# If I Had a Hammer

For the past five years or so, I have been working on adaptations of algebraic or spectral graph theory to study geomorphic, pedological, and ecological systems. My most recent development (unpublished, for reasons that will become clear in a moment) is some methods for measuring the complexity of historical sequences in Earth surface systems.

The idea is that a historical sequence represents a series of different states or stages—for example, vegetation communities along a successional trajectory; river channel morphological states; different soils in a paleosol sequence; depositional environments in a stratigraphic sequence, nodes of phylogenetic trees in biological evolution, etc. These are treated as directed graphs. The states or stages are the graph nodes or vertices, and the historical transitions are the edges or links between the nodes.

These kinds of sequences are most often conceptualized as linear progressions (A-->B-->C--> . . . .) or as cycles ((A-->B-->C--> . . . -->A). If that accurately represents the system, great—those are the simplest graph structures! However, in some cases the evolutionary sequence is divergent—it splits or forks, as in a biological evolutionary tree, cladogram, or phylogenetic sequence. Divergent evolution has also been documented recently in geomorphic and pedologic systems. Or in some cases, previously existing states reoccur in ways other than a simple cycle. In yet others, more complex mesh-type networks of various transitions among system states may evolve (this is best illustrated by the more complex state-and-transition models).

Algebraic graph theory allows us to measure the structural complexity of the historical sequence (via the graph spectral radius), the (inferential) synchronicity and convergence properties (via algebraic connectivity) and the extent that a subgraph is representative of the overall pattern of historical change, which can be useful if you know or suspect that not all relevant transitions have been observed or inferred. I can also determine the extent to which graph complexity is due to the number of possible transitions vs. the specific way the transitions are wired.

To me this seems like some great stuff. But, on the other hand, I am not happy with simply measuring or quantifying something because I can. What information or insight can quantifying the complexity of historical sequences of Earth surface system development gain us? What geoscience or ecological problems could it solve, or at least address?

That’s what I’m not sure about.

*(http://ux.stackexchange.com/)*

Abraham Maslow is often credited with the saying that if all you have is a hammer, every problem looks like a nail. Some geoscientists are guilty of this, I suppose, as are other scientists. More often, however, we get or devise a new hammer, metaphorically speaking. We realize that not every scientific problem is a nail, so we grab the hammer and go looking for a nail. That’s what I feel like now. If you know of a nail I might take a whack at with this hammer, or otherwise have thoughts on how this particular hammer might be useful, I’d love to hear from you (at jdp@uky.edu).

Some of my previous algebraic graph-theory based work on historical networks is available here and here.