Heterogeneous Connectivity in Sparse Networks: Fan-in Profiles, Gradient Hierarchy, and Topological Equilibria

Profiled Sparse Networks (PSN) replace uniform connectivity with deterministic, heterogeneous fan-in profiles defined by continuous, nonlinear functions, creating neurons with both dense and sparse receptive fields. We benchmark PSN across four classification datasets spanning vision and tabular domains, input dimensions from 54 to 784, and network depths of 2--3 hidden layers. At 90% sparsity, all static profiles, including the uniform random baseline, achieve accuracy within 0.2-0.6% of dense baselines on every dataset, demonstrating that heterogeneous connectivity provides no accuracy advantage when hub placement is arbitrary rather than task-aligned. This result holds across sparsity levels (80-99.9%), profile shapes (eight parametric families, lognormal, and power-law), and fan-in coefficients of variation from 0 to 2.5. Internal gradient analysis reveals that structured profiles create a 2-5x gradient concentration at hub neurons compared to the ~1x uniform distribution in random baselines, with the hierarchy strength predicted by fan-in coefficient of variation ($r = 0.93$). When PSN fan-in distributions are used to initialise RigL dynamic sparse training, lognormal profiles matched to the equilibrium fan-in distribution consistently outperform standard ERK initialisation, with advantages growing on harder tasks, achieving +0.16% on Fashion-MNIST ($p = 0.036$, $d = 1.07$), +0.43% on EMNIST, and +0.49% on Forest Cover. RigL converges to a characteristic fan-in distribution regardless of initialisation. Starting at this equilibrium allows the optimiser to refine weights rather than rearrange topology. Which neurons become hubs matters more than the degree of connectivity variance, i.e., random hub placement provides no advantage, while optimisation-driven placement does.