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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \section{Convergence}  When do the opinions of a group of individuals influencing each other stabilize? Conditions have been given, in the literature, for the general paradigms which our BDPs are limit cases of. We briefly recall those results below, before giving In this section, we give  a characterization of convergence for BDPs. \subsection{Some graph-theoretic \subsection{Graph-theoretic  preliminaries} We start with some preliminary graph-theoretic remarks.   Let us first recall some vocabulary.   Let $G = \tuple{\N, R}$ be a graph and $R^*$ be the transitive and symmetric closure of $R$.  
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\item[]\emph{aperiodic} if the greatest common divisor of the lengths of its cycles is $1$.  \end{itemize}  For the general class of DeGroot processes, a graph guarantees that any distribution of opinion will converge if and only if ''every set of nodes that is strongly connected and closed is aperiodic" (Jackson, p.233). In propositional opinion diffusion setting, two main results give sufficient conditions for stabilization. First, on graphs containing cycles of size at most one (i.e, only self-loops), for agents using any aggregation function satisfying ballot-monotonicity, opinions will always converge in at most at most $k+1$ steps, where $k$ is the diameter of the graph.   Before turning to the convergence conditions for BDPs, note Note  that the class of graphs BDPs rely on (finite, serial and functional) present a special shape: \begin{fact} \label{fact:uniquecycle}  Let $G$ be an influence graph and $C$ be a connected component of $G$.   Then $C$ contains exactly one cycle, and this cycle forms a closed set.  
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Assume that $C$ contains more than one cycle. Since each cycle forms a closed set, there is no path connecting any node inside the cycle to any node outside the cycle, which contradicts connectedness. So $C$ contains a unique cycle, which forms a closed set.   \end{proof}  This means thatconnected components of  influence graphs will look like confluent chains aiming together towards cyclical tails. \subsection{Convergence of BDPs} 
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Convergence results have been given both for DeGroot processes \cite{Golub_2010} and for propositional opinion diffusion for specific aggregation procedures \cite{Grandi:2015:POD:2772879.2773278}. Let us briefly recall those results.   In the case of DeGroot processes, a graph guarantees that any distribution of opinion will converge if and only if ''every set of nodes that is strongly connected and closed is aperiodic" (Jackson, p.233).  %\footnote{A set of nodes is said to be ``strongly connected" if there is a directed path from any node to any other node in the set, ``closed" if there is no link from nodes in the set to nodes outside the set, and aperiodic if the greatest common divisor of all directed cycles is $1$}   %This result applies of course to our limit cases, the BDGs.  In propositional opinion diffusion setting, two main results give sufficient conditions for stabilization.   First, on graphs containing cycles of size at most one (i.e, only self-loops), for agents using any aggregation function satisfying ballot-monotonicity, opinions will always converge in at most at most $k+1$ steps, where $k$ is the diameter of the graph.   (THIS IS THM 2 IN THE PAPER. NOTE THAT THE PROPERTY OF ALLOWING ONLY CYCLES OF LENGTH 1 GUARANTEES THAT THE GRAPH IS APERIODIC, ALREADY MEETING DEGROOT'S CONDITION FOR STABILIZATION.)  Second, graphs which contain no cycle of length one and only vertex-disjoint cycles, such that for each cycle there exists an agent who has at least two influencers, when agents aggregate using the unanimity rule, opinions converge after at most $\N$ steps.   (THIS IS THM 6 IN THEIR PAPER, AND I AM STILL UNSURE WHY IT EXCLUDES SELF-LOOPS. WE CANNOT USE THIS RESULT, BECAUSE IT INVOLVES NODES HAVING TWO INFLUENCERS, WHICH WE EXCLUDE ANYWAY BY FUNCTIONALITY.)