Network Theory and Discrete Calculus – Electrical Networks

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Network Theory and Discrete Calculus

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

$G\circ dV = [G,V] = I$ and $\partial I = 0.$

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

$\begin{aligned} G = \sum_{i,j} \sum_{\epsilon\in[i,j]} G_{i,j}^\epsilon \mathbf{e}^{i,j}_\epsilon \end{aligned}$

is a discrete 1-form representing conductance,

$\begin{aligned} V = \sum_i V_i \mathbf{e}^i \end{aligned}$

is a discrete 0-form representing voltage, and

$\begin{aligned} I = \sum_{i,j} \sum_{\epsilon\in[i,j]} I_{i,j}^\epsilon \mathbf{e}^{i,j}_\epsilon. \end{aligned}$

In components, this becomes

$G_{i,j}^\epsilon \left(V_j - V_i\right) = I^\epsilon_{i,j}.$

The second equation is a charge conservation law which simply says

$I_{*,i} = I_{i,*},$

where

$\begin{aligned} I_{*,i} = \sum_j \sum_{\epsilon\in[j,i]} I^\epsilon_{j,i}\end{aligned}$

is the sum of all currents into node $i$ and

$\begin{aligned} I_{i,*} = \sum_j \sum_{\epsilon\in[i,j]} I^\epsilon_{i,j}\end{aligned}$

is the sum of all currents out of node $i$ . 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

$\partial I = 0$

is related to the corresponding Maxwell’s equation

$d^\dagger j = 0,$

where $d^\dagger$ is the adjoint exterior derivative and $j$ is the 4-current 1-form

$j = j_x dx + j_y dy + j_z dz + \rho dt.$

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

$[G,V] = GV - VG = I$

is curious because it implies that $[G,\cdot]$ is a derivative, i.e.

$[G,V_1 V_2] = [G,V_1] V_2 + V_1 [G, V_2].$

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

$\begin{aligned} \mathbf{G} = \sum_{i,j} \sum_{\epsilon\in[i,j]} \mathbf{e}_\epsilon^{i,j}\end{aligned}$

and showed that for any directed graph and any discrete 0-form $\phi$ that

$d\phi = [\mathbf{G},\phi].$

Is it possible that $G$ and $\mathbf{G}$ 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

$dV = I$

and

$\partial d V = 0.$

In components, the above becomes

$\begin{aligned} \sum_j \left(V_j - V_i\right) \left(N_{i,j} + N_{j,i} \right) = 0.\end{aligned}$

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

$G^*_{i,j} = N_{i,j} + N_{j,i}.$

If the $G^\epsilon_{i,j}$ ‘s appearing in the conductance 1-form $G$ are themselves effective conductances resulting from multiple more fundamental directed edges, then we do in fact have

$G = \mathbf{G}.$

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

Written by Eric

December 10, 2011 at 9:23 pm

Posted in Directed Graphs, Discrete Calculus, John Baez, Network Theory

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Network Theory and Discrete Calculus – Quantized Conductance « Phorgy Phynance

December 17, 2011 at 10:24 pm

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