Measuring What the Crawler Sees: Discovery Curves, Core Persistence, and Shell Dynamics in Longitudinal Web Crawls

A longitudinal web crawl is a sequence of partial samples of an evolving URL population. Pairwise containment between two crawls is the standard probe; under a simple \emph{urn} model of the crawl -- each round samples a fraction of the URLs and replaces a fraction -- it recovers two interpretable rates, per-round survival $\alpha$ and coverage $c$, but treats the population as uniform and consumes one pair at a time. In this work, we define a formal language for talking about a crawl. We extend this analysis with the \emph{discovery curve} $U(s, T)$, the cumulative URL footprint over a sliding window of $T$ crawls starting at $s$, which under the same urn model is also a closed-form function of $(\alpha, c)$. Containment and the discovery curve are then two projections of one process: independent fits agree on $(\alpha, c)$ when the urn is homogeneous, so any disagreement is itself a measurement. Applied to Common Crawl (2020--2025, domain granularity) and to the German Academic Web (GAW, URL granularity), the two projections disagree on both archives, and a two-component urn with a persistent core fraction $\kappa$ alongside shell parameters $(\alpha_\partial, c_\partial)$ reconciles the disagreement. A residual on $c_\partial$ remains, signaling that the shell itself is not homogeneous; $\kappa$ is recorded as the scalar entry point to a rank-resolved generalization, which is left to follow-up work. \keywords{web archive \and crawl coverage \and discovery curve \and urn model \and two-component model \and URL lifetime}