\mathcal{F}$ by $$\label{def:Tcal}
            \mathcal{T}(f,t) = \Lambda^t f.$$ We can then define the *one step* deterministic *idealistic process* as $$\label{def:idealProc}
            K^t_{\mathcal{T}} := \mathcal{T}(k^t,t), \quad  K^0_{\mathcal{T}} = k^0 \in \mathcal F^N.$$

Using these idealistic processes, the time evolution of the knowledge-like function is given by $$\label{eq:decompkt}
    k^t = \Delta k^t + K_{\mathcal{T}}^t,$$ where $\Delta k^t = k^t - K_{\mathcal{T}}^t$.

Let $\mathcal M_\mathcal{F} = \{(k, ...,k), k \in \mathcal{F}\}$ be the space of all the common shared conceptualizations.

We first prove the contraction of the idealized process $K_{\mathcal{T}}^t$ from in the space ${\mathcal{M}}_\mathcal{F}$ using some algebraic properties of primitive matrices, as well as results about inhomogeneous Markov chains. Secondly, we shall prove the convergence of the process $\Delta k^t$ with high probability using learning theory. Then, under certain hypothesis (such as $\tau = 0$), we prove the convergence of $k^t$ towards the set ${\mathcal{M}}_\mathcal{F}$ with high probability.

Primitive matrices and their applications
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[The behavior of processes is mainly driven by the influence matrix $\Lambda$. In this Section, we study the relationship between the properties of the influence matrix and the interactions taking place within the population.]{}

A matrix $A \in \mathcal{M}_N(\mathbb{R})$ is said to be primitive if $A\geq0$ and if $\exists k 