deletions | additions       diff --git a/section_Unanimity_and_2_colorability__.tex b/section_Unanimity_and_2_colorability__.tex  index 6447db8..e14889a 100644  --- a/section_Unanimity_and_2_colorability__.tex  +++ b/section_Unanimity_and_2_colorability__.tex 
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\end{definition}  We give a sufficient condition for non-convergence of UP:  \begin{lemma} \begin{lemma}\label{lemma:suff.UP}  Let $G$ be a (serial and non-necessarily functional) influence profile and $\O$ be an opinion profile, such that, for some $p\in \Atoms $, for all $i,j\in C$, where $C$ is a connected component of $G_p$: if $i\in R_p(j)$, then $O_i(p)\neq O_j(p)$.   Then $\O$ does not converge in UP.  \end{lemma} 
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The possibility of such a distribution of opinions on $p$ relies on the graph $G_p$ being $2$-colorable, which is again a requirement about the length of cycles, since a graph is 2-colorable if and only if it contains no cycle of odd length. However, non $2$-colorability is not a sufficient condition for convergence of UPs in general: a simple cycle of three agents, for instance, is not $2$-colorable but does not guarantee convergence either (as illustrated above with the convergence conditions for BDPs).   Nevertheless, there is a class of influence profiles for which being $2$-colorable is a necessary condition of non-convergence of UPs, the \emph{symmetric} ones:  \begin{lemma}\label{lemma:influenceUP} \begin{lemma}\label{lemma:symm.opinionUP}  Let $\G$ be a symmetric (serial and non-necessarily functional) influence profile and $\O$ be an opinion profile. $\O$ converges in UP on $\G$ if and only if:  \begin{itemize}  \item[] For some $p\in\Atoms $, for all $i,j\in C$, where $C$ is a connected component of $G_p$: if $i\in R_p(j)$, then $O_i(p)\neq O_j(p)$.   \end{lemma}  \begin{proof}  Assume that for any $p\in\Atoms$, for any connected component $C$ of $G_p$, there exist $i,j\in C$, such that $ R_p(j) and O_i(p)= O_j(p)$. By definition of UP, this implies that $O_i(p)$ is stable, and that all agents with distance $\leq k$ will be stable after at most $k$ steps. The above \ref{lemma:suff.UP} provides the other direction.   \end{proof}  \begin{lemma}\label{lemma:symm.influenceUP}  Let $\G$ be a symmetric (serial and non-necessarily functional) influence profile.  All opinion profiles $\O$, converge in UPs on $\G$ iff:  \begin{itemize} 
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\end{itemize}  \end{theorem}  \begin{proof}  This follows from \ref{lemma:influenceUP} \ref{lemma:symm.influenceUP}  and \end{proof}