### Discrete network dynamics. Part 1: Operator theory

This posting is a brief overview of my recent paper entitled "Discrete network dynamics. Part 1: Operator theory", which can be found in arXiv at cs.NE/0511027.

The basic idea is that an MCMC simulation (see a description of MCMC algorithms in a previous posting of mine here) of a joint probability Pr(

**), where**

*x***is the joint state of an**

*x**N*-dimensional system (e.g.

*x*is the state of the

_{i}*i*

^{th}node of an

*N*-node network), generates a sequence of

*'s (i.e.*

**x****(1),**

*x***(2), ... ,**

*x***(**

*x**t*), ...) that are sampled from Pr(

**).**

*x***(**

*x**t*+1) is generated from

**(**

*x**t*) by an operation that holds some of the components of

**(**

*x**t*) fixed, whilst stochastically updating the rest of the components of

**(**

*x**t*). The MCMC update operation is chosen so that if Pr(

**(**

*x**t*)) has the correct joint probability then so also does Pr(

**(**

*x**t*+1)). This was discussed in detail here.

The MCMC update operation is particularly simple if the state of only

*one*of the nodes of the network is updated at a time, so that

**(**

*x**t*+1) differs from

**(**

*x**t*) in only

*one*of its vector components. Let us assume that it is the

*i*

^{th}component that is updated, then MCMC update operation consists of two simpler operations:

- Erase the state
*x*. This is an_{i}*unconditional*erase operation, in which the state is erased with no memory of what it used to be. - Randomly rewrite the state
*x*. This is a_{i}*conditional*rewrite operation, in which the state is drawn from the conditional probability Pr(*x*│_{i}\*x**x*)._{i}

The diagram shows an MCMC update in a visually intuitive way for a 2-node network (i.e. *N*=2). Node 1 is at the top of the diagram, node 2 is at the bottom of the diagram, and time reads from left to right across the diagram. The initial state *x*_{1} of node 1 is changed to final state *x*_{1}´ by the influence of the state *x*_{2} of node 2 (which is itself *unchanged* in going from initial to final state). The solid lines from left to right indicate the evolution of the states of nodes 1 and 2. The vertical dashed line indicates the influence of the state of node 2 on the state of node 1, and the probability of the final state of node 1 is indicated as Pr(*x*_{1}´│*x*_{2}), which is memoryless because it does *not* depend on *x*_{1}.

The above diagram is a Feynman-like diagram describing the annihilation of the state of node 1, and then subsequent creation of a new state for node 1, and the "amplitude" for this process is actually a probability given by Pr(*x*_{1}´│*x*_{2}).

This identification allows the MCMC update operator to be written as

*H _{a}* ≡ ∑

_{i}

*a*

_{i}† (∑

*) (∑*

_{j}a_{j}*Pr(*

_{k}*i*│

*k*)

*b*†

_{k}*b*)

_{k}where *a _{i}*† is the creation operator for state

*i*at node 1,

*b*† is the creation operator for state

_{k}*k*at node 2,

*b*†

_{k}*b*is the number operator for state

_{k}*k*at node 2, ∑

*Pr(*

_{k}*i*│

*k*)

*b*†

_{k}*b*is the operator for computing the conditional probability factor that weights the MCMC update, and ∑

_{k}*unconditionally annihilates the state at node 1.*

_{j}a_{j}Application of the MCMC update operator *H _{a}* automatically generates the correct probability-weighted ensemble of possible next states of the 2-node network in which the state of node 1 is updated. The full MCMC update operator is the sum of

*H*+

_{a}*H*, where

_{b}*H*is obtained from

_{b}*H*by interchanging the "a" and "b" operators (and appropriately changing the conditional probability factor). The Feynman-like diagram version of

_{a}*H*+

_{a}*H*is thus a

_{b}*sum*of two diagrams as shown below.

The MCMC update operator correctly implements the 2-step MCMC update procedure described earlier. Each node of the network has a state with occupancy 1 that is erased (i.e. annhilated) during step 1 of the MCMC update to become the "vacuum" state with occupancy 0, and the state is rewritten (i.e. created) during step 2 of the MCMC update to become a state with occupancy 1 again. This update process is memoryless because its intermediate state is the "vacuum".

However, the MCMC update operator goes beyond the standard MCMC updates, because it can *also* be applied to initial states that have an occupancy of 2 or more. The algebra of the creation and annihilation operators automatically does this generalisation for you, and thus defines a unique generalisation of the MCMC algorithm for multiply occupied states of the network nodes.

The above discussion has focussed on a network with only 2 nodes. The generalisation to an *N*-node network is straightforward when the probability Pr(** x**) of the joint state of the network is expanded using the Hammersley-Clifford expansion, as is shown in my paper.

The structure of this MCMC algorithm is similar to a quantum field theory of bosons:

- Each state of node
*i*of the network can have a variable occupancy, which corresponds to each state of bosons of type*i*having a variable occupancy. - An MCMC update causes the multiply occupied state of node
*i*of the network to change by using the neighbouring nodes to influence the hopping of an occupant from one state of node*i*to another state of node*i*, which corresponds to a boson of type*i*scattering from one state to another under the influence of bosons of various other types.

This completes the reformulation of MCMC algorithms using the language of creation and annihilation operators, and opens up a large number of possible lines of enquiry.

One particularly interesting consequence of multiple occupancy MCMC algorithms is self-organising networks, in which the state space of a multiply occupied network node can split into 2 or more smaller state spaces, each of which inherits some of the original node's multiple occupants, so that overall the original node splits into several daughter nodes. I will be saying *much* more about this later.

## 2 Comments:

So far you have a suggestive and elegant form of book-keeping; are we about to see analogues of the physics described by QFT addressing systems with infinite degrees of freedom? And does the connection you have made offer insights 'the other way round', into what is going on in QFT behind its calculational prescriptions

DND1 is only the first of several papers, and it describes what amounts to a method of book-keeping.

Equation 61 is a general mixed state with

anyoccupancy. If you substitute in the solution in equation 66 and then sum the series you get the state Ψ = exp(p1 a1† + ... + pm am†)|0>, which is a coherent state. However, as noted in equation 62, the MCMC update operator is number-conserving, so it keeps the various occupancies in Ψseparate. Thus the physically significant behaviours under the MCMC update operator are thefixedoccupancy states given in equation 62, which arenotcoherent states.One

coulduse other types of update operator, including ones that mixed together different occupancies by invoking birth and death processes. However, for practical implementations of this type of computation (e.g. for use in information processing) you must restrict yourself tofiniteoccupancy, because MCMC algorithms must operate on finitely-occupied histograms.As for understanding what QFT is about, I find that if QFT is put on a lattice and stripped of everything except its basic commutation relations, then it is much easier to see what is going on. Counting samples in and out of histogram bins is a really concrete way of understanding the combinatoric aspects of the QFT of bosons.

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