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BDPs are also limit cases of processes that have recently been proposed in the multi-agent systems literature as \emph{propositional opinion diffusion} processes \cite{Grandi:2015:POD:2772879.2773278}, i.e., cases where 1) the aggregation rule is the unanimity rule (an agent adopts an opinion if and only if all her influencers agree on it), and 2) each agent has exactly one influencer.   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \section{Convergence of BDPs} \section{Convergence}  \subsection{Graph-theoretic preliminaries} 
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%which is a closed set.   \end{proof}  This means thatour  influence graphs will look like parallel chains aiming together  towardsa  cyclical tail.   \comment{  \subsection{Backdrop: extra graph theoretical remark (WE DON'T NEED THIS ONE)}  First, for functional graphs, no node in a cycle can have any successor outside of the cycle. Therefore, all cycles have to be vertex-disjoint:  \begin{fact} \label{fact:vertex}  Let $G$ be an influence graph. All cycles in $G$ are vertex-disjoint.  \end{fact}  \begin{proof}  Let $G$ be an influence graph, such that it contains two cycles which are not vertex disjoint. Then there is a vertex $i$ contained in both cycles, such that $i$ has a different successor in each cycle. But since $G$ is functional, $i$ has at most one successor. Contradiction.  \end{proof}  } tails.  \subsection{Convergence} \subsection{Convergence of BDPs}  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 $\O^\star$. 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.  It must be intuitively clear that non-convergence in a BDP is linked to the existence of loops. However, from the above observation   %(Facts \ref{fact:vertex} and   (\ref{fact:connected}), (\ref{fact:uniquecycle}),  we know that nodes in a cycle cannot have any guru outside this cycle, cycles (including self-loops) can only occur at the `tail' of the influence graph.%be at the ``tail"(s)/root of the graph (i.e WHAT WE CALLED A SINK AND WHAT UMBERTO CALLS A %``SOURCE" GIVEN THAT THE ARROWS GO IN THE OPPOSITE DIRECTION).  Hence, if the opinions in the (unique) cycle do not converge, which can only happen in a cycle of length $\geq 2$, the opinions of the whole population in the same connected component will not converge. The above implies that for any influence graphs with a cycle of length $\geq 2$, there can be exists  a distribution of opinions which loops. This brings us back to convergence result for general (not necessarily Boolean) DeGroot processes. Indeed, for functional and serial influence graphs, a closed connected component is aperiodic if and only if its cycle is of length $1$. \begin{lemma}\label{lemma:influence}  Let $\G=(G_{p_1},\dots,G_{p_m})$ be an influence profile. Then the following are equivalent: 
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\ldots  \end{proof}  The above gives a characterization of the class of influence profiles on which \emph{all} opinion streams converge. But it is also interesting to characterize the class of opinion profiles on which \emph{any} influence graph would make the resulting opinion stream converge: \begin{lemma}\label{lemma:opinion}  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: 
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What do these results correspond to when applied to our ``follow your guru" policy, that is, when imposing functionality and seriality on graphs and focusing solely on the unanimity rule? They imply the following: if, for all $i\in\{1,\ldots,m\}$, the influence graph $\G_i$ contains only cycles of length $1$, then all opinions profiles on $\G$ converge. Note that this follows directly both from the DeGroot convergence result stated earlier (since graphs with only cycles of length one are trivially aperiodic) and from the propositional opinion diffusion theorem above (since the unanimity rule is trivially monotonic). This illustrates how the BDPs are limit cases both of the opinion diffusion and of the DeGroot stochastic opinion diffusion.  \subsection{Backdrop: extra graph theoretical remark (WE DON'T NEED THIS ONE)}  First, for functional graphs, no node in a cycle can have any successor outside of the cycle. Therefore, all cycles have to be vertex-disjoint:  \begin{fact} \label{fact:vertex}  Let $G$ be an influence graph. All cycles in $G$ are vertex-disjoint.  \end{fact}  \begin{proof}  Let $G$ be an influence graph, such that it contains two cycles which are not vertex disjoint. Then there is a vertex $i$ contained in both cycles, such that $i$ has a different successor in each cycle. But since $G$ is functional, $i$ has at most one successor. Contradiction.  \end{proof}  %%%%%%%%%%%%%%%%%%%%