# Complexity of Earth Surface System Evolutionary Pathways

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## Abstract

Evolution of Earth surface systems (ESS) comprises sequential transitions between system states. Treating these as directed graphs, algebraic graph theory was used to quantify complexity of archetypal structures, and empirical examples of forest succession and alluvial river channel change. Spectral radius measures structural complexity and is highest for fully connected, lowest for linear sequential and cyclic graphs, and intermediate for divergent and convergent patterns. The irregularity index \(\beta \) represents the extent to which a subgraph is representative of the full graph. Fully connected graphs have \(\beta = 1\). Lower values are found in linear and cycle patterns, while higher values, such as those of divergent and convergent patterns, are due to a few highly connected nodes. Algebraic connectivity (\(\mu (\mathrm{G}))\) indicates inferential synchronization and is inversely related to historical contingency. Highest values are associated with fully connected and strongly connected mesh graphs, whereas forking structures and linear sequences all have \(\mu (G)\) = 1, with cycles slightly higher. Diverging vs. converging graphs of the same size and topology have no differences with respect to graph complexity, so complexity change is dependent on whether development results in increased or reduced richness. Convergent-divergent mode switching, however, would generally increase ESS complexity, decrease irregularity, and increase algebraic connectivity. As ESS and associated graphs evolve, none of the possible trends reduces complexity, which can only remain constant or increase. Algebraic connectivity may increase, however. As improving shortcomings in ESS evolution models generally result in elaborating possible state changes, this produces more structurally complex but less historically contingent models.

## Keywords

Complexity Evolutionary trajectory Directed graph Algebraic graph theory## References

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