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

Triangle Preferential Attachment Has Power-law Degrees and Eigenvalues; Eigenvalues Are More Stable to Network Sampling