I'm quite confused on how does Kirchhoff's law work in a closed network (seen as a graph), if current could only flow from a higher potential node to a lower potential node.
In my linear algebra textbook there is a section about applying matrix operations to matrices that represent electrical circuits. The graph representing the current flows between nodes is directed and has directions going from the higher to lower potential node. Then after grounding a node it is possible to solve for the magnitude of the current flows.
What's surprising is that it's possible for the current flows to be negative, which means that it will be in the opposite direction as stated, and the current would be flowing in the wrong direction.
I also don't quite understand how could Kirchhoff's law hold if current could only flow from high potential to low potential nodes. If this is so, then the highest potential node will have no in/out current since current can only flow out and the sum of all current flows must be 0, then inducting we get that all nodes must have no current.
Kirchhoffs current law (KCL) states that the sum of all currents flowing into a node equals zero. KCL deals with the sum of currents at a node and not potential. KVL deals with potential and says that the total voltage around a loop equals zero. Matrix operations are used to solve for system variables.