Generative Model

We model a page as a set of independent content regions, where the temporal behavior of each region falls into one of three categories:

• Static behavior: content essentially remains unchanged throughout the page's lifetime, such as templates.
  
• Churn behavior: content is overwritten repeatedly, such as advertisements or ``quote of the day.''
  
• Scroll behavior: content is appended periodically, such as blog postings or ``what's new'' items. Typically, as new items are appended, very old items are removed.

Page A in Figure 1 consists of a small static region (templates) and a larger churn region (advertisements). Page B consists of a large static region (templates), a churn region (advertisements) and a scroll region (recipe postings).

In our generative model, each churn or scroll region $R$ has an associated Poisson update process with rate parameter $\lambda_R$ (in our data sets updates closely follow a Poisson distribution, which is consistent with previous findings). In a churn region, each update completely replaces the previous region content, yielding the fragment lifetime distribution:

$$\mathbb{P}(\mathit{lifetime} \leq t) = 1-e^{(-\lambda_R \cdot t)}$$

In a scroll region each update appends a new content item and evicts the item that was appended $K$ updates previously, such that at any given time there are $K$ items present. The fragment lifetime distribution is:

$$\mathbb{P}(\mathit{lifetime} \leq t) = 1-\sum_{i=0}^{K-1} \frac{(\lambda_R \cdot t)^i \cdot e^{-\lambda_R \cdot t}}{{i}!}$$

Figure 4 plots the lifetime distributions for a churn region and a scroll region with $K=10$, where both regions have the same update rate $\lambda_R = 0.25$. The two distributions are quite different. Fragment lifetimes tend to be much longer in the scroll case. In fact, in the scroll case it is unlikely for a fragment to have a short lifetime because it is unlikely for ten updates to occur in rapid succession, relative to $\lambda_R$.

Figure 4: Cumulative distribution of fragment lifetimes, under two different content evolution models.

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