Hoping for some help with approaching the following problem:
As shown in the graphic, there are five nodes A,B,C,D,E and a set of directed edges. The "weights" are not scalars though but 2-dimensional vectors.
Let's image the nodes to be cities and the edges to be roads that connect the cities. Each road/edge has a length and a width (e.g. from A to B 1/1 and from B to C 1/0.8). The length is the (discrete) time it takes the cars to travel to the other city and the width is the proportion of cars that travel along this road (assuming the cars can be infinitely divided into pieces).
For example, let's assume 10 cars start from city A (and will never leave the "graph"). All 10 cars travel along the edge from A to B and it takes them one unit of time. At city B the 10 cars are divided and 0.8*10=8 cars travel along the edge B->C and it takes 1 time unit and 0.2*10=2 cars travel along edge B->D and it takes 0.5 time units.
Let's assume in each city there is some person who tracks how many cars travel through the city at each time unit (graphs beside the nodes, forget the graph for node D, ups).
If cars arrive at the same time in a city they add up again. So at any given time the sum of all cars must be 10.
How to formulate this problem?
Is there an analytical solution?
What is the long-term behavior of this "system? In other words, how would the "protocol" of the person in a city that tracks the number of cars look like for a given amount of time e.g. 100 time units?
How to generalize to n-dimensional "edge vectors"?