  We consider the number of paths that must pass through a subset $X$ of
vertices of a network $N$ in a maximum sequence of arc-disjoint paths
connecting two vertices $y$ and $z$. We show that when $X$ is a singleton, that
number equals the difference between the maximum flow value from $y$ to $z$ in
$N$ and the maximum flow value from $y$ to $z$ in the network obtained by $N$
setting to zero the capacities of arcs incident to $X$. That fact theoretically
justifies the common identification of those two concepts in network
literature. We also show that the same equality does not hold when $|X|\geq 2.$
Consequently, two conceptually different group centrality measures involving
paths and flows can naturally be defined, both extending the classic flow
betweenness centrality.
