This is a question about properties of large directed graphs which are preserved when we randomly sample edges. Imagine I have an infinite sequence of positively weighted directed graphs.  The graphs are represented by adjacency matrices$$M_1,M_2,...,M_n,...$$where each $M_n \in \mathbb{R}^{n \times n}$. The sup-norm $\|M_n\|_{\infty} = \max_{i} \sum_{j=1}^n |M_{n}(i,j)|$ is uniformly bounded by some bound $B$  for all $n \in \mathbb{N}$. Furthermore, for any $\ell < n $ we have $M_{\ell}(i,j) = M_n(i,j)$ as long as $i,j \leq \ell$. Many times we do not observe the full graph. Instead of observing $M_n$, we observe a random variable $\tilde{M_n}$where $$\tilde{M_n}(i,j) = \begin{cases} M_n(i,j) & \text{ with probability } \pi \\ 0   & \text{ with probability } 1-\pi.\end{cases}$$Let $\tilde{y_n} = f(\tilde{M_n})$ be a real-valued function of the matrix $\tilde{M_n}$. Using a Chernoff bound, I know that if $f(\tilde{M_{n}}) = \sum_{i,j} a_{ij} \tilde{M_n(i,j)}$ is a linear function, then the value of $f(\tilde{M_{n}})$ is tightly concentrated around its expectation, with the probability of it being ``far away'' form $E[f(\tilde{M_n})]$ decreasing exponentially with $n$.Is this true for non-linear properties? For example, if $f(\tilde{M_n})$ is a convex function, do we have a similar concentration bound?
Concentration Bounds for functions of matrices
graph theory;graph algorithms;chernoff bound