Network Theory and Discrete Calculus – Electrical Networks
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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
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.
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
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!