  In Part I, we studied the communication for omniscience (CO) problem and
proposed a parametric (PAR) algorithm to determine the minimum sum-rate at
which a set of users indexed by a finite set $V$ attain omniscience. The
omniscience in CO refers to the status that each user in $V$ recovers the
observations of a multiple random source. It is called the global omniscience
in this paper in contrast to the study of the successive omniscience (SO),
where the local omniscience is attained subsequently in user subsets. By
inputting a lower bound on the minimum sum-rate for CO, we apply the PAR
algorithm to search a complimentary subset $X_* \subsetneq V$ such that if the
local omniscience in $X_*$ is reached first, the global omniscience whereafter
can still be attained with the minimum sum-rate. We further utilize the outputs
of the PAR algorithm to outline a multi-stage SO approach that is characterized
by $K \leq |V| - 1$ complimentary subsets $X_*^{(k)}, \forall k \in
\{1,\dotsc,K\}$ forming a nesting sequence $X_*^{(1)} \subsetneq \dotsc
\subsetneq X_*^{(K)} = V$. Starting from stage $k = 1$, the local omniscience
in $X_*^{(k)}$ is attained at each stage $k$ until the final global omniscience
in $X_*^{(K)} = V$. A $|X_*{(k)}|$-dimensional local omniscience achievable
rate vector is also derived for each stage $k$ designating individual users
transmitting rates. The sum-rate of this rate vector in the last stage $K$
coincides with the minimized sum-rate for the global omniscience.
