# HCBC - Hard Cross Betweenness Centrality

#### Definition

This centrality is suited for a $MIoT$ and, more in general, for a scenario consisting of a set of related $IoT$. $HCBC$ specialized to measure the betweenness centrality of nodes by privileging paths involving more $IoT$ of the $MIoT$ and, therefore, $c$-nodes over $i$-nodes.

Let $n_{jk}\in N_k$ be the node corresponding to the instance $l_{jk}$ of the object $o_j$ in the $IoT I_k$. The Hard Cross Betweenness Centrality $HCBC(n_{jk})$ is defined as:

$$HCBC(n_{jk})={\underset{n_{s_u}\in N_u, n_{t_v}\in N_v, k\ne u, k\ne v, u\ne v}{\sum}} {\bar \sigma_{n_{s_u}n_{t_v}} (n_{jk})\over {\bar \sigma_{n_{s_u}n_{t_v}}}}$$

$HCBC$ is analogously to $SCBC(n_{jk})$, it computes the centrality of a node by selecting only the shortest paths between nodes belonging to different networks. Furthermore, differently from the definition of $SCBC$, the node $n_{jk}$ is constrained to belong to a network different from the ones of the source and the destination nodes of the path.

$HCBC$ can be considered as an evolution of $BC$ along the same direction as $SCBC$. The only difference between $SCBC$ and $HCBC$ is that the latter is capable of detecting central $c$-nodes and $i$-nodes linking at least three $IoT$. $HBCB$ capables of overcoming the limits characterizing the classic $BC$ in a $MIoT$.

Let $n_{jk}\in N_k$ be the node corresponding to the instance $l_{jk}$ of the object $o_j$ in the $IoT I_k$. The Hard Cross Betweenness Centrality $HCBC(n_{jk})$ is defined as:

$$HCBC(n_{jk})={\underset{n_{s_u}\in N_u, n_{t_v}\in N_v, k\ne u, k\ne v, u\ne v}{\sum}} {\bar \sigma_{n_{s_u}n_{t_v}} (n_{jk})\over {\bar \sigma_{n_{s_u}n_{t_v}}}}$$

$HCBC$ is analogously to $SCBC(n_{jk})$, it computes the centrality of a node by selecting only the shortest paths between nodes belonging to different networks. Furthermore, differently from the definition of $SCBC$, the node $n_{jk}$ is constrained to belong to a network different from the ones of the source and the destination nodes of the path.

$HCBC$ can be considered as an evolution of $BC$ along the same direction as $SCBC$. The only difference between $SCBC$ and $HCBC$ is that the latter is capable of detecting central $c$-nodes and $i$-nodes linking at least three $IoT$. $HBCB$ capables of overcoming the limits characterizing the classic $BC$ in a $MIoT$.