Abstract
Multiple pathways and outcomes are common in evolutionary sequences for biological and other environmental systems due to nonlinear complexity, historical contingency, and disturbances. From any starting point, multiple evolutionary pathways are possible. From an endpoint or observed state, multiple possibilities exist for the sequence of events that created it. However, for any observed historical sequence-e.g., ecological or soil chronosequences, stratigraphic records, or lineages-only one historical sequence actually occurred. Here, a measure of the embedded complexity of historical sequences based on algebraic graph theory is introduced. Sequences are represented as system states S(t), such that S(t - 1) ≠ S(t) ≠ S(t + 1). Each sequence of N states contains nested subgraph sequences of length 2, 3, …, N - 1. The embedded complexity index (which can also be interpreted in terms of embedded information) compares the complexity (based on the spectral radius λ(1)) of the entire sequence to the cumulative complexity of the constituent subsequences. The spectral radius is closely linked to graph entropy, so the index also reflects information in the sequence. The analysis is also applied to ecological state-and-transition models (STM), which represent observed transitions, along with information on their causes or triggers. As historical sequences are lengthened (by the passage of time and additional transitions or by improved resolutions or new observations of historical changes), the overall complexity asymptotically approaches λ(1) = 2, while the embedded complexity increases as N(2.6). Four case studies are presented, representing coastal benthic community shifts determined from biostratigraphy, ecological succession on glacial forelands, vegetation community changes in longleaf pine woodlands, and habitat changes in a delta.