deletions | additions       diff --git a/section_Colorability_and_unanimity_In__.tex b/section_Colorability_and_unanimity_In__.tex  index 17de44d..a18effb 100644  --- a/section_Colorability_and_unanimity_In__.tex  +++ b/section_Colorability_and_unanimity_In__.tex 
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Recall that a graph is properly two-colored if no node has a successor of the same color. The above result can be reformulated as follows: if for some $j$, $O$ properly colors $G_{p_j}$, then $O$ does not converge. In fact, it is easy to see that in such a case, all agents will change their opinion on $p_j$ at every step, entering an oscillation of size $2$. Note that this limit case of opinion distribution is yet another special case of DeGroot processes, another intersection between the frameworks propositional opinion diffusion and DeGroot.   The possibility of this extreme distribution of opinions relies on the graph $G_{p_j}$ being $2$-colorable. $2$-colorable, which is equivalent to $G_{p_j}$ not containing any cycle of odd length.  However, this non $2$-colorability  is not a necessary condition for non-convergence of UPs: UPs in general:  a simple cycle of three agents, for instance, is not $2$-colorable but (as we have shown above with BDPs) does not necessarily converge. guarantee convergence.  Nevertheless, there is a class of (non-necessarily functional) influence profiles for which begin $2$-colorable is a necessary condition of non-convergence: non-convergence,  theclass of  \emph{symmetric} influence graphs. profiles:  \begin{proposition}  Let $\G=(G_{p_1},\dots,G_{p_m})$ be a symmetric (and non-necessarily functional)  influence profile and $\O=(O_1,\dots,O_n)$ be an opinion profile. The opinion diffusion under the unanimity rule converges for $\O$ on $\G$ iff:  \begin{itemize}  \item[] For all $j\in\{1,\ldots,m\}$, $G_pj$ is not $2$-colorable, OR