Efficient generation of graph structured correlated random variables via MCMC/Gibbs

Sometime back I had asked this question about generating correlated random draws based on the correlation structure given by a graph. Link Here

The solution there requires to create $n\times n$ correlation matrix which can be prohibitive for large $n$ and also doing the Cholesky decomposition can be challenging.

So, I am contemplating a more MCMC style approach. Say, the graph looks like:

a --> b --> c

Then, first I generate "a" from $\mathcal{N}(0,1)$. Next, I generate "b" conditional on "a" e.g. as $a*\rho*\sqrt{1-\rho^2}$ etc and so on.

Or more generally generate a random variable for a node conditional on its "Markov Blanket".

So, here's my question

1). Has anyone tried to generate random variables in this iterative MCMC style fashion?

2). The graph that I drew above is very simple. If we have to draw a random variable conditioned on (say) 10 other variables' fixed values i.e. it has 10 neighbors in the graph. Then how would we do it? Do we still follow the standard procedure as here

generating random variables by fixing some