In a fork,
A <- C -> B, A and B are independent given C. We can say that A and B are d-separated or the path between them is blocked by C, given C. So information does not flow (why?). In a chain,
A -> C -> B, A and B are independent given C. Again, we can say that A and B are d-separated or the path between A and B is blocked by C, given C. However, in a collider, the situation is somehow the opposite. In a collider,
A -> C <- B, A and B are independent NOT given C, but if C (or any descendant of C) is given they are dependent.
What exactly is a path in a casual model? A path can apparently have edges not all of the same direction, but why? I read that a path somehow tells us if information flows. Which info? Why does a path would allow information to flow (or not)? In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)? In which sense information flows or not flows along these paths?