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Thus, based on this idea and works in literature \cite{Kuhnle2018,zhang2016least}, we define a multiplex OSNs.
\textbf{Definition 1 :} A multiplex OSNs with $n$ networks is a set \mathbb{G}^n = (I,G^n ) ,
where $I=(V,C)$ is the set of individuals represented in the center of Fig. \ref{fig:fig1} \todo{\textbf{Answer2.2:} The caption figures have been consolidated with addition information explaining the main idea behind each figure.} where for each individual $i\in I$ is represented by a node $v \in V$ and a set of characteristics $c \in C$.
The characteristic of an individual defines its behaviors toward a rumor which will be defined in the following section( see section\ref{sec:32}).
The set $G^n=\{ G_1=(V,E_1), $ $G_2=(V,E_2), ... ,G_n=(V,E_n)\}$ is a set of $n$ graphs, where $G_i=(V,E_i)$ is a directed graph representing an OSN; for example in Fig. \ref{fig:fig1}, $G^3=\{ G_1=(V,E_1), $ $G_2=(V,E_2),G_3=(V,E_3)\}$ are respectively Instagrame, Twitter and FaceBook networks represented by directed graph. .
Without loss of generality, we consider each network of the multiplex has the same number of nodes.
Therefore, if a node $v \in G_i$ does not belong to $G_j$, we add this node to $G_j$ as an isolated node presented in black colors in Fig. \ref{fig:fig1}.}
\subsection{Individual Behavior Toward a Rumor Formulation}
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