This paper is also relevant.
There seems to be some claims that biologists are interested in finding network motifs (i.e. subgraphs) that occur more often than expected in large networks -- thus suggesting a demand for software packages such as FANMOD, which finds (or estimates) the number of times each non-isomorphic subgraph on n vertices appears in a given network.
Question: What are some examples of where large networks arise in biological systems and where motifs play a role (or are suspected to play a role)?
If possible, I would also like to be able to access these networks directly (i.e. download them) and test them myself.
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You'll probably want to have a look at Uri Alon's work, who is the author of the theory of network motifs, which is basically what you are referring to in your question.
Uri studied the structure of the yeast's gene regulatory network and postulated that it can be decomposed into a set of motifs, some of which are more frequent than others. I recommend you to look at his homepage (there you will find some downloadable datasets) and his books.
Thanks, I found a few things there I can take a look through.
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Stay tuned for a paper by Shane Neph on this subject.