• Network Theory and Discrete Calculus

In the previous post, we introduced discrete calculus on a binary tree. In particular, we introduced two sets of basis 1-forms we’ll refer to as

1. Graph bases and
2. Coordinate bases

related by

and saw that discrete 1-forms can be expressed in either left- or right-components forms

where, in general, the left- and right-component 0-forms do not coincide, i.e.

and

due to the noncommutativity of discrete 0-forms and discrete 1-forms.

Product Rule

Recall the exterior derivative of a discrete 0-form may be expressed in left-component form as

where

and

Although the product rule is satisfied, i.e.

note the discrete 0-form is on the right of the discrete 1-form and the discrete 0-form is on the left of the discrete 1-form . Attempting to express the product rule in left components, we find

In order to move to the left of the coordinate bases above, we need to know the commutation relations

and

These commutation relations may be determined by noting that for any two discrete 0-forms and , we have

Therefore,

and

where use has been made of the coordinate commutation relations in the previous post.

Putting everything together, we find the product rule above implies the left components satisfy

and

Change of Variables

Change of variables is something straightforward, yet has many applications, so it is worth writing it down here for future reference.

Let be a discrete 0-form with

If is invertible, we can rewrite this as

Given any other discrete 0-form , we have

From this, we can read off the discrete chain rules

and

• Network Theory and Discrete Calculus

So far in this series we’ve touched on a few applications of discrete calculus, but these were still at a fairly high level of abstraction. In this post, we lay some foundations for some very concrete applications that will allow us to actually start calculating things.

The Binary Tree

A particularly nice directed graph with many applications is the binary tree – a portion of which is illustrated below:

A node in the binary tree is labelled , where the first integer denotes the “spatial” position, i.e. its location at a given time, and the second integer denotes the “temporal” position.

Discrete Forms

A general discrete 0-form on a binary tree is written as usual as

where the sum is only over nodes of the binary tree and not over all integers. For instance, if is in the binary tree, then and are not.

Due to the special nature of the binary tree, a general discrete 1-form may also be reduced to a single sum over nodes, but in two distinct ways. First, we can group edges directed away from a given node. Second, we can group edges directed toward a given node.

In the first case, we can write

which is referred to as the left-component form and in the second case, we can write

which is referred to as the right-component form. These are two equivalent ways of expressing the same general discrete 1-form with

and

To see why these are referred to as left- and right-component forms, denote

and

and define a pair of basis 1-forms

and

Next, we can define left- and right-component 0-forms

and

respectively so that a discrete 1-form may be expressed in left-component form as

or equivalently in right-component form as

In other words, the left- and right- component forms of the bases allow us to express a general discrete 1-form form in terms of left- or right-component discrete 0-forms.

Differentials

The exterior derivative of a general discrete 0-form on a binary tree is given in left-component form as

From this, we can read off the left-components which we’ll denote as

and

so that

Similarly, the right-components are given by

and

so that

Noncommutative Coordinates

Although, strictly speaking, coordinates (other than node labels) are not necessary for performing computations in discrete calculus, it is helpful when comparing to continuum calculus to introduce coordinate 0-forms to the binary tree

where is the spatial distance between endpoints of a directed edge at successive time steps, and

where is the temporal spacing between successive temporal nodes.

In this special case, we have

and

so that

and

These relations can be inverted resulting in

and

so that

where

and

and the discrete calculus begins to resemble the continuum calculus.

Commutation Relations

The coordinates and were referred to as “noncommutative” above because although discrete 0-forms commute, i.e.

in general, discrete 0-forms and discrete 1-forms do not commute. A straightforward computation results in the following commutation relations

and

from which it follows that

From here, there are two continuum limits one could consider that lead to different calculi. In the first, we could set

In this case, all commutation relation vanish and the continuum is also a commutative limit, i.e. the coordinates commute in this limit and we’re left with the usual deterministic continuum calculus.

In the second limit, we could set

In this case, the second and third commutation relations vanish, but the first one remains

This limit gives rise to stochastic calculus (or a very close cousin). Motivated by this, the discrete calculus on a binary tree when setting

but keeping finite is referred to as discrete stochastic calculus.

• Network Theory and Discrete Calculus

As stated in the Introduction, one of the motivations for this series is to work in parallel with John Baez’ series on network theory to highlight some applications of discrete calculus. In this post, I reformulate some of the material in Part 11 pertaining to Noether’s theorem.

The State-Time Graph

The directed graphs associated with discrete stochastic mechanics are described in the post The Discrete Master Equation, where the simple state-time graph example below was presented

Conceptually, the thing to keep in mind is that any transition from one state to another requires a time step. Therefore a transition from node to node is more precisely a transition from node to node .

On a state-time graph, a discrete 0-form can be written as

and a discrete 1-form can be written as

The Master Equation

The master equation for discrete stochastic mechanics can be expressed simply as

where is a discrete 0-form representing the state at all times with

and is a discrete 1-form representing transition probabilities with

for all . When expanded into components, the master equation becomes

Observables and Expectations

A general discrete 0-form on a state-time graph is defined over all states and all time. However, occasionally, we would like to consider a discrete 0-form defined over all states at a specific point in time. To facilitate this in a component-free manner, denote

so the identity can be expressed as

The discrete 0-form is a projection that projects a general discrete 0-form to a discrete 0-form defined only at time . For instance, given a discrete 0-form , let

so that

In discrete stochastic mechanics, an observable is nothing more than a discrete 0-form

The expectation of an observable with respect to a state is given by

where was defined in a previous post. Note:

Some Commutators

In preparation for the discrete Noether’s theorem, note that

and

For these commutators to vanish, we must have

This implies if and only if is constant on each connected component of the state-time graph.

Constant Expectations

In this section, we determine the conditions under which the expectation of an observable is constant in time, i.e.

for all . This is a fairly straightforward application of the discrete master equation, i.e.

indicating the condition we’re looking for is

Noether’s Theorem

In this section, we demonstrate that when both and are constant in time, this implies

which, in turn, implies . To do this, we first expand

The condition for this trace to vanish is the same as the condition for the commutators above to vanish, i.e.

Expanding the trace further results in

Summing over and when and are constants results in

while summing and in the third term results in

by definition of the transition 1-form. Consequently, when and are constants, it follows that

Finally, this implies if and only if and are constant in time.

• Network Theory and Discrete Calculus

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

where

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.

Graph Conductance

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.

where

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

where

Equivalence

Could it be that ?

Let denote a partition of the set of directed edges from node to node and express the graph operator as

where

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.

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

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.

• 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

and

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

where

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.

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

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

and

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!

• Network Theory and Discrete Calculus

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

and

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.

Graph Divergence

The graph divergence was introduced here as a boundary operator, but the relation to divergence was mentioned here.

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.

Graph Laplacian

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

• Network Theory and Discrete Calculus

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

we have

and

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.

• Network Theory and Discrete Calculus

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 .

Network Theory and Discrete Calculus – Introduction

To give the result first, the master equation can be expressed in terms of discrete calculus simply as

where  is a discrete 0-form representing the states of a Markov chain (at all times), is a discrete 1-form representing transition probabilities, and is the boundary operator, i.e. a kind of graph divergence.

The rest of this post explains the terms in this discrete master equation and how it works.

The State-Time Graph

When working with a finite (or countable) number of states, there is nothing new in considering states to be associated to nodes and the transition probabilities to be associated to directed edges of a bi-directed graph. A simple 2-state example is given below

The directed graphs we work with for discrete stochastic calculus are slightly different and could be referred to as “state-time” graphs, which are supposed to make you think of “space-time”. A state at time is considered a different node than the state at time . An example 2-state, 2-time directed graph is illustrated below:

There are four directed edges in this state-time graph, which will be labelled

*
*
*
*

For states, the state-time graph will look similar but with more states appended horizontally.

The Discrete Master Equation

A discrete 0-form representing the states at all times can be expressed as

and a discrete 1-form representing the transition probabilities can be expressed as

The product of the 0-form and the 1-form is given by

The boundary of a directed edge is given by

Now for some gymnastics, we can compute

This is zero only when the last term in brackets is zero, i.e.

or

Since is right stochastic, we have

so that

In other words, when is right stochastic and , we get the usual master equation from stochastic mechanics

Parting Thoughts

The master equation is a boundary. This makes me wonder about homology, gauge transformations, sources, etc. For example, since

does this imply

for some discrete 2-form ?

If is a discrete 2-form whose boundary does not vanish, then

gives the same dynamics because This would be a kind of gauge transformation.

There are several directions to take this from here, but that is about all the energy I have for now. More to come…

Discrete differential geometry on causal graphs

Shortly after that, I wrote an informal paper applying the theory to finance in

Financial modeling using discrete stochastic calculus

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:

Discrete Burgers equation revisited

Motivated by these results, I started looking at a reformulation of the Navier-Stokes equation in

Towards Navier-Stokes from noncommutative geometry

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

Discrete stochastic mechanics

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.