Do I influence myself more than my neighbors? Consider relations between people is defined by a weighted symmetric undirected graph $W$, and $w_{ij}$ shows amount of weight $i$ has for $j$. Assume all weights are non-negative and less than $1$ i.e. $$0\leq w_{ij}<1, \forall{i,j}$$ and symmetric $w_{ij}=w_{ji}$. We say $i$ and $j$ are friends if $w_{ij}>0$.
Define the influence matrix $\Psi=[\text{I}+W]^{-1}$ (Assume it is well-defined). Is it always the case that my influence on myself is greater than my influence on my friends? $$\Psi_{i,i}>\sum_{j\in N(i)}|\Psi_{i,j}|$$
where $N(i)$ is a set of friends of $i$.

Look at the inverse of $$\left[\begin{array}{ccc}1 & 0 & .9\\ 0 & 1 & .9\\ 0 & .9 & 1\end{array}\right]$$
