Wave propagation in "Stitched Continua"

Transport in complex systems, such as the brain, emerges from a continuous local geometry stitched with a sparse network of long-range connections. However, existing models rarely couple these descriptions, and are unable to uncover the interplay between geometry and topology, and their influence on brain dynamics. Here, we introduce a framework that embeds sparse long-range connections within a continuous wave equation. We find solutions for signal propagation in these inhomogeneous spaces and investigate how long-range connections, scaled by propagation speed, shape the dynamics. We demonstrate the emergence of spatially heterogeneous response frequencies, similar to those observed in the human cortex. We further show that partially localized eigenmodes mediate this behavior, which provides a way to “hear the topology” of a system — that is, to infer the topological and geometrical structure that best explains observed spatiotemporal patterns. Together, this framework unifies network and continuum perspectives, links brain structure to function, and generalizes to other systems where continuous transport is influenced by sparse long-range interactions, such as traffic flow or epidemic spreading.