# Enhancing Cache Robustness in Information-Centric Networks: Per-Face Popularity Approaches
## 3 Enhancing Cache Robustness via Per-Face Popularity
### Per-Face Popularity with \(\beta\)-Parameterization  
PFP-DA yields very good results, as we will see later in this article. However, even slight improvements on top of PFP-DA would be welcome. Therefore, we take advantage of the fact that the overall popularity calculation looks like the formula for the Zipf distribution. Therefore, we considered the impact that introducing a parameter similar to the \(\alpha\) value used in Zipf-like distributions would have on the cache, including pollution and hit rates. Thus, our formula for calculating the overall popularity became the following:  
\[P(C,\ \beta)=\sum_{\begin{subarray}{c}i=1\\ 0\leq\beta\leq 1\end{subarray}}^{n}\frac{1}{r_{i}\beta} \tag{4}\]  
The parameter, \(\beta\), is introduced as being between 0 and 1, inclusive. Thus, our regular PFP-DA would be considered a special case of the above equation with \(\beta=1\). This new parameterized scheme is designated PFP-Beta, or PFP-\(\beta\). It still maintains dynamic aging, as in PFP-DA, but with the additional parameter introduced.  
As we would expect, the numeric distance between the top-ranked item and subsequently ranked items is reduced as \(\beta\) is reduced. To obtain a more intuitive feel for this behavior, let us consider an example. If we consider the regular PFP-DA case, with \(\beta=1\), the top-ranked item from a face will contribute \(1/1=1\), the second-ranked item will contribute \(1/2=0.50\), and the third-ranked item will contribute \(1/3\approx 0.33\). So, the respective distances are shown in Table 1.  
As \(\beta\) is reduced, thus yielding a small distance between each item and the top item, as seen in the above tables, the overall contributions for the ranked items themselves are larger (e.g., for \(\beta=1\), the 4 th item contributes 0.25; for \(\beta=0.5\), the contribution for the fourth item is 0.50; and for \(\beta=0.1\), the contribution is 0.87). What this means is that the popular items are all contributing more, and the difference between a fourth or a fifth item is not as significant for a \(\beta=0.1\) as it would be for items ranked with \(\beta=1\). Therefore, with these larger numbers, it stands to reason that a PFP scheme using a small \(\beta\) will favor items to be cached that have contributions from multiple faces even more than a PFP scheme using a larger \(\beta\).  
We will see that this holds true as we discuss the experimental results later in this article. However, we will also see that a small \(\beta\) value in a PFP scheme also works best  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Rank**} & **Numeric Distance from Top Item** \\ \hline
1 & 0 \\
2 & 0.5 \\
3 & 0.67 \\
4 & 0.75 \\
5 & 0.80 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numeric distance between items and top item, \(\beta=1\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Rank** & **Numeric Distance from** & **Rank** & **Numeric Distance from** \\ \cline{2-5}  & **Rank** & **Top Item** & **Top Item** \\ \cline{2-5}  & 1 & 0 \\ (a) & 2 & 0.29 \\  & 3 & 0.42 \\  & 4 & 0.50 \\  & 5 & 0.55 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numeric distance between items and top item, (a) \(\beta=0.5\) and (b) \(\beta=0.1\).  
under certain circumstances and far worse in others, involving situations with a higher percentage of attackers.