deletions | additions       diff --git a/section_Unanimity_and_2_colorability__.tex b/section_Unanimity_and_2_colorability__.tex  index 9dcc856..12dd162 100644  --- a/section_Unanimity_and_2_colorability__.tex  +++ b/section_Unanimity_and_2_colorability__.tex 
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The possibility of this extreme distribution of opinions relies on the graph $G_p$ being $2$-colorable, which is equivalent to $G_p$ not containing any 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} \begin{lemma}\label{lemma: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}  \item[] For all $p\in\Atoms$, $G_p$ is not $2$-colorable.   \end{lemma}  \begin{proof}  Let $p\in\Atoms$ and $G_p$ be not $2$-colorable. This means that for some connected set component  of $G_p$ containing the  nodes $S\subseteq\N$ from $C\subseteq\N$  with diameter $k$ in $G_p$, $k$,  there is a cycle of length $c$ for $c$ odd. Let $O$ be an arbitrary opinion profile. Since $k$ is odd, there exist $i,j\in S$ such that $j\in R_p(i)$ 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 lemma above gives the other direction. \end{proof}  The above characterizes the influence profiles which guarantee that any opinion profile converge, but, as we have done above for the BDPs, we can also characterize the pairs of influence and opinion profiles which converge:  \begin{theorem}  Let $\G$ be a symmetric (serial and non-necessarily functional) influence profile and $\O$ be an opinion profile.  The UP converges for $\O$ on $\G$ iff: