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\section{Introduction}  %%%%%%%%%%%%%%%%%%%%%%%%%  \section{Preliminaries}  \subsection{Boolean DeGroot Processes}  In \cite{DeGroot}, DeGroot proposes a simple model of step by step opinion change under social influence. The model combines two types of matrices. Assuming a group of $n$ agents, a first $n\times n$ matrix represents the weighted influence network (who influences whom and how much), and a second $n \times m$ matrix represents the probability assigned by agents to $m$ different alternatives.   Both the agents' opinion and the influence weights are taken within $[0,1]$ and are (row) stochastic (each row sums up to $1$).  \subsection{Dynamics}  \subsection{Modal $\mu$-calculus}  %%%%%%%%%%%%%%%%%%%%  \section{Stabilization}  \subsection{Sufficient conditions for convergence}  Limit cases of theorems by Jackson and the Toulousians.  \subsection{Colorability}  %\subsection{STABILITY and 2-COLORABILITY}  There are two stability-related questions, given an initial model:   \begin{enumerate}  \item Is the model stable in the sense that it is a limit point of the dynamics (that is it is a fixpoint of $\Box$, if my thoughts above are correct)? YES!  \item Is the model such that it will eventually reach a limit point, or oscillate instead?  \end{enumerate}  As to 1), if the validity of the fixpoint equation $p \leftrightarrow \Box p $ fails in the model, then we are not in a stable model. Notice that your Proposition 2, if one assumes the frame is functional, states precisely that the model is unstable whenever $p \leftrightarrow \Box p $ is not valid in the model.  As to 2), if neither p nor ~p is a post-fixpoint of $Box$ then the frame is 2-colorable (and vice versa). And that means the model cannot possibly stabilize. @DAVIDE: NO, AS MENTIONED THIS MORNING, THIS DEPENDS ON THE FRAMES BEING SYMMETRIC, SO I NEED TO THINK A BIT BETTER INTO WHAT THE FORMULA MEANS WITHOUT SYMMETRY IMPOSED. I WILL COME BACK TO THIS VERY SOON. (we have seen that for instance a simple triangle can carry non-stabilizing opinions).   The Boolean case of the theorem by Jackson should state conditions under which, no matter what the initial assignment for $p$ is, the model will always stabilize (that is either $p$ or $\lnot p$ is a post-fixpoint of $\Box$). So Jackson’s theorem should state conditions that make a graph non-2-colorable.   Indeed, Jackson's theorem excludes 2-colorable graphs, since they are not aperiodic (the greater divisor of all cycles lengths is 2). Jackson results implies that 2-colorable graphs do not guarantee stabilization for any distribution of opinions (and of influence weights). However, more can be said.   For instance, in the case of \emph{symmetric} frames, all and only models which are not proper two-coloring of the graphs will stabilize. This gives us a full characterization of stabilization. There exists only 2 such colorings (for each connected component) of a graph. And the loop is always of size two.NOTE THAT THIS LAST PARAGRAPH DOES NOT RELY ON THE FRAME BEING FUNCTIONAL, SO THIS IS A MORE GENERAL RESULT ABOUT UNANIMITY RULE IN SYMMETRIC GRAPHS. MAYBE WE SHOULD ALSO INTRODUCE IT AS SUCH THEN.   QUESTIONS:   WHAT PROPERTY DOES THE FORMULA CHARACTERIZING STABILIZATION CORRESPOND TO ON NON-SYMMETRIC FRAMES?   WHAT ARE THE CONDITIONS FOR STABILIZATION IN THE RESTRICTED CASE OF BDG (COMPARED TO THE GENERAL CASE OF JACKSON'S THEOREM)  %%%%%%%%%%%%%%%%%%%%  \section{An application to liquid democracy}  Liquid Democracy (or ``proxy voting") is a way of organizing voting. On each issue submitted to vote, each agent can either cast his own vote, or he can delegate it to another agent (a ``proxy"), who he considers to be better equipped to make the best decision. As such, proxy voting stands in between direct democracy and standard representative democracy.   It has been argued that liquid democracy leads to collective outcomes which are closer to individual opinions than representative democracy \cite{}.   It has also been argued that liquid democracy would be too unstable to be implemented in practice as a replacement of representative democracy \cite{}.  \begin{Conclusions}  %%%%%%%%%%%%%% NOTES %%%%%%%%%%%%%%%%%%%  \appendix  \section{Irrational individuals?}