We first sought to test the extent to which cost conservation shapes the topol- ogy of the human connectome by implementing a pure geometric model (i.e. K ( u, v ) = 1). For each participant we tuned the free parameter, η , to a range where the geometric model consistently produced synthetic networks with near-minimal energies (Figure 1B) and analyzed the top 1% lowest-energy synthetic networks (100 networks/participant). At this point in parameter space ( η = − 4 . 01 ± 0 . 31; sample mean ± standard error; see Figure1C), synthetic networks achieved an average energy of E = 0 . 29 ± 0 . 02 with KS statistics KS$_{k}$ = 0 . 15 ± 0 . 03, KS$_{b}$ = 0 . 18 ± 0 . 04, KS$_{e}$ = 0 . 27 ± 0 . 03, and KS$_{c}$ = 0 . 29 ± 0 . 02 (Figure 1B). To contextualize these scores, we compared them to KS statistics obtained from a null generative model where connections were formed with uniform probability. We found that, with the exception of KS$_{e}$ ( p ≈ 0 . 4; Wilcoxon signed-rank test (Wilcoxon, 1945)), the geometric model produced significantly lower energy and smaller KS statistics (maximum p ≈ 10 − $^{5}$). Interestingly, the point at which energy is minimized deviates from the re- spective minima of KS$_{e}$ and KS$_{c}$ , demonstrating that even the-best fitting synthetic networks generated by the geometric model cannot simultaneously match observed connectomes in terms of clustering and edge length distribu- tions. The reason for this is intuitive: A strong distance penalty is required to generate highly clustered networks, which inadvertently penalizes the formation of long-distance connections. Conversely, realistic edge length distributions arise when the distance penalty is relatively weak, at which point synthetic networks become vastly under-clustered. The energy minimum occurs at a point situated between these two extremes, trading off accuracy along one dimension with the other though never simultaneously minimizing both (Figure 1D). 9