles & 1 & 27 (13, 14) \\ \hline UM\_1 & University of Michigan & 1 & 110 (55, 55) \\ \hline UM\_2 & University of Michigan & 1 & 35 (13, 22) \\ \hline USM & University of Utah School of Medicine & 1 & 101 (58, 43) \\ \hline YALE & Yale Child Study Center & 1 & 56 (28, 28) \\ \hline BNI\_1 & Barrow Neurological Institute & 2 & 58 (29, 29) \\ \hline EMC\_1 & Erasmus University Medical Center Rotterdam & 2 & 54 (27, 27) \\ \hline ETH\_1 & ETH Zurich & 2 & 37 (13, 24) \\ \hline GU\_1 & Georgetown University & 2 & 106 (51, 55) \\ \hline IP\_1 & Institute Pasteur & 2 & 56 (22, 34) \\ \hline IU\_1 & Indiana University & 2 & 40 (20, 20) \\ \hline KKI\_1 & Kennedy Krieger Institute & 2 & 211 (56, 155) \\ \hline KUL\_3 & Katholieke Universiteit Leuven & 2 & 28 (28, 0) \\ \hline NYU\_1 & NYU Langone Medical Center & 2 & 78 (48, 30) \\ \hline NYU\_2 & NYU Langone Medical Center & 2 & 27 (27, 0) \\ \hline OHSU\_1 & Oregon Health and Science University & 2 & 93 (37, 56) \\ \hline OILH\_2 & Qin Institute of Living Hartford & 2 & 59 (24, 35) \\ \hline SDSU\_1 & San Diego State University & 2 & 58 (33, 25) \\ \hline SU\_2 & Stanford University & 2 & 42 (21, 16) \\ \hline TCD\_1 & Trinity Centre for Health Sciences & 2 & 42 (21, 21) \\ \hline U\_MA\_1 & University of Miami & 2 & 28 (13, 15) \\ \hline
**Total** & All sites & 1 and 2 & 2226 (1060, 1161) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of subjects for each site of the ABIDE I and ABIDE II collections. The site IDs and the extended site names are reported, together with the sample composition in terms of the number of subjects with autism spectrum disorders (ASD) and the typically developing controls (TD), and the total number of subjects. Because of the male vs. female unbalance in the ASD condition [16,17], which also occurs in the composition of the ABIDE datasets, we restricted our analysis to the cohort of male subjects. Moreover, due to the under-representation of subjects older than 40 years in the ABIDE datasets, only subjects with ages between 5 and 40 years were considered. We also excluded the sites NYU_2 and KUL_3 since, as noticeable from Figure 1, they did not provide controls but only ASD subjects. In fact, without control data, it is not possible to run the harmonization procedure illustrated in Section 2.4. The list of the IDs of subjects of the ABIDE dataset used in this study is provided in the Supplementary Materials (Table S1). After adopting the selection criteria above, we obtained a dataset consisting of 1470 male subjects coming from ABIDE I and ABIDE II, of which 737 are TD controls and 733 ASD, from 36 sites. Figure 1: Bar diagram showing the number of subjects acquired at each site for each diagnostic group (**a**). Box plots showing the age distribution of subjects per site (**b**). In both plots, the sites are sorted by increasing median age.