deletions | additions       diff --git a/section_Unanimity_and_2_colorability__.tex b/section_Unanimity_and_2_colorability__.tex  index edf28a3..91bbd3e 100644  --- a/section_Unanimity_and_2_colorability__.tex  +++ b/section_Unanimity_and_2_colorability__.tex 
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\begin{align}  \O_i^{n+1}(p) & = \left\{  \begin{array}{ll}  \O_i^{n}(p) & \mbox{if for some $j,k\in R_p(i)$,$\O_j^{n}(p)\neq \O_k^{n}(p)$ R_p(i)$,$O_j^{n}(p)\neq O_k^{n}(p)$  } \\ \O_j^{n}(p) & \mbox{otherwise, where $j \in R_p(i)$ }   \end{array}  \right. 
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We give a sufficient condition for non-convergence of UPs:  \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 $\O_i(p)\neq \O_j(p)$. $O_i(p)\neq O_j(p)$.  Then $\O$ does not converge in UP.  \end{lemma}  \begin{proof}  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 $i,j\in C$ with $C$ a connected component of $G_p$: if $i\in R_p(j)$, then $\O_i(p)\neq \O_j(p)$. $O_i(p)\neq O_j(p)$.  Then, by definition of UPs, for all $i\in C$, $\O^1_i(p)\neq \O_i(p)$, $O^1_i(p)\neq O_i(p)$,  and by repeating the same argument, for all $n\in\mathbb{N}$, $\O^{n+1}_i(p)\neq \O^n_i(p)$. $O^{n+1}_i(p)\neq O^n_i(p)$.  \end{proof}  Intuitively, the above condition for non-convergence corresponds to a situation of global maximal disagreement: \emph{all} agents (of a connected component) disagree with \emph{all} their influencers.  
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Let $\G$ be a symmetric (serial and non-necessarily functional) influence profile and $\O$ be an opinion profile. The following are equivalent:   \begin{enumerate}[noitemsep]  \item $\O$ converges in UP on $\G$.  \item For all $p\in\Atoms $, for all connected component $C$ of $G_p$, there are $i,j\in C$, such that $i\in R_p(j)$, and $\O_i(p)= \O_j(p)$. $O_i(p)= O_j(p)$.  \end{enumerate}  \end{lemma}  \begin{proof}  \fbox{$2) \Rightarrow 1)$} 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 $\O_i(p)= \O_j(p)$. $O_i(p)= O_j(p)$.  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. \fbox{$1) \Rightarrow 2)$} This follows from Lemma~\ref{lemma:suff.UP}. \end{proof}  This means that opinions on a given $p$ will converge if and only if two agents influencing each other on $p$ already agree on it. We can therefore, as we did for BDPs, characterize the class of influence profiles for which all (symmetric) opinion profiles converge in UPs: 
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\end{enumerate}  \end{theorem}  \begin{proof}  \fbox{$2) \Rightarrow 1)$} Let $p\in\Atoms$ and $C$ be connected component of $G_p$ with diameter $k$. Let $C$ contain a cycle of length $c$, with $c$ odd. Let $\O$ be an arbitrary opinion profile. Since $c$ is odd, there exist $i,j\in S$ such that $j\in R_p(i)$ and $\O_i(p)=\O_j(p).$ $O_i(p)=O_j(p).$  By definition of UP, this implies that $\O_i(p)$ $O_i(p)$  is stable, and that all agents with distance $\leq k$ will be stable after at most $k$ steps. Hence, $\O$ converges. \fbox{$1) \Rightarrow 2)$} This follows from Lemma~\ref{lemma:symm.opinionUP}. \end{proof}