The image above shows the "flows" of some arbitrary entities along certain paths $(i = 1,2...6)$. The value of the flow over a path is equal to $x_i$. Each dot is a "node". I need to find a system of equations such that the net flow at each node is zero. Then, I need to find the solutions to this system such that the flow along each path is either $1$, $0$, or $-1$.
I've found that the following system has all net flows at each node equal to zero:
$x_1 + x_6 -x_2 = 0$ ..... (describes the flow at the node between paths 1 and 2)
$x_2 - x_3=0$ ...... (the flow at node between paths $2$ and $3$, etc.)
$x_3-x_4=0$
$x_4-x_5-x_6=0$
$x_5-x_1=0$
I am pretty sure I could also represent this situation via the following system:
$x_1 + x_2 + x_3 + x_4 + x_5 = x_6$
$x_1 + x_2 + x_3 + x_4 + x_6 = x_5$
$x_1 + x_5 = x_6$
However, I'm not sure how to represent a "solution" here, or how to determine the solutions such that the flow ($x_i$) along each path $(i=1,2...6)$ is either $1$, $0$, or $-1$.
Any tips would be greatly appreciated. Please don't give me the entire answer, though, as I would like to arrive at the answer myself.
Thanks!