## Network Theory and Discrete Calculus – Electrical Networks

This post is part of a series

### Basic Equations

In Part 16 of John Baez’ series on Network Theory, he discussed electrical networks. On the day he published his article (November 4), I wrote down the following in my notebook

and

The first equation is essentially the discrete calculus version of Ohm’s Law, where

is a discrete 1-form representing conductance,

is a discrete 0-form representing voltage, and

In components, this becomes

The second equation is a charge conservation law which simply says

where

is the sum of all currents into node and

is the sum of all currents out of node . This is more general than it may first appear. The reason is that directed graphs are naturally about spacetime, so the currents here are more like 4-dimensional currents of special relativity. The equation

is related to the corresponding Maxwell’s equation

where is the adjoint exterior derivative and is the 4-current 1-form

This also implies the discrete Ohm’s Law appearing above is 4-dimensional and actually a bit more general than the usual Ohm’s Law.

### Some Thoughts

I’ve been thinking about this off and on since then as time allows, but questions seem to be growing exponentially.

For one, the equation

is curious because it implies that is a derivative, i.e.

Further, although by pure coincidence, in my paper with Urs, we introduced the graph operator

and showed that for any directed graph and any discrete 0-form that

Is it possible that and are related?

I think they are. This brings thoughts of spin networks and Penrose, but I’ll try to refrain from speculating too much beyond mentioning it.

If they were related, this would mean that the discrete Ohm’s Law above simplifies even further to

and

In components, the above becomes

This expresses an effective conductance in terms of the total number of directed edges connecting the two nodes in either direction, i.e.

If the ‘s appearing in the conductance 1-form are themselves effective conductances resulting from multiple more fundamental directed edges, then we do in fact have

Applications from here can go in any number of directions, so stay tuned!

[…] my last post, I mentioned the graph […]

Network Theory and Discrete Calculus – Quantized Conductance « Phorgy PhynanceDecember 17, 2011 at 10:24 pm