  The problem of sending the maximum amount of flow $q$ between two arbitrary
nodes $s$ and $t$ of complex networks along links with unit capacity is
studied, which is equivalent to determining the number of link-disjoint paths
between $s$ and $t$. The average of $q$ over all node pairs with smaller degree
$k_{\rm min}$ is $<q>_{k_{\rm min}} \simeq c k_{\rm min}$ for large $k_{\rm
min}$ with $c$ a constant implying that the statistics of $q$ is related to the
degree distribution of the network. The disjoint paths between hub nodes are
found to be distributed among the links belonging to the same edge-biconnected
component, and $q$ can be estimated by the number of pairs of edge-biconnected
links incident to the start and terminal node. The relative size of the giant
edge-biconnected component of a network approximates to the coefficient $c$.
The applicability of our results to real world networks is tested for the
Internet at the autonomous system level.
