# Phorgy Phynance

## Network Theory and Discrete Calculus – Notation Revisited

This post is part of a series

As stated in the Introduction to this series, one of my goals is to follow along with John Baez’ series and reformulate things in the language of discrete calculus. Along the way, I’m coming across operations that I haven’t used in any of my prior applications of discrete calculus to mathematical finance and field theories. For instance, in the The Discrete Master Equation, I introduced a boundary operator

\begin{aligned} \partial \mathbf{e}^{i,j} = \mathbf{e}^j-\mathbf{e}^i.\end{aligned}

Although, I hope the reason I call this a boundary operator is obvious, it would be more precise to call this something like graph divergence. To see why, consider the boundary of an arbitrary discrete 1-form

\begin{aligned}\partial \alpha = \sum_{i,j} \alpha_{i,j} \left(\mathbf{e}^j - \mathbf{e}^i\right) = \sum_i \left[ \sum_j \left(\alpha_{j,i} - \alpha_{i,j}\right)\right] \mathbf{e}^i.\end{aligned}

A hint of sloppy notation has already crept in here, but we can see that the boundary of a discrete 1-form at a node $i$ is the sum of coefficients flowing into node $i$ minus the sum of coefficients flowing out of node $i$. This is what you would expect of a divergence operator, but divergence depends on a metric. This operator does not, hence it is topological in nature. It is tempting to call this a topological divergence, but I think graph divergence is a better choice for reasons to be seen later.

One reason the above notation is a bit sloppy is because in the summations, we should really keep track of what directed edges are actually present in the directed graph. Until now, simply setting

$\mathbf{e}^{i,j} = 0$

if there is no directed edge from node $i$ to node $j$ was sufficient. Not anymore.

Also, for applications I’ve used discrete calculus so far, there has always only been a single directed edge connecting any two nodes. When applying discrete calculus to electrical circuits, as John has started doing in his series, we obviously would like to consider elements that are in parallel.

I tend to get hung up on notation and have thought about the best way to deal with this. My solution is not perfect and I’m open to suggestions, but what I settled on is to introduce a summation not only over nodes, but also over directed edges connected those nodes. Here it is for an arbitrary discrete 1-form

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

where $[i,j]$ is the set of all directed edges from node $i$ to node $j$. I’m not 100% enamored, but is handy for performing calculations and doesn’t make me think too much.

For example, with this new notation, the boundary operator is much clearer

\begin{aligned} \partial \alpha &= \sum_{i,j} \sum_{\epsilon\in [i,j]} \alpha_{i,j}^{\epsilon} \left(\mathbf{e}^{j}-\mathbf{e}^i\right) \\ &= \sum_i \left[\sum_j \left( \sum_{\epsilon\in[j,i]} \alpha_{j,i}^{\epsilon} - \sum_{\epsilon\in[i,j]} \alpha_{i,j}^{\epsilon} \right)\right]\mathbf{e}^i.\end{aligned}

As before, this says the graph divergence of $\alpha$ at the node $i$ is the sum of all coefficients flowing into node $i$ minus the sum of all coefficients flowing out of node $i$. Moreover, for any node $j$ there can be one or more (or zero) directed edges from $j$ into $i$.