Invariance property for extended means

e study the properties of the mean-type mappings $\mathbf{M}\text{\hspace{0.17em}⁣}:I^p\to I^p$ of the form $$\mathbf{M}(x_1,\dots ,x_p):=(M_1(x_{{\alpha }_{1,1}},\dots ,x_{{\alpha }_{1,d_1}}),\dots ,M_p(x_{{\alpha }_{p,1}},\dots ,x_{{\alpha }_{p,d_p}})),$$ where $p$ and $d_i$-s are positive integers, each $M_i$ is a $d_i$-variable mean on an interval $I\subset \mathbb{R}$, and ${\alpha }_{i,j}$-s are elements from $\{1,\dots ,p\}$. We show that, under some natural assumption on $M_i$-s, the problem of existing the unique $\mathbf{M}$-invariant mean can be reduced to the ergodicity of the directed graph with vertexes $\{1,\dots ,p\}$ and edges $\{({\alpha }_{i,j},i)\text{\hspace{0.17em}⁣}:i,j\text{ admissible}\}$.