# IBC - Inner Betweenness Centrality

#### Definition

The $IBC$ has been conceived for measuring the betweenness centrality with a focus on a single $IoT$ of the $MIoT$ and it privileges $i$-nodes over $c$-nodes. It does not coincide with the classical betweenness centrality because, it considers paths which connect two nodes of the same $IoT$ but, at the same time, involve nodes belonging to other $IoT$ of the $MIoT$.

Let $n_{jk}\in N_k$ be the node corresponding to the instance $l{jk}$ of the object $o_j$ in the $IoT I_k$ of the $MIoT M$. The Inner Betweenness Centrality $IBC(n_{jk})$ is defined as:

$$IBC(n_{jk})={\underset{n_{s_k}\in N_k,n_{s_k}\ne n_{j_k},n_{t_k}\ne n_{j_k}}{\sum}} {\bar \sigma_{n_{s_k}n_{t_k}}(n_{j_k})\over \bar \sigma_{n_{s_k}n_{t_k}}}$$

where $\bar \sigma_{n_{s_k}n_{t_k}}$ is the total number of the shortest paths from $n_s$ to $n_t$ that involve also nodes of the $MIoT$ not belonging to $N_k$, and ${\bar \sigma_{n_{s_k}}n_{t_k}(n_{j_k})}$ is the total number of these shortest paths that pass through $n_{jk}$.

$IBC$ can be considered as an evolution of $BC$, capable of evaluating inner central nodes taking into account the fact that the network $I_k$ is not alone but it is part of a $MIoT$. As a consequence, if all the paths connecting $n_{s_k}$ to $n_{t_k}$ include at least one node belonging to networks different from $I_k$ but inside the $MIoT$, then $BC$ does not capture them and considers $n_{s_k}$ and $n_{t_k}$ unconnected. By contrast, in a more precise way, $IBC$ considers that there may exist one or more connections between them in the $MIoT$, even if they require the intervention of nodes belonging to other networks.

Let $n_{jk}\in N_k$ be the node corresponding to the instance $l{jk}$ of the object $o_j$ in the $IoT I_k$ of the $MIoT M$. The Inner Betweenness Centrality $IBC(n_{jk})$ is defined as:

$$IBC(n_{jk})={\underset{n_{s_k}\in N_k,n_{s_k}\ne n_{j_k},n_{t_k}\ne n_{j_k}}{\sum}} {\bar \sigma_{n_{s_k}n_{t_k}}(n_{j_k})\over \bar \sigma_{n_{s_k}n_{t_k}}}$$

where $\bar \sigma_{n_{s_k}n_{t_k}}$ is the total number of the shortest paths from $n_s$ to $n_t$ that involve also nodes of the $MIoT$ not belonging to $N_k$, and ${\bar \sigma_{n_{s_k}}n_{t_k}(n_{j_k})}$ is the total number of these shortest paths that pass through $n_{jk}$.

$IBC$ can be considered as an evolution of $BC$, capable of evaluating inner central nodes taking into account the fact that the network $I_k$ is not alone but it is part of a $MIoT$. As a consequence, if all the paths connecting $n_{s_k}$ to $n_{t_k}$ include at least one node belonging to networks different from $I_k$ but inside the $MIoT$, then $BC$ does not capture them and considers $n_{s_k}$ and $n_{t_k}$ unconnected. By contrast, in a more precise way, $IBC$ considers that there may exist one or more connections between them in the $MIoT$, even if they require the intervention of nodes belonging to other networks.