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$\newcommand{\pop}{\operatorname{pop}}$Assuming $ G = \{g_1, g_2, g_3, \ldots , g_{N_G}\}$ is set of all the globally existing data objects. The total request rate for $G$ is shown by $\lambda_G$. The popularity of data objects in $G$ follows a Zipf distribution with Zipf paramters shown as $\alpha_G$. We show the global popularity of $\text{object}_i$ as follows: $\pop(g_{i},\alpha_G) = \frac{(1/i)^{\alpha_G}}{\sum (1/j)^{\alpha_G}}$. The request rate for $\text{object}_i$ (shown as $\lambda_{g_i}$)in G is calculated as $\lambda_{g_i} = \lambda_G \times \pop(g_i,\alpha_G) = \lambda_G \times \frac{(1/i)^{\alpha_G}}{\sum (1/j)^{\alpha_G}}$. Then, we suppose there are two regions in which the order of data objects' local popularity in each region is different from the global one.

Assuming $\lambda_F$ and $\lambda_H$ as the request rates for the data objects in the two regions, we are looking for the order of data objects' popularity in these two regions complying with the following premises:

  1. $\exists F \subset G , \ \exists \lambda_F, \ F = \{f_{1}, f_{2}, f_{3}, \ldots , f_{N_F}\} \ $ $\forall i \ \exists l : f_{i} = g_l, \pop(f_i,\alpha_{F})= \frac{(1/i)^{\alpha_F}}{\sum (1/j)^{\alpha_F}}$
  2. $\exists H \subset G , \exists \lambda_H, \ H = \{h_1, h_2, h_3, \ldots , h_{N_H}\} \ $ $\forall i \ \exists l : h_i = g_l, \pop(h_i,\alpha_H)= \frac{(1/i)^{\alpha_H}}{\sum (1/j)^{\alpha_H}}$
  3. $F \cup H = G$
  4. $\lambda_F + \lambda_H = \lambda_G$

As a result, we have only one of the following cases standing for each $g_i$:

  • $\forall \ g_i\ \exists\ f_j\ \exists\ h_l\ : g_i = f_j = h_l \Rightarrow \lambda_{g_i} = \lambda_{f_i} + \lambda_{h_i}$

  • $\forall \ g_i\ \exists\ f_j\ \nexists\ h_l\ : g_i = f_j = h_l \Rightarrow \lambda_{g_i} = \lambda_{f_i} $

  • $\forall \ g_i\ \nexists\ f_j\ \exists\ h_l\ : g_i = f_j = h_l \Rightarrow \lambda_{g_i} = \lambda_{h_i} $

The question is now, how we can prove that there are such $F$ and $H$?

Alireza Montazeri Gh
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  • You have $F\cup H\subseteq G$. Are you assuming $F\cup H=G$, i.e. the two regions fill up the whole space and not just part of it? Also, am I right in surmising that you intend $F\cap H=\varnothing$? ${}\qquad{}$ – Michael Hardy Jul 10 '15 at 19:21
  • You use $F$ and $H$ to refer to the two regions, but also to refer to the sets of objects requested in the two regions. Perhaps you could address my comment above with both interpretations. ${}\qquad{}$ – Michael Hardy Jul 10 '15 at 19:23
  • Do you have some actual data for which this model seems like a good fit? How closely do the popularity ranks of objects in one region resemble those of the other? – Michael Hardy Jul 10 '15 at 19:26
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    @MichaelHardy Thanks for your comments. I added one more premise to my question. However, F and H could have common members. Regarding your third question, I should say that the study shows that not only the global popularity of Internet objects (like YouTube videos) follows a Zipf distribution, but also their local popularity (like subset of YouTube videos on a campus) follows a Zipf distribution. – Alireza Montazeri Gh Jul 10 '15 at 19:35
  • Is the study published? ${}\qquad{}$ – Michael Hardy Jul 10 '15 at 19:50
  • @MichaelHardy they are not in a single paper. But there are some papers showing that global popularity of videos in YouTube follows Zipf distribution. On the other hand, there are some other papers studying the distribution of local YouTube requests like on a campus showing the local requests also follow Zipf distributions. It is worth mentioning that although the global and local requests have Zipf-like distributions, the Zipf distributions have different parameters. – Alireza Montazeri Gh Jul 10 '15 at 20:10
  • I need this for my simulation in which different regions have different Zipf-like request patterns while the sum of all requests in all regions still follow a Zipf distribution. – Alireza Montazeri Gh Jul 10 '15 at 20:12

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