I am trying to solve a graph optimisation problem but find it difficult in how to approach it. Say we have a directed weighted (cyclic) graph where the weight of each edge is represented as a function f(x). How can I now find the cycle of a node which maximises the flow of the path?
The problem is essentially maximising, e.g., f(f(f(f(x)))). I have been looking at various algorithms such as Bellman-Ford for cycle-detecting and how the maximum flow problem is approached, but failing to incorporate these dynamic weights. Maximum flow is also more oriented towards the complete network instead of finding a single optimal cycle.
Not a full answer, but too long for a comment here. Trying to help, but frankly this question is quite vaguely worded.
It appears that what you are trying to do is quite hard. If you are allowed to set the capacities of arcs dynamically as specified here, then couldn't you reduce finding a Hamiltonian cycle to finding a maximum-flow cycle containing a vertex $v$ in the graph? Consider the following dynamic rule for setting arc weights: Let $P$ be the path you found so far and let $y$ be the ending vertex of $P$.
Then, if $P$ has no more than $n-1$ vertices, where $n$ is the number of vertices in $G$, do the following: If $y'$ is not in any vertex in $P$, set the weight [capacity?] $c(yy')$ of arc $yy'$ to be $n$, and if $y'$ is on $P$, then set $c(yy')$ to $1$.
If $P$ has $n$ vertices, then do the following: If $yy'$ completes a Hamiltonian cycle then set $c(yy')$ to $n$, otherwise set $c(yy')$ to $1$.
Then, by this arc weighting scheme, there is a cycle with maximum flow $n$ if $G$ has a Hamiltonian cycle. So an algorithm that finds a cycle that allows maximum flow with this weighting scheme, finds Hamiltonian cycle in any digraph $G$ that has one, in polynomial time. This is impossible unless $P=NP$.