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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?

I have one more question. According to the definition of Possion Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed in a restricted area) similar to PPP(An infinite number of nodes distributed in an infinite area)?

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?

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area as a Poisson point process of density $\lambda$.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?

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?

I have one more question. According to the definition of Possion Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed in a restricted area) similar to PPP(An infinite number of nodes distributed in an infinite area)?