Fig. 4

Relationship to rich-club and physical distance of functional connections. The top left a illustrates the results of the gradient comparisons and rich-club taxonomy, showing gradient reductions in feeder and rich-club nodes in ASD compared to controls (based on Student’s t-tests) but increases in local nodes. The box plot indicates the mean ± SD. The top right a shows the vertex-wise association of between-group differences in gradient scores (t-statistic of the map from Fig. 1b; shown on the y-axis) and the average geodesic distance of functional connections of a given vertex (x-axis). The positive association indicates that regions with long-range connections tend to show gradient reductions (Pearson correlation coefficient r = 0.6). The bottom left b panel shows the proportion of visited nodes as a function of SFC steps in controls and ASD, illustrating the lower proportion of active rich-club vs local nodes from 100–200 steps. The bottom right b shows significantly shorter average connectivity distances in autism at these higher steps, established using Student’s t-tests with multiple comparisons correction