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

## TL;DR

This paper introduces a Triangle Generalized Preferential Attachment (TGPA) model that exhibits power-law degree and eigenvalue distributions, demonstrating that eigenvalues are more stable under sampling, explaining their frequent observation in real networks.

## Contribution

The paper presents a new TGPA model with non-trivial clustering and proves it has power-law spectra, also showing eigenvalues are more robust to sampling than degrees.

## Key findings

- Eigenvalues show power-law behavior even with limited network sampling.
- Degree distributions are less reliably power-law under sampling.
- TGPA model clearly demonstrates the stability of eigenvalue spectra.

## Abstract

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.