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The above gives a characterization of the class of influence profiles on which all opinion profiles converge. But we can also give a full characterization of convergence, i.e, we can also characterize the class of opinion profiles which converge for graphs that do \emph{not} belong on this class:  \begin{proposition}{}  Let $\G=(G_{p_1},\dots,G_{p_m})$ be an influence profile and $\O=(O_1,\dots,O_n)$ be an opinion  profile. Then the following are equivalent: \begin{itemize}  \item[] The BDP converges foran opinion profile  $\O$ on $\G$. \item[] For some $l\in \{1,\ldots,m\}$, there is a set of agents $C\subseteq\N$ such that: $C$ is a cycle in $G_{p_l}$ and there are two agents $i,j\in C$ such that $O_i(p_l)\neq O_j(p_l)$.  \end{itemize}  \end{proposition} 
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Below is yet another way to put it:  \begin{proposition}{}  Let $\G=(G_{p_1},\dots,G_{p_m})$ be an influence profile and $\O$ $\O=(O_1,\dots,O_n)$ be  an opinion profile. Then the following are equivalent: \begin{itemize}  \item[] The BDP converges for $\O$ on $\G$.  \item[] For all $j\in \{1,\ldots,m\}$, $G_{p_j}$ contains no cycle of length $\geq 2$ or for some $l\in \{1,\ldots,m\}$, there is a set of agents $C\subseteq\N$ such that: $C$ is a cycle in $G_{p_l}$ and there are two agents $i,j\in C$ such that $O_i(p_l)\neq O_j(p_l)$.