  We establish the conditions under which several algorithmically exploitable
structural features hold for random intersection graphs, a natural model for
many real-world networks where edges correspond to shared attributes.
Specifically, we fully characterize the degeneracy of random intersection
graphs, and prove that the model asymptotically almost surely produces graphs
with hyperbolicity at least $\log{n}$. Further, we prove that in the parametric
regime where random intersection graphs are degenerate an even stronger notion
of sparseness, so called bounded expansion, holds with high probability. We
supplement our theoretical findings with experimental evaluations of the
relevant statistics.
