# SCBC - Soft Cross Betweenness Centrality

#### Definition

This centrality 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 $IoT Ik$. The Soft Cross Betweenness Centrality $SCBC(n_{jk})$ is defined as:

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

$SCBC(n_{jk})$ computes the centrality of a node by selecting only the shortest paths between nodes belonging to different networks. $SCBC$ can be considered as an evolution of $BC$ capable of detecting central (in the betweenness centrality sense) $c$-nodes and $i$-nodes by taking into account that these nodes do not belong to a single-$IoT$ scenario but that they are part of a $MIoT$, and this fact can influence the shortest paths considered in the computation of betweenness centrality.

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

$SCBC(n_{jk})$ computes the centrality of a node by selecting only the shortest paths between nodes belonging to different networks. $SCBC$ can be considered as an evolution of $BC$ capable of detecting central (in the betweenness centrality sense) $c$-nodes and $i$-nodes by taking into account that these nodes do not belong to a single-$IoT$ scenario but that they are part of a $MIoT$, and this fact can influence the shortest paths considered in the computation of betweenness centrality.