  The multicast capacity of a directed network is closely related to the
$s$-$t$ maximum flow, which is equal to the $s$-$t$ minimum cut capacity due to
the max-flow min-cut theorem. If the topology of a network (or link capacities)
is dynamically changing or have stochastic nature, it is not so trivial to
predict statistical properties on the maximum flow. In this paper, we present a
coding theoretic approach for evaluating the accumulate distribution of the
minimum cut capacity of weighted random graphs. The main feature of our
approach is to utilize the correspondence between the cut space of a graph and
a binary LDGM (low-density generator-matrix) code with column weight 2. The
graph ensemble treated in the paper is a weighted version of
Erd\H{o}s-R\'{e}nyi random graph ensemble. The main contribution of our work is
a combinatorial lower bound for the accumulate distribution of the minimum cut
capacity. From some computer experiments, it is observed that the lower bound
derived here reflects the actual statistical behavior of the minimum cut
capacity.
