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Motivated by the problem of detecting a change in the evolution of a network,we consider the preferential attachment random graph model with atime-dependent attachment function. Our goal is to detect whether theattachment mechanism changed over time, based on a single snapshot of thenetwork and without directly observable information about the dynamics. We castthis question as a hypothesis testing problem, where the null hypothesis is apreferential attachment model with a constant affine attachment parameter$\delta_0$, and the alternative hypothesis is a preferential attachment modelwhere the affine attachment parameter changes from $\delta_0$ to $\delta_1$ atan unknown changepoint time $\tau_n$. For our analysis we focus on the regimewhere $\delta_0$ and $\delta_1$ are fixed, and the changepoint occurs close tothe 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 weaim to detect the changepoint shortly after it has happened.  We present two tests based on the number of vertices with minimal degree, andshow that these are asymptotically powerful when $\tfrac{1}{2}<\gamma<1$. Weconjecture that there is no powerful test based on the final network snapshotwhen $\gamma < \tfrac{1}{2}$. The first test we propose requires knowledge of$\delta_0$. The second test is significantly more involved, and does notrequire the knowledge of $\delta_0$ while still achieving the same performanceguarantees. Furthermore, we prove that the test statistics for both tests areasymptotically normal, allowing for accurate calibration of the tests. This isdemonstrated by numerical experiments, that also illustrate the finite sampletest properties.

Detecting a late changepoint in the preferential attachment model