Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less than $\mathcal{R}$. The adjacent matrix is $A$. We define an operation on adjacent matrix "$\circ$". For adjacent matrix $A$ and $B$
$$C=A\circ B$$
$$ c_{ij}=\left\{
\begin{array}{rcl}
0       &      & {\sum_{k=1}^{N}a_{ik}b_{kj}=0}\\
\\
1       &      & {\sum_{k=1}^{N}a_{ik}b_{kj}>0}
\end{array} \right. $$
$A^{[1]}=A$, $A^{[i+1]}=A^{[i]}\circ A$.
Define the intial adjacent matrix $A_0=A$. We select a pair of elements randomly from $A_i$ :$A_i(m,n)$ and $A_i(n,m)$. $A_{i+1}(m,n)=A_{i+1}(n,m)=1-A_{i}(m,n)=1-A_{i}(n,m)$.
Other element of $A_{i+1}$ is equal to the corresponding element of $A_i$.
Then we can get a series of matrix $A_0, A_1, \cdots, A_s$.
There comes the question: $A_0^{[\infty]} \circ A_1^{[\infty]} \circ A_2^{[\infty]} \circ \cdots \circ  A_s^{[\infty]}$=$?$
I have waited for more than a week? Could anyone help?