Network Theory and Discrete Calculus – Quantized Conductance

The Graph Operator

In my last post, I mentioned the graph operator

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

and the fact the exterior derivative of a discrete 0-form can be expressed as a commutator

$\begin{aligned} dV = [\mathbf{G},V] = \sum_{i,j} (V_j - V_i) \mathbf{e}^{i,j}. \end{aligned},$

where

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

I then let myself speculate that the graph conductance 1-form

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

could be nothing more than the graph operator. In this post, I hope to explain a bit more how that might work.

Graph Conductance

Recall that the discrete Ohm’s Law

$[G,V] = I$

gives the total current

$\begin{aligned} I = \sum_{i,j} \sum_{\epsilon\in[i,j]} I^\epsilon_{i,j} \mathbf{e}^{i,j}_{\epsilon} = \sum_{i,j} (V_j-V_i) \sum_{\epsilon\in[i,j]} G^\epsilon_{i,j} \mathbf{e}^{i,j}_{\epsilon}. \end{aligned}$

If we did not need to probe the current in any one of the individual parallel directed edges, it would be tempting to replace them with a single effective directed edge representing the total current flowing them, i.e.

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

where

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

In doing so, we could also replace the conductances with a single effective conductance

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

where

$\begin{aligned} G_{i,j} = \sum_{\epsilon\in[i,j]} G^\epsilon_{i,j}.\end{aligned}$

Equivalence

Could it be that $\mathbf{G} = G$ ?

Let $P[i,j]$ denote a partition of the set $[i,j]$ of directed edges from node $i$ to node $j$ and express the graph operator as

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

where

$\begin{aligned} N^E_{i,j} \mathbf{e}_E^{i,j} = \sum_{\epsilon\in E} \mathbf{e}^{i,j}_\epsilon \end{aligned}$

and $N^E_{i,j}$ is the number of directed edges in the subset $E$ . This would only make sense if we were not going to probe into any single directed edge within any element of the partition.

Comparing this to the conductance

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

we see that the graph conductance can be interpreted as the graph operator where each directed edge of our electric network is actually composed of a number $G^\epsilon_{i,j}$ of fundamental directed edges, i.e. conductance is simply counting the number of sub-paths within each directed edge.

Thoughts

As before, thinking about this (as time allows) raises more questions than answers. For example, if the above makes any sense and is in any way related to nature, this would imply a fundamental unit of conductance and that conductance should be quantized, i.e. come in integer multiples of the fundamental unit. For completely unrelated (?) reasons, conductance is observed to be quantized due to the waveguide like nature of small, e.g. nano, wires and the fundamental unit of conductance is given by

$G_0 = \frac{2 e^2}{h},$

where $e$ is the electron charge and $h$ is Planck constant.

This also makes me think of the geometric origin of inhomogeneous media. In vacuum, I would expect there to be just a single directed edge connecting any two nodes. Hence, I would expect $G^\epsilon_{i,j} = G_0$ in vacuum. In the presence of matter, e.g. components of an electrical network, there should be bunches of directed edges between any two nodes.

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Written by Eric

December 17, 2011 at 10:24 pm

Posted in Azimuth Project, Directed Graphs, Discrete Calculus, Network Theory

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