Synchronization and scale in geomorphic systems

Geomorphic systems consist of coupled subsystems with traits of small-world networks (SWN), characterized by tightly connected clusters of components, with fewer connections between the clusters. Geomorphic systems based on scale hierarchies often exhibit a connected caveman small-world network (CCSWN) structure. SWNs are efficient for linking a large number of components with a relatively small number of links; but effects of CCSWN structure on synchronization and scale linkage have not been examined. Synchronization is analyzed via graph theory and applied to: (1) relationships among three levels of form–process interaction in stream channels; (2) hierarchical relationships of weathering systems at scales from weathering profiles to landscapes; and (3) interactions in fluviokarst systems at the scale of flow processes and of landscape evolution. Relationships among system components are represented as simple unweighted graphs. The largest eigenvalue of the adjacency matrix (spectral radius) reflects the critical coupling strength required to synchronize the system. The second-smallest eigenvalue of the Laplacian of the adjacency matrix (algebraic connectivity) is a measure of the synchronizability. In all examples both are much less than the maximum for networks of the same number of nodes. The sparseness of the networks is the major contributor to the low synchronization, but the specific pattern of connections (“wiring”) is also significant. Where CCSWN structures arise naturally, they help explain how geomorphic effects are transmitted between disparate scales in the absence of obvious scale linkage. Where CCSWNs are an option for representation of geomorphic systems in models and data structures, they will not improve scale linkage, despite the efficiency of SWNs in other respects. Methods developed here can be applied to evaluating alternative spatial data structures or mapping strategies which either increase synchronization, supporting a lumping or aggregation approach, or decrease synchronization, indicating disaggregation or splitting into scale hierarchies.