Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter $\delta_0$, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $\delta_0$ to $\delta_1$ at an unknown changepoint time $\tau_n$. For our analysis we focus on the regime where $\delta_0$ and $\delta_1$ are fixed, and the changepoint occurs close to the observation time of the network (i.e., $\tau_n = n - c n^\gamma$ with $c>0$ and $\gamma \in (0, 1)$). This corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree, and show that these are asymptotically powerful when $\tfrac{1}{2}<\gamma<1$. We conjecture that there is no powerful test based on the final network snapshot when $\gamma < \tfrac{1}{2}$. The first test we propose requires knowledge of $\delta_0$. The second test is significantly more involved, and does not require the knowledge of $\delta_0$ while still achieving the same performance guarantees. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. This is demonstrated by numerical experiments, that also illustrate the finite sample test properties.

翻译：受网络演化过程中变点检测问题的启发，我们考虑具有时变连接函数的优先连接随机图模型。研究目标是基于网络单次快照且无直接动态观测信息的情况下，检测连接机制是否随时间发生变化。我们将该问题构建为假设检验问题：原假设为具有恒定仿射连接参数δ₀的优先连接模型，备择假设为仿射连接参数在未知变点时间τₙ处从δ₀变为δ₁的优先连接模型。分析聚焦于δ₀与δ₁固定，且变点发生在接近网络观测时刻（即τₙ = n - c n^γ，其中c>0，γ∈(0,1)）的情形，这与我们旨在检测变点发生后近期变化的实际场景相符。我们提出两种基于最小度顶点数的检验方法，证明当1/2<γ<1时这些方法具有渐近检验功效，并推测当γ<1/2时不存在基于最终网络快照的有效检验。第一种检验需已知δ₀，第二种检验更为复杂但在不依赖δ₀的前提下仍能达到相同性能保证。此外，我们证明两种检验统计量均具有渐近正态性，可实现检验的精确校准。数值实验验证了理论结果，并展示了有限样本条件下的检验性质。