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Archive for October 2011

Network Theory and Discrete Calculus – The Discrete Master Equation

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This post is a follow up to

Network Theory and Discrete Calculus – Introduction

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

\partial(\psi P) = 0,

where \psi is a discrete 0-form representing the states of a Markov chain (at all times), P is a discrete 1-form representing transition probabilities, and \partial 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 \psi_i to be associated to nodes and the transition probabilities P_{i,j} 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 i at time t is considered a different node than the state i at time t+1. 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

* \mathbf{e}^{(i,t)(i,t+1)}
* \mathbf{e}^{(i,t)(j,t+1)}
* \mathbf{e}^{(j,t)(i,t+1)}
* \mathbf{e}^{(j,t)(j,t+1)}

For N 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

\psi = \sum_i \sum_t \psi_i^t \mathbf{e}^{(i,t)}

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

P = \sum_{i,j} \sum_t P_{i,j}^t \mathbf{e}^{(i,t)(j,t+1)}.

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

\psi P = \sum_{i,j} \sum_t \psi_i^t P_{i,j}^t \mathbf{e}^{(i,t)(j,t+1)}.

The boundary of a directed edge is given by

\partial \mathbf{e}^{(i,t)(j,t+1)} = \mathbf{e}^{(j,t+1)} - \mathbf{e}^{(i,t)}.

Now for some gymnastics, we can compute

\begin{aligned} \partial(\psi P)  &= \sum_{i,j} \sum_t \psi_i^t P_{i,j}^t \left[\mathbf{e}^{(j,t+1)} - \mathbf{e}^{(i,t)}\right] \\  &= \sum_{i,j} \sum_t \left[\psi_j^t P_{j,i}^t \mathbf{e}^{(i,t+1)} - \psi_i^t P_{i,j}^t \mathbf{e}^{(i,t)}\right] \\  &= \sum_i \sum_t \left[\sum_j \left(\psi_j^t P_{j,i}^t - \psi_i^{t+1} P_{i,j}^{t+1}\right)\right] \mathbf{e}^{(i,t+1)}.  \end{aligned}

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

\sum_j \left(\psi_j^t P_{j,i}^t - \psi_i^{t+1} P_{i,j}^{t+1}\right) = 0

or

\psi_i^{t+1} \sum_j P_{i,j}^{t+1} = \sum_j \psi_j^t P_{j,i}^t.

Since P is right stochastic, we have

\sum_j P_{i,j}^{t+1} = 1

so that

\psi_i^{t+1} = \sum_j \psi_j^t P_{j,i}^t.

In other words, when P is right stochastic and \partial(\psi P) = 0, we get the usual master equation from stochastic mechanics

\partial(\psi P) = 0\implies \psi_i^{t+1} = \sum_j \psi_j^t P_{j,i}^t.

Parting Thoughts

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

\partial(\psi P) = 0,

does this imply

\psi P = \partial F

for some discrete 2-form F?

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

\psi P + \partial G

gives the same dynamics because \partial^2 = 0. 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…

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Written by Eric

October 29, 2011 at 10:04 pm

Network Theory and Discrete Calculus – Introduction

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I’ve enjoyed applying discrete calculus to various problems since Urs Schreiber and I wrote our paper together back in 2004

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.

Written by Eric

October 28, 2011 at 9:12 am