Are there well-known or interesting applied problems where large time series of small (say, $\dim \le 30$) sparse matrices arise naturally?
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Now, that sounds innocuous enough, but the key reason I ask is the following
Background: a new tool of my software company maps network traffic to (real-) time series of small sparse matrices (the same sort of thing could be done for many different types of transactions, though) and allows these time series to be visualized and interactively sampled quite effectively. I would like to see it get some nontrivial use from the academic community and for suitable projects (meaning among other things having absolutely nothing to do with network traffic or commercial endeavors) I could arrange for premium licenses free of charge.
Also, I might put something in about random walks on the root lattice $A_n$, which can naturally be mapped to this sort of framework (though the resulting matrices may not be sparse, that strictly doesn't matter).