Scale-free behavior of weight distributions of connectomes

To determine the precise link between anatomical structure and function, brain studies primarily concentrate on the anatomical wiring of the brain and its topological properties. In this work, we investigate the weighted degree and connection length distributions of the KKI-113 and KKI-18 human connectomes, the fruit fly, and of the mouse retina. We found that the node strength (weighted degree) distribution behavior differs depending on the considered scale. On the global scale, the distributions are found to follow a power-law behavior, with a roughly universal exponent close to 3. However, this behavior breaks at the local scale as the node strength distributions of the KKI-18 follow a stretched exponential, and the fly and mouse retina follow the lognormal distribution, respectively which are indicative of underlying random multiplicative processes and underpins non-locality of learning in a brain close to the critical state. However, for the case of the KKI-113 and the H01 human (1mm$^3$) datasets, the local weighted degree distributions follow an exponentially truncated power-law, which may hint at the fact that the critical learning mechanism may have manifested at the node level too.