# P-CBC - Popularity-Weighted Content-Based Centrality

#### Definition

The logic behind this centrality is that although computing centrality on the topological graph yields interesting insights. Yet it fails to capture that, in network of caches, the consumer is interested in connecting to the content, not to a specific node. If a node is well connected, but holds content of little value, it is not useful to the user. It has low centrality, in the context of content delivery, despite having a high centrality within the topological graph.

for a node $v$, and for a probability distribution $p_x$ for the content $x\in X$ and if $\sigma_v(u, x)$ is the number of shortest paths from user $u$ to content $x$ going through node $v$ and $\sigma (u, x)$ the total number of shortest paths between $u$ and$x$, then:

$$P-CBC(v)={\underset{u,x}{\sum}} {\sigma_u (u,x)\over \sigma_u (u,x)}\times p_x$$

The content popularity is represented by the probability $p_x= {\lambda (l, \bar t)\over \Lambda (l, \bar t)}$ as a measure of the user interests for content $x$ at location $l$ and time $\bar t$. Here, $\lambda (l, \bar t)$ represents the number of interest for the content $x$ at location $l$ and time $\bar t$ and $\Lambda = {\underset {X_v}{\sum}} \lambda (l,\bar t)$ is the total interests for all contents items.

$P-CBC$ takes into account how well a node is connected to the content the network is delivering, rather than to the other nodes in the network

for a node $v$, and for a probability distribution $p_x$ for the content $x\in X$ and if $\sigma_v(u, x)$ is the number of shortest paths from user $u$ to content $x$ going through node $v$ and $\sigma (u, x)$ the total number of shortest paths between $u$ and$x$, then:

$$P-CBC(v)={\underset{u,x}{\sum}} {\sigma_u (u,x)\over \sigma_u (u,x)}\times p_x$$

The content popularity is represented by the probability $p_x= {\lambda (l, \bar t)\over \Lambda (l, \bar t)}$ as a measure of the user interests for content $x$ at location $l$ and time $\bar t$. Here, $\lambda (l, \bar t)$ represents the number of interest for the content $x$ at location $l$ and time $\bar t$ and $\Lambda = {\underset {X_v}{\sum}} \lambda (l,\bar t)$ is the total interests for all contents items.

$P-CBC$ takes into account how well a node is connected to the content the network is delivering, rather than to the other nodes in the network