High-Dimensional Sparsity is Necessary for Brain-Scale Information Reachability

We show that sparse connectivity in the human brain enables full network reachability with orders of magnitude fewer connections than a fully dense graph of equivalent size. Using simple k-hop reachability as a proxy for functional information propagation, we compare a brain-like network with N ~8.6 x 10^10 neurons, each connected to d, ~10^4 other neurons, to a fully dense graph of the same node count. Although both networks reach the same number of unique nodes, the dense network requires 10^7 times more edges. This demonstrates that high-dimensional sparsity is a practical necessity for brain-scale information systems.