Direction of current flow in resistor networks
if we adopt a "positive" direction for the current (I guess it doesn't matter which way is chosen), how do we determine the current direction along each edge? In other words, having now induced a current flow through the resistor network, my originally undirected graph will become a directed one, but I don't know how the edge directionality ought to be defined consistently.


You can't necessarily know which direction current is flowing without completing the analysis of the circuit (calculating the value of every current, magnitude as well as direction). This is because every current path in a circuit influences the others, except in specific short or open-circuit cases.

(In complex circuits we assume that — or rather, engineer them so that — some influences are minimal and approximate the short or open cases; for example, we assume that the power supply "rails" have a constant voltage (and design the power supply and distribution so that this assumption is almost correct) regardless of the current taken from them, so we can analyze each powered sub-circuit independently, and insofar as they aren't independent we call these interactions “noise” rather than trying to analyze/simulate the entire system except in unusual circumstances.)



Therefore, don't insist on getting the correct direction of current flow as an early step in analyzing a circuit. Instead, pick a direction arbitrarily (or guess if you're doing it by hand), then complete your analysis. If the answer turns out to be negative, then you can reverse the edge of your directed graph if you care.

Note also that if you analyze a circuit under different conditions (simple example: a battery or capacitor may be either charging or discharging) then the current flow may be reversed, but if you want to see how the circuit behaves it's more useful to have a consistent edge direction that sometimes results in a negative current (or voltage or power) than to have only positive values.