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We say that the stream of opinion profiles $\O^0, \O^1, \ldots, \O^n, \ldots$ {\em converges} if there exists $n \in \mathbb{N}$ such that $\O^n = \O^{n+1}$.   %Such limit profile, when it exists, is denoted $\O^\star$.   We will also say that a stream of opinion profiles converges {\em for issue} $p$ if there exists $n \in \mathbb{N}$ such that $\O^n(p) = \O^{n+1}(p)$.  Given a stream of opinion profiles starting at $\O$ we say that agent $i \in \N$ stabilizes in that stream for issue $p$ if there exists $n \in \mathbb{N}$ such that $\O^n_i(p) $O^n_i(p)  = \O^{m}_i(p)$ O^{m}_i(p)$  for any $m > n$. So a BDP on influence graph $\G$ starting with the opinion profile $\O$ is said to converge if the stream $\O^0, \O^1, \ldots, \O^n, \ldots$ generated according to Definition \ref{def:BDP} where $\O = \O^0$ converges. Similarly, A BDP is said to converge for issue $p$ if its stream converges for $p$, and an agent $i$ in the BDP is said to stabilize for $p$ if it stabilizes for $p$ in the stream generated by the BDP.   \subsubsection{Two results}