Weight geometry governs functional memory in complex systems

Abstract: Complex systems, from gene regulatory networks to neural circuits and transportation infrastructures, exhibit rich functional behaviour that topology alone does not capture. Here we show that functional memory exhibits a universal organisational regularity: in every biological, ecological, social, and technological domain studied, real interaction strengths organise memory at greater hierarchical depth than random weight assignment on the same topology, across thirty-four networks spanning several orders of magnitude in size and density. Using a thermodynamic description of multiscale information flow, we quantify how memory is distributed across path lengths and show that functional memory organisation collapses onto four recurrent dynamical organisations, revealing an intrinsically low-dimensional structure. Comparing each network against null models that selectively perturb weighted transport geometry, mesoscale structure, and directionality reveals that these ingredients contribute distinct and non-equivalent roles: weight geometry systematically governs memory depth, mesoscale structure shapes memory organisation across scales, and directionality modulates the sensitivity of the cascade to structural perturbation. The same comparison provides an operational criterion for whether network weights encode genuine functional interaction structure. These results establish weighted transport geometry as a primary organiser of functional memory and show that weighted interactions carry dynamical structure that binary topology alone cannot recover.