Weight geometry, not topology, governs functional memory in complex systems

A new preprint on arXiv by Elkaïoum M. Moutuou and Habib Benali challenges the conventional focus on binary network topology by demonstrating that weight geometry—the actual strength of interactions—plays a primary role in organizing functional memory across complex systems. Analyzing 34 diverse networks spanning biology, ecology, social systems, and technology, the researchers applied a thermodynamic description of multiscale information flow to quantify how memory is distributed across path lengths. They found that real interaction strengths systematically create deeper hierarchical memory structures compared to random weight assignments on the same topology. The results collapsed onto just four recurrent dynamical organizations, revealing an intrinsically low-dimensional structure underlying functional memory in all studied domains.

By comparing each network against null models that selectively perturb weight geometry, mesoscale structure, and directionality, the authors showed these ingredients contribute distinct and non-equivalent roles. Weight geometry systematically governs memory depth; mesoscale structure shapes how memory is organized across scales; and directionality modulates the cascade's sensitivity to structural perturbations. The work provides an operational criterion for determining whether network weights encode genuine functional interaction structure, rather than random noise. This establishes weighted transport geometry as a fundamental organizer of memory—suggesting that binary topology alone cannot capture the dynamical richness of complex systems, from neural circuits to gene regulatory networks.