  The identification of motifs--subgraphs that appear significantly more often
in a particular network than in an ensemble of randomized networks--has become
a ubiquitous method for uncovering potentially important subunits within
networks drawn from a wide variety of fields. We find that the most common
algorithms used to generate the ensemble from the real network change subgraph
counts in a highly correlated manner, so that one subgraph's status as a motif
may not be independent from the statuses of the other subgraphs. We demonstrate
this effect for the problem of 3- and 4-node motif identification in the
transcriptional regulatory networks of E. coli and S. cerevisiae in which
randomized networks are generated via an edge-swapping algorithm (Milo et al.,
Science 298:824, 2002). We show that correlations among 3-node subgraphs are
easily interpreted, and we present an information-theoretic tool that may be
used to identify correlations among subgraphs of any size.
