Suppose that we have a fully connected graph of size $N$, each unit with $k$ neighbors. This can be written as an adjacency matrix $A$. Then, let $I$ be an identity matrix of size $N$, with $c \in [0,1]$ a constant.
If we define:
$$ V = (I-cA)^{-1} $$
is there a way to approximate a general formula for $V$ in terms of $k$ and $N$? Or is there also a way to characterize $V_{ij}$ for $i\neq j$ in closed form? Thanks.
There is no form in terms of $N$, $k$ and $c$.
Take two admissible matrices A. Note that $f_A(c)=(I-cA)^{-1} = \sum_{n=0}^\infty c^n A^n$ so if $f_A = f_B$ then $A^n$=$B^n$ for all $n$ (because Taylor expansions are unique) but just take two non-isomorphic admissible graphs with the same $N$ and $k$ values.
Relevant: $A^n_{(i,j)}$ is the number of $n$-length walks from $i$ to $j$. Imagine a bug starting at $i$ and moving around the graph. At each edge he has probability $p$ of dying. What is the $\mu$, the expected number of times the bug visits $j$? Well, $\sum_{n=0}^\infty P_n$ where $P_n$ is the probability of the bug being at $j$ at $n$ steps, surviving. There are $k^n$ paths from $i$ and $A^n_{(i,j)}$ paths from $i$ to $j$ so $P_n=p^n A^n_{(i,j)}/k^n$. $\mu = \sum_{n=0}^\infty (p/k)^n * A^n_{(i,j)} = (I-(p/k)A)^{-1}$