  Preferential attachment models are a common class of graph models which have
been used to explain why power-law distributions appear in the degree sequences
of real network data. One of the things they lack, however, is higher-order
network clustering, including non-trivial clustering coefficients. In this
paper we present a specific Triangle Generalized Preferential Attachment Model
(TGPA) that, by construction, has nontrivial clustering. We further prove that
this model has a power-law in both the degree distribution and eigenvalue
spectra. We use this model to investigate a recent finding that power-laws are
more reliably observed in the eigenvalue spectra of real-world networks than in
their degree distribution. One conjectured explanation for this is that the
spectra of the graph is more robust to various sampling strategies that would
have been employed to collect the real-world data compared with the degree
distribution. Consequently, we generate random TGPA models that provably have a
power-law in both, and sample subgraphs via forest fire, depth-first, and
random edge models. We find that the samples show a power-law in the spectra
even when only 30\% of the network is seen. Whereas there is a large chance
that the degrees will not show a power-law. Our TGPA model shows this behavior
much more clearly than a standard preferential attachment model. This provides
one possible explanation for why power-laws may be seen frequently in the
spectra of real world data.
