My interest in electrical networks began with the 2013 summer REU at the University of Washington with James Morrow. I continued researching under Morrow through my undergrad days and gave talks at subsequent REUs. Now I’m collaborating with Avi Levy, a grad student at UW.
The best write-up of my results so far is on arXiv.
Setup of the Problem:
We have a graph with vertices and edges which represents an electrical network. Each vertex represents a “node” or “resistor” and each edge is “wire.” Each edge has a positive weight or conductance . The current flowing from node to node is the conductance of the edge times the difference in voltage from to . The net current at each node is the sum of the currents exiting that node.
Some of the nodes are declared to be interior and some of them are declared to be boundary. The forward problem is this: We are given certain voltages on the boundary nodes; can we find voltages on the interior nodes that will make the net current at each interior vertex zero? Once we show that such a voltage function exists, we can find the net currents on the boundary vertices.
Now we come to the inverse problem. We know the graph (the shape of the network). We do not know the conductances. For any voltages on the boundary nodes, we know what the net currents on the boundary nodes will be. Can we find the conductances?
Electrical networks are a mathematical model used by engineers and physicists, but they are also studied by probabilists because of their connection to random walks on graphs.
Harmonic functions can be defined when the edge-weights are in an arbitrary ring and the potentials are in an -module . Electrical networks over rings are related to the sandpile group, and can be used to assign interesting algebraic invariants to graphs with boundary.