Degree-biased diffusion geometry of networks. Implications for network neurosciences

Degree-biased diffusion gives unequal directional influence to the two ends of an undirected edge. Its propagator is therefore nonsymmetric, and it generates two geometries. The column geometry compares the responses to perturbations at different sources, and the row geometry compares the observations across the same family of perturbations. We show that both geometries are Euclidean at finite times, but they can differ in their transient behavior and in their long-time limits. In contrast with standard diffusion, individual distances can expand temporarily even while the dynamics approach consensus. Degree-biased diffusion conserves a degree-weighted quantity instead of the ordinary network mass. An exact transport-and-reservoir decomposition identifies the local exchange responsible for this difference. Degree-biased Laplacians also arise exactly as the linearization of the dynamic mean-field model of large-scale brain activity when the long-range synaptic coupling is modulated by inverse receiver and sender gains. This gives them a neurobiological interpretation and a physical time scale. On a 76-region human structural connectome, degree bias changes the pairwise proximity, the geometrically optimal routes, and the regions visited by those routes. Hubs-attracting diffusion produces an earlier and higher maximum of column signature information than standard diffusion, while it reduces the row maximum. Hubs-repelling diffusion increases both maxima, but it reaches them later. The number of source responses needed to reconstruct the row kernel also depends strongly on the sampling order. In particular, high-degree sources are favored under attraction and low-degree sources under repulsion. These results characterize how a fixed network can generate distinct relational geometries under different propagation laws. Here, we treat geometric routes as distinct from literal signal trajectories, and mathematical reconstruction as distinct from evidence of conscious access.