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Let us first recall some vocabulary.   Let $G = \tuple{\N, R}$ be a graph and $R^*$ be the transitive and symmetric closure of $R$.   \begin{itemize}  \item[]A A  \emph{path} is a sequence of nodes $$, such that, for all $l\in\{1,\dots,k\}$, $i_lRi_{l+1}$;  \item[]A $i_lRi_{l+1}$.A  \emph{cycle} is a path of length $k$ such that $i_1=i_k$; \item[]A A  set of nodes $S\subseteq \N$ is said to be: \begin{itemize}  \item[]\emph{connected} if for any $i,j \in \S$: $iR^*j$,   \item[]\emph{strongly connected} if for any $i,j \in \S$: $iRj$,   \item[]\emph{closed} if for any $i\in S$, $j\notin S$, it is not the case that $iRj$,  \item[]a \emph{connected component} if for any $i,j \in \N$: $iR^*j$ if and only if $i,j\in C$, S$,  \item[]\emph{aperiodic} if the greatest common divisor of its cycles is $1$.  \end{itemize}  \end{itemize}  Note that the graphs we are interested in (finite, serial and functional) come with a special shape: each of their connected components contains exactly one cycle, and this cycle forms the ``tail" of the component:  \begin{fact} \label{fact:connected}  Let $G$ be an influence graph and $C$ be a connected component of $G$.