Archive for the ‘Azimuth Project’ Category
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The Graph Operator
In my last post, I mentioned the graph operator
and the fact the exterior derivative of a discrete 0-form can be expressed as a commutator
I then let myself speculate that the graph conductance 1-form
could be nothing more than the graph operator. In this post, I hope to explain a bit more how that might work.
Recall that the discrete Ohm’s Law
gives the total current
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.
In doing so, we could also replace the conductances with a single effective conductance
Could it be that ?
Let denote a partition of the set of directed edges from node to node and express the graph operator as
and is the number of directed edges in the subset . 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
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 of fundamental directed edges, i.e. conductance is simply counting the number of sub-paths within each directed edge.
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
where is the electron charge and 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 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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Another Note on Notation
In a previous post, I introduced a slightly generalized notation in order to deal with directed graphs with multiple directed edges between any two nodes, e.g. parallel elements in electrical networks. However, the revised notation now makes some simpler calculations look more cumbersome. This is an example of what my adviser called the conservation of frustration. For example, the coboundary is now given by:
Applied to a general discrete 0-form, this becomes
To re-simplify the notation while maintaining the advantages of the new generalized notation, we can define
and we’re back to
as before. Furthermore, we have
where is the number of directed edges from node to node .
Trace and Inner Products
Given a discrete 0-form , we define its trace via
Similarly, given a discrete 1-form , its trace is given by
With the trace, we can define the inner product of discrete 0-forms via
and the inner product of discrete 1-forms via
where is the edge product.
With the inner products defined above, a simple calculation shows
so the graph divergence is the adjoint of the coboundary.
In relating discrete calculus to algebraic topology, typically, in algebraic topology you would have a coboundary operator for cochains and a boundary operator for chains. With discrete calculus, we have both and for discrete forms.
The graph Laplacian of a discrete 0-form is given by
More generally, we could define a graph Laplace-Beltrami operator
Graph Dirac Operator
The graph Dirac operator is essentially the “square root” of the graph Laplace-Beltrami operator. Since and , we have
so the/a graph Dirac operator is given by
This post is part of a series
In my last post, I noted that in following John Baez’ series, I’m finding the need to introduce operators that I haven’t previously used in any applications. In this post, I will introduce another. It turns out that we could get away without introducing this concept, but I think it helps motivate some things I will talk about later.
In all previous applications, the important algebra was a noncommutative graded differential algebra. The grading means that the degree of elements add when you multiply them together. For example, the product of two nodes (degree 0) is a node (degree 0+0), the product of a node (degree 0) and a directed edge (degree 1) is a directed edge (degree 0+1), and the product of a directed edge (degree 1) with another directed edge is a directed surface (degree 1+1).
Note the algebra of nodes is a commutative sub-algebra of the full noncommutative graded algebra.
There is another related commutative edge algebra with corresponding edge product.
The edge product is similar to the product of nodes in that it is a projection given by
It is a projection because for an arbitrary discrete 1-form
The product of two discrete 1-forms is
I have not yet come across an application where the full edge algebra is needed. When the product does arise, one of the discrete 1-forms is usual the coboundary of a discrete 0-form, i.e.
When this is the case, the edge product can be expressed as a (graded) commutator in the noncommutative graded algebra, i.e.
An example of this will be seen when we examine electrical circuits.
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
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
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 is the sum of coefficients flowing into node minus the sum of coefficients flowing out of node . 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
if there is no directed edge from node to node 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
where is the set of all directed edges from node to node . 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
As before, this says the graph divergence of at the node is the sum of all coefficients flowing into node minus the sum of all coefficients flowing out of node . Moreover, for any node there can be one or more (or zero) directed edges from into .
I’ve enjoyed applying discrete calculus to various problems since Urs Schreiber and I wrote our paper together back in 2004
Shortly after that, I wrote an informal paper applying the theory to finance in
From there I wrote up some private notes laying the foundations for applying a higher-dimensional version of discrete calculus to interest rate models. However, life intervened, I went to work on Wall Street followed by various career twists leading me to Hong Kong where I am today. The research has laid fairly dormant since then.
I started picking this up again recently when my friend, John Baez, effectively changed careers and started the Azimuth Project. In particular, I’ve recently developed a discrete Burgers equation with corresponding discrete Cole-Hopf transformation, which is summarized – including numerical simulation results – on the Azimuth Forum here:
Motivated by these results, I started looking at a reformulation of the Navier-Stokes equation in
This is still a work-in-progress, but sorting this out is a necessary step to writing down the discrete Navier-Stokes equation.
Even more recently, John began a series of very interesting Azimuth Blog posts on network theory. I knew that network theory and discrete calculus should link up together naturally, but it took a while to see the connection. It finally clicked one night as I laid in bed half asleep in one of those rare “Eureka!” moments. I wrote up the details in
There is much more to be said about the connection between network theory and discrete calculus. I intend to write a series of subsequent posts in parallel to John’s highlighting how his work with Brendan Fong can be presented in terms of discrete calculus.