Network Theory (Part 31)

Last time we came up with a category of labelled graphs and described circuits as ‘cospans’ in this category.

Cospans may sound scary, but they’re not. A cospan is just a diagram consisting of an object with two morphisms going into it:

We can talk about cospans in any category. A cospan is an abstract way of thinking about a ‘chunk of stuff’ with two ‘ends’ and It could be any sort of stuff: a set, a graph, an electrical circuit, a network of any kind, or even a piece of matter (in some mathematical theory of matter).

We call the object the apex of the cospan and call the morphisms the legs of the cospan. We sometimes call the objects and the feet of the cospan. We call the input and the output. We say the cospan goes from to though the direction is just a convention: we can flip a cospan and get a cospan going the other way!

If you’re wondering about the name ‘cospan’, it’s because a span is a diagram like this:

Since a ‘span’ is another name for a bridge, and this looks like a bridge from to category theorists called it a span! And category theorists use the prefix ‘co-‘ when they turn all the arrows around. Spans came first historically, and we will use those too at times. But now let’s think about how to compose cospans.

Composing cospans is supposed to be like gluing together chunks of stuff by attaching the output of the first to the input of the second. So, we say two cospans are composable if the output of the first equals the input of the second, like this:

We then compose them by forming a new cospan going all the way from to :

The new object and the new morphisms are built using a process called a ‘pushout’ which I’ll explain in a minute. The result is cospan from to called the composite of the cospans we started with. Here it is:

So how does a pushout work? It’s a general construction that you can define in any category, though it only exists if the category is somewhat nice. (Ours always will be.) You start with a diagram like this:

and you want to get a commuting diamond like this:

which is in some sense ‘the best’ given the diagram we started with. For example, suppose we’re in the category of sets and is a set included in both and Then we’d like to be the union of and There are other choices of that would give a commuting diamond, but the union is the best. Something similar is happening when we compose circuits, but instead of the category of sets we’re using the category of labelled graphs we discussed last time.

How do we make precise the idea that is ‘the best’? We consider any other potential solution to this problem, that is, some other commuting diamond:

Then is ‘the best’ if there exists a unique morphism from to the ‘competitor’ making the whole combined diagram commute:

This property is called a universal property: instead of saying that is the ‘best’, grownups say it is universal.

When has this universal property we call it the pushout of the original diagram, and we may write it as Actually we should call the whole diagram

the pushout, or a pushout square, because the morphisms matter too. The universal property is not really a property just of but of the whole pushout square. But often we’ll be sloppy and call just the object the pushout.

Puzzle 1. Suppose we have a diagram in the category of sets

where and the maps are the inclusions of this intersection in the sets and Prove that is the pushout, or more precisely the diagram

is a pushout square, where are the inclusions of and in the union

More generally, a pushout in the category of sets is a way of gluing together sets and with some ‘overlap’ given by the maps

And this works for labelled graphs, too!

Puzzle 2. Suppose we have two circuits of resistors that are composable, like this:

and this:

These give cospans in the category where

(Remember from last time that is the category of graphs with edges labelled by elements of some set ) Show that if we compose these cospans we get a cospan corresponding to this circuit:

If you’re a mathematician you might find it easier to solve this kind of problem in general, which requires pondering how pushouts work in Alternatively, you might find it easier to think about this particular example: then you can just check that the answer we want has the desired property of a pushout!

If this stuff seems complicated, well, just know that category theory is a very general, powerful tool and I’m teaching you just the microscopic fragment of it that we need right now. Category theory ultimately seems very simple: I can’t really think of any math that’s simpler! It only seem complicated when it’s unfamiliar and you have a fragmentary view of it.

So where are we? We know that circuits made of resistors are a special case of cospans. We know how to compose cospans. So, we know how to compose circuits… and in the last puzzle, we saw this does just what we want.

The advantage of this rather highbrow approach is that a huge amount is known about composing cospans! In particular, suppose we have any category where pushouts exist: that is, where we can always complete any diagram like this:

to a pushout square. Then we can form a category where:

• an object is an object of

• a morphism from an object to an object is an equivalence classes of cospans from to

• we compose cospans in the manner just described.

Why did I say ‘equivalence class’? It’s because the pushout is not usually unique. It’s unique only up to isomorphism. So, composing cospans would be ill-defined unless we work with some kind of equivalence class of cospans.

To be precise, suppose we have two cospans from to :

Then a map of cospans from one to the other is a commuting diagram like this:

We say that this is an isomorphism of cospans if is an isomorphism.

This gives our equivalence relation on cospans! It’s an old famous theorem in category theory—so famous that it’s hard to find a reference for the proof—that whenever is a category with pushouts, there’s a category where:

• an object is an object of

• a morphism from an object to an object is an isomorphism class of cospans from to

• we compose isomorphism classes of cospans by picking representatives, composing them and then taking the isomorphism class.

This takes some work to prove, but it’s true, so this is how we get our category of circuits!

Next time we’ll do something with this category. Namely, we’ll cook up a category of ‘behaviors’. The behavior of a circuit made of resistors just says which currents and potentials its terminals can have. If we put a circuit in a metaphorical ‘black box’ and refuse to peek inside, all we can see is its behavior.

Then we’ll cook up a functor from the category of circuits to the category of behaviors. We’ll call this the ‘black box functor’. Saying that it’s a functor mainly means that

Here and are circuits that we can compose, and is their composite. The black square is the black box functor, so is the behavior of the circuit There’s a way to compose behaviors, too, and the equation above says that the behavior of the composite circuit is the composite of their behaviors!

This is very important, because it says we can figure out what a big circuit does if we know what its pieces do. And this is one of the grand themes of network theory: understanding big complicated networks by understanding their pieces. We may not always be able to do this, in practice! But it’s something we’re always concerned with.

Network Theory Seminar (Part 1)

 

Check out this video! I start with a quick overview of network theory, and then begin building a category where the morphisms are electrical circuits. These lecture notes provide extra details:

• Network theory (part 30).

With luck, this video will be the first of a series. I’m giving a seminar on network theory at U.C. Riverside this fall. I’ll start by sketching the results here:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

But this is a big paper, and I also want to talk about other papers, so I certainly won’t explain everything in here—just enough to help you get started! If you have questions, don’t be shy about asking them.

I thank Blake Pollard for filming this seminar, and Muhammad “Siddiq” Siddiqui-Ali for providing the videocamera and technical support.

The Large-Number Limit for Reaction Networks (Part 3)

joint with Arjun Jain

We used to talk about reaction networks quite a lot here. When Arjun Jain was visiting the CQT, we made a lot of progress understanding how the master equation reduces to the rate equation in the limit where there are very large numbers of things of each kind. But we never told you the end of the story, and by now it’s been such a long time that you may need a reminder of some basic facts!

So…

The rate equation treats the number of things of each kind as continuous—a nonnegative real number—and says how it changes in a deterministic way.

The master equation treats the number of things of each kind as discrete—a nonnegative integer—and says how it changes in a probabilistic way.

You can think of the master equation as the ‘true’ description, and the rate equation as an approximation that’s good in some limit where there are large numbers of molecules — or more precisely, where the probability distribution of having some number of molecules of each kind is sharply peaked near some large value.

You may remember that in the master equation, the state of a chemical system is described by a vector in a kind of ‘Fock space’, while time evolution is described with the help of an operator on this space, called the ‘Hamiltonian’ :

The Hamiltonian is built from annihilation and creation operators, so all of this looks very much like quantum field theory. The details are here, and we won’t try to review them all:

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics.

The point is this: the ‘large-number limit’ where the master equation reduces to the rate equation smells a lot like the ‘classical limit’ of quantum field theory, where the description of light in terms of photons reduces to the good old Maxwell equations. So, maybe we can understand the large-number limit by borrowing techniques from quantum field theory!

How do we take the classical limit of quantum electromagnetism and get the classical Maxwell equations? For simplicity let’s ignore charged particles and consider the ‘free electromagnetic field': just photons, described by the quantum version of Maxwell’s equations. When we take the classical limit we let Planck’s constant go to zero: that much is obvious. However, that’s not all! The energy of each photon is proportional to so to take the classical limit and get a solution of the classical Maxwell’s equations with nonzero energy we also need to increase the number of photons. We cleverly do this in such a way that the total energy remains constant as

So, in quantum electromagnetism the classical limit is also a large-number limit!

That’s a good sign. It suggests the same math will also apply to our reaction network problem.

But then we hit an apparent roadblock. What’s the analogue of Planck’s constant in chemical reaction networks? What should go to zero?

We told you the answer to that puzzle a while ago: it’s the reciprocal of Avogadro’s number!

You see, chemists measure numbers of molecules in ‘moles’. There’s a large number of molecules in each mole: Avogadro’s number. If we let the reciprocal of Avogadro’s number go to zero, we are taking a limit where chemistry becomes ‘continuous’ and the discreteness of molecules goes away. Of course this is just a mathematical trick, but it’s a very useful one.

So, we got around that roadblock. And then something nice happened.

When taking the classical limit of quantum electromagnetism, we focus attention on certain quantum states that are the ‘best approximations’ to classical states. These are called ‘coherent states’, and it’s very easy to study how the behave as we simultaneously let and let the expected number of photons go to infinity.

And the nice thing is that these coherent states are also important in chemistry! But because chemistry involves probabilities rather than amplitudes, they have a different name: ‘Poisson distributions’. On this blog, Brendan Fong used them to give a new proof of a great result in mathematical chemistry, the Anderson–Craciun–Kurtz theorem.

So, we have most of the concepts and tools in place, and we can tackle the large-number limit using quantum techniques.

You can review the details here:

• The large-number limit for reaction networks (part 1).

• The large-number limit for reaction networks (part 2) .

So, after a quick refresher on the notation, we’ll plunge right in.

As you’ll see, we solve the problem except for one important technical detail: passing a derivative through a limit! This means our main result is not a theorem. Rather, it’s an idea for how to prove a theorem. Or if we act like physicists, we can call it a theorem.

Review of notation

The rate equation says

where:

• is a vector describing concentrations of different species at time In chemistry these species could be different kinds of molecules.

• Each is a transition, or in chemistry, a reaction.

• is a vector of natural numbers saying how many items of each species appear as inputs to the reaction This is called the source of the reaction.

• is a vector of natural numbers saying how many items of each species appear as outputs of the reaction This is called the target of the reaction. So, says the net change of the number of items of each species in the reaction

• The rate at which the reaction occurs is proportional to the rate constant times the number

Here we are raising a vector to a vector power and getting a number, using this rule:

where is any vector of natural numbers and is any vector of nonnegative real numbers. From now on we’ll call a vector of natural numbers a multi-index.

In this paper:

• John Baez, Quantum techniques for reaction networks.

it was shown that the master equation implies

Here:

• is the stochastic state saying the probability of having any particular number of items of each species at each time We won’t review the precise details of how this work; for that reread the relevant bit of Part 8.

• is the th number operator, defined using annihilation and creation operators as in quantum mechanics:

For the annihilation and creation operators, see Part 8.

• is the expected number of items of the th species at time

• Similarly, is the expected value of a certain product of operators. For any multi-index we define the falling power

and then we define

The large-number limit

Okay. Even if you don’t understand any of what we just said, you’ll see the master and rate equation look similar. The master equation implies this:

while the rate equation says this:

So, we will try to get from the first to the second second with the help of a ‘large-number limit’.

We start with a few definitions. We introduce an adjustable dimensionless quantity which we call This is just a positive number, which has nothing to do with quantum theory except that we’re using a mathematical analogy to quantum mechanics to motivate everything we’re doing.

Definition. The rescaled number operators are defined as This can be thought of as a rescaling of the number of objects, so that instead of counting objects individually, we count them in bunches of size

Definition. For any multi-index we define the rescaled falling power of the number operator by:

and also define

for any multi-index

Using these, we get the following equation:

where for any multi-index we set

This suggests a way to rescale the rate constants in the master equation:

Definition. The rescaled rate constants are

From here onwards, we change our viewpoint. We consider the rescaled rate constants to be fixed, instead of the original rate constants So, as we decrease we are studying situations where the original rate constants change to ensure that the rescaled rate constants stays fixed!

So, we switch to working with a rescaled master equation:

Definition. The rescaled master equation is:

This is really a one-parameter family of equations, depending on We write a solution of the rescaled master equation as but it is really one solution for each value of

Following the same procedure as above, we can rescale the rate equation, using the same definition of the rescaled rate constants:

Definition. The rescaled number of objects of the species is defined as where is the original number of objects of the species. Here again, we are counting in bunches of

Using this to rescale the rate equation, we get

Definition. The rescaled rate equation is

where

Therefore, to go from the rescaled master equation to the rescaled rate equation, we require that

as If this holds, we can identify with and get the rate equation from the master equation!

To this end, we introduce the following crucial idea:

Definition. A semiclassical family of states, is defined as a one-parameter family of states depending on such that for some we have

for every as

In particular, this implies

for every index

Intuitively, a semiclassical family is a family of probability distributions that becomes more and more sharply peaked with a larger and larger mean as decreases. We would like to show that in this limit, the rescaled master equation gives the rescaled rate equation.

We make this precise in the following propositions.

Proposition 1. If is a semiclassical family as defined above, then in the limit, we have as well.

Proof. For each index

By the definition of a semiclassical family,

since every term but the first approaches zero. Thus, we have

A similar but more elaborate calculation shows that

or in other words

as   █

Proposition 2. If is a semiclassical family of states, then

for any multi-index

Proof. Consider the centered moment of the number operator:

Taking the limit as goes to zero, this becomes

For a general multi-index we can prove the same sort of thing with a more elaborate calculation. First note that

The right-hand side can be expanded as

We can write this more tersely as

where for any multi-index we define

and for any multi-indices we define

Now using the definition of a semiclassical state, we see

But this equals zero, as the last expression, expanded, is

where each individual sum is zero.   █

Here is the theorem that would finish the job if we could give a fully rigorous proof:

“Theorem.” If is a solution of the rescaled master equation and also a semiclassical family for the time interval then is a solution of the rescaled rate equation for

Proof sketch. We sketch a proof that relies on the assumption that we can pass the limit through a time derivative. Of course, to make this rigorous, we would need to justify this. Perhaps it is true only in certain cases.

Assuming that we can pass the limit through the derivative:

and thus

and thus

As expected, we obtain the rescaled rate equation.   █

Another question is this: if we start with a semiclassical family of states as our initial data, does it remain semiclassical as we evolve it in time? This will probably be true only in certain cases.

An example: rescaled coherent states

The best-behaved semiclassical states are the coherent states.
Consider the family of coherent states

using the notation developed in the earlier mentioned paper. In that paper it was shown that for any multi-index and any coherent state we have

Using this result for we get

Since equals plus terms of order as we have

showing that our chosen is indeed a semiclassical family.

Network Theory (Part 30)

The network theory series is back! You may have thought it died out, but in fact it’s just getting started. Over the last year my grad students have made huge strides in working out the math of networks. Now it’s time to explain what they’ve done.

In the last three episodes I explained how electrical circuits made of resistors, inductors and capacitors are a great metaphor for many kinds of complex systems made of interacting parts. And it’s not just a metaphor; there are mathematically rigorous analogies—in other words, isomorphisms—between such electrical circuits and various other kinds of ‘passive linear networks’.

I showed you a chart of these analogies last time:

	displacement:	flow:	momentum:	effort:
Mechanics: translation	position	velocity	momentum	force
Mechanics: rotation	angle	angular velocity	angular momentum	torque
Electronics	charge	current	flux linkage	voltage
Hydraulics	volume	flow	pressure momentum	pressure
Thermal Physics	entropy	entropy flow	temperature momentum	temperature
Chemistry	moles	molar flow	chemical momentum	chemical potential

But what do I mean by a ‘passive linear network’? Let me explain this very roughly at first, since we’ll be painfully precise later on.

Right now by ‘network’ I mean a graph with gizmos called ‘components’ on the edges. For example:

In a network there is some kind of ‘flow’ running along each edge, and also some kind of ‘effort’ across that edge. For example, in electronics the flow is electrical current and the effort is voltage. The chart shows the meaning of flow and effort in other examples.

‘Passivity’ means roughly that none of the components put out energy that didn’t earlier come in. For example, resistors lose energy (which goes into heat, which we’re not counting). Capacitors can store energy and later release it. So, resistors and capacitors are passive—and so are inductors. But batteries and current sources actually put out energy, so we won’t allow them in our networks yet. For now, we’re just studying how passive components respond to a source of flow or effort.

For some subtleties that show up when you try to make the concept of passivity precise, try:

• Passivity (engineering), Wikipedia.

Finally, ‘linearity’ means that the flow along each edge of our network is linearly related to the effort across that edge. Here are the key examples:

• For electrical resistors, linearity is captured by Ohm’s law. If an edge in our network is labelled by a resistor of resistance usually drawn like this:

then Ohm’s law says:

where is the voltage across that edge and is the current along that edge.

• If our edge is labelled by an inductor of inductance :

we have

Here we need to think of the voltage and current as functions of time.

• If our edge is labelled by a capacitor of capacitance :

we write the equation

where again we think of the voltage and current as functions of time.

Both linearity and passivity are simplifying assumptions that we eventually want to drop. If we include batteries or current sources, we’re dropping passivity. And if include transistors, we’re dropping linearity. Obviously both these are important!

However, there is a lot to do even with these simplifying assumptions. And now it’s time to get started!

In what follows, I will not use the terms ‘flow’ and ‘effort’, which are chosen to be neutral and subject-independent. Instead, I’ll use the vocabulary from electronics, e.g. ‘current’ and ‘voltage’. The reason is that we’ve all heard of resistors, capacitors, Ohm’s law and Kirchhoff’s laws, and while these have analogues in every row of the chart, it seems pointless to make up weird new ‘neutral’ terms for all these concepts.

But don’t be fooled by the jargon! We’re not merely studying electrical circuits. We’ll be studying passive linear networks in full generality… with the help of category theory.

Linear passive networks as morphisms

To get going, let’s think about circuits made of resistors. We can do this without harm, because we’ll later include capacitors and inductors using a simple effortless trick. Namely, we’ll generalize the ‘resistance’ of a resistor, which is a real number, to something called ‘impedance’, which is an element of some larger field. Everything will be so abstract that replacing resistances with impedances will be as easy as snapping our fingers.

Right now I want to define a category where the morphisms are circuits made of resistors. Any morphism will go from some ‘inputs’ to some ‘outputs’, like this:

So a morphism is roughly a graph with edges labelled by numbers called ‘resistances’, with some special nodes called ‘inputs’ and some special nodes called ‘outputs’.

What can do with morphisms? Compose them! So, suppose we have a second morphism whose inputs match the outputs of the first:

Then we can compose them, attaching the outputs of the first to the inputs of the second. We get this morphism as the result:

So, composing morphisms is a way to build big electrical circuits—or other ‘linear passive networks’—out of little ones.

This seems pretty simple, but let’s try to formalize it and see why we have a category. In fact it takes a bit of thought. To check that we get a category, we need to check that composition is associative:

and that each object has an identity morphism that does what an identity should:

All these equations seem obviously true in our example… until you try to prove them.

You might think an identity morphism should be a bunch of straight pieces of wire—a bunch of edges each with an input node and an output node—but that doesn’t quite work, since sticking an extra edge onto a graph gives a new graph with an extra edge!

Also, we are composing circuits by ‘sticking them together’. This process is formalized in category theory using a pushout, and pushouts are only defined ‘up to canonical isomorphism’. The very simplest example is the disjoint union of two sets. We all know what it means, but if you examine it carefully, you’ll see it’s only defined up to canonical isomorphism, because it involves a choice of how we make the two sets disjoint, and this choice is somewhat arbitrary.

All this means the category we’re after is a bit subtler than you might at first expect; in fact, it’s most naturally thought of as a bicategory, meaning roughly that all the equations above hold only ‘up to canonical isomorphism’.

So, we proceed like this.

First we define a concept of ‘labelled graph’, where (for now) only the edges are labelled. We do this because we want our circuits to have edges labelled by ‘resistances’, which are real numbers. But we do it in greater generality because later we’ll want the edges to be labelled by ‘impedances’, which are elements of some other field. And since we’re studying electrical circuits just as examples of networks, later still we will probably want graphs whose edges are labelled in still other ways.

So:

Definition. A graph consists a finite set of edges, a finite set of nodes, and two functions

Thus each edge will have some node as its source and some node as its target:

Definition. Given a set we define an -labelled graph to be a graph together with a function This assigns to each edge its label We call the label set.

We use the letter because for circuits of resistors we will take the label set to be

the positive real numbers, and will be the resistance of the edge For circuits that also contain inductors and capacitors we will take the label set to be the positive elements of some larger field… but more about that later!

Now we want to give our -labelled graph a set of nodes called ‘inputs’ and a set of nodes called ‘outputs’. You might think the set of inputs should be disjoint from the set of outputs, but that’s a tactical error! It turns out an identity morphism in our category should have the inputs being exactly the same as the outputs… and no edges at all:

To handle this nicely, we need to make a category of -labelled graphs. This works in the obvious way, if you’re used to this stuff. A morphism from one -labelled graph to another sends edges to edges, nodes to nodes, and preserves everything in sight:

Definition. Given -graphs and a morphism of -labelled graphs from to is a pair of functions

such that the following diagrams commute:

There is a category where the objects are -labelled graphs and the morphisms are as we’ve just defined them.

Warning: the morphisms in are not the morphisms of the kind we really want, the ones that look like this:

They are just a step along the way. A morphism of the kind we really want is a diagram like this in :

where is an -labelled graph and are -labelled graphs with no edges!

You see, if and have no edges, all they have is nodes. We call the nodes of the inputs and those of the outputs. The morphisms and say how these nodes are included in is our circuit made of resistors.

In general, any diagram shaped like this is called a cospan:

If we turned the arrows around it would be called a span. Cospans are good whenever you a thing with an ‘input end’ and an ‘output end’, and you want to describe how the ends are included in the thing. So, they’re precisely what we need for describing a circuit made of resistors, like this:

This makes us want to cook up a category where:

• an object is an -labelled graph with no edges. We can alternatively think of it as a finite set: a set of nodes.

• a morphism from to is a cospan of -labelled graphs:

We still need to say how to compose these morphisms. We know it will amount to attaching the outputs of one circuit to the inputs of the next—that’s all there is to it! But we need to make this precise and prove we get a category. And as I’ve hinted, we will actually get something bigger and better: a bicategory! This will come as no surprise to if you’re familiar with span and cospan bicategories–but it may induce a heart attack otherwise.

This bicategory can then be ‘watered down’ to give our category And when we take

we’ll get the category where morphisms are circuits made of resistors! We’ll call this

Then I’ll explain what we can do with this category! There’s no end of things we could do with it. But the main thing Brendan does is study the ‘black-boxing’ operation, where we take a circuit, forget its inner details, and only keep track of what it does. This turns out to be quite interesting.

References

I thank Brendan Fong for drawing some of the pictures of circuits here. For the details of what I’m starting to explain here, read our paper:

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

You can learn more about the underlying ideas here:

• Dean C. Karnopp, Donald L. Margolis and Ronald C. Rosenberg, System Dynamics: a Unified Approach, Wiley, New York, 1990.

• Forbes T. Brown, Engineering System Dynamics: a Unified Graph-Centered Approach, CRC Press, Boca Raton, 2007.

• Francois E. Cellier, Continuous System Modelling, Springer, Berlin, 1991.

Network Theory News

 

You may be wondering, somewhere deep in the back of your mind, what happened to the Network Theory series on this blog. It’s nowhere near done! I plan to revive it, since soon I’ll be teaching a seminar on network theory at U.C. Riverside. It will start in October and go on at least until the end of March.

Here’s a version of the advertisement I sent to the math grad students:

Network Theory Seminar

I’ll be running a seminar on Network Theory on Mondays from 3:10 to 4:30 pm in Surge 268 starting on October 6th.

Network theory uses the tools of modern math—categories, operads and more—to study complex systems made of interacting parts. The idea of this seminar is to start from scratch, explain what my grad students have been doing, and outline new projects that they and other students might want to work on. A lot has happened since I left town in January.

I hope to see you there!

If you want more detail, here is a sketch of what’s been happening.

1) Franciscus Rebro has been working on “cospans”, a very general way to treat a physical system with inputs and outputs and treat it as a morphism in category. This underlies all the other projects.

2) Brendan Fong, a student of mine at Oxford, is working on a category where the morphisms are electrical circuits, and composing morphisms is sticking together circuits:

• Brendan Fong, A compositional approach to control theory.

• John Baez and Brendan Fong, A compositional framework for passive linear networks.

The first paper here is an outline of the project; the second one actually carries out the project, or at least a large chunk of it.

3) Blake Pollard, a student in the physics department, has been studying Markov processes. In a Markov process, things randomly hop from one vertex of a graph to another along edges. Blake has created a category where morphisms are ‘open’ Markov process, in which things flow in and out of certain special vertices called ‘terminals’.

4) Jason Erbele has been working on categories in control theory, the branch of math used to study physical systems that interact with the outside world via inputs and outputs. After finishing this paper:

• John Baez and Brendan Fong, Categories in control.

he’s been taking more concept from control theory and formalizing them using categories.

5) Oh yeah, and what about me? I gave a series of lectures on network theory at Oxford, and you can see videos of them here:

• John Baez, Network Theory.

Jacob Biamonte and I have also more or less finished a book on chemical reaction networks and Petri nets:

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics.

But I won’t be talking about that; I want to talk about the new work my students are doing!

Quantum Frontiers in Network Science

guest post by Jacob Biamonte

There’s going to be a workshop on quantum network theory in Berkeley this June. The event is being organized by some of my collaborators and will be a satellite of the biggest annual network science conference, NetSci.

A theme of the Network Theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. Remember the first post by John which outlined the goal of a general theory of networks? Well, everyone’s been chipping away at this stuff for a few years now and I think you’ll agree that this workshop seems like an excellent way to push these topics even further, particularly as they apply to complex networks.

The event is being organized by Mauro Faccin, Filippo Radicchi and Zoltán Zimborás. You might recall when Tomi Johnson first explained to us some ideas connecting quantum physics with the concepts of complex networks (see Quantum Network Theory Part 1 and Part 2). Tomi’s going to be speaking at this event. I understand there is even still a little bit of space left to contribute talks and/or to attend. I suspect that those interested can sort this out by emailing the organizers or just follow the instructions to submit an abstract.

They have named their event Quantum Frontiers in Network Science or QNET for short. Here’s their call.

Quantum Frontiers in Network Science

This year the biggest annual network science conference, NetSci will take place in Berkeley California on 2-6 June. We are organizing a one-day Satellite Workshop on Quantum Frontiers in Network Science (QNET).

A grand challenge in contemporary complex network science is to reconcile the staple “statistical mechanics based approach” with a theory based on quantum physics. When considering networks where quantum coherence effects play a non-trivial role, the predictive power of complex network science has been shown to break down. A new theory is now being developed which is based on quantum theory, from first principles. Network theory is a diverse subject which developed independently in several disciplines to rely on graphs with additional structure to model complex systems. Network science has of course played a significant role in quantum theory, for example in topics such as tensor network states, chiral quantum walks on complex networks, categorical tensor networks, and categorical models of quantum circuits, to name only a few. However, the ideas of complex network science are only now starting to be united with modern quantum theory. From this respect, one aim of the workshop is to put in contact two big and generally not very well connected scientific communities: statistical and quantum physicists.

The topic of network science underwent a revolution when it was realized that systems such as social or transport networks could be interrelated through common network properties, but what are the relevant properties to consider when facing quantum systems? This question is particularly timely as there has been a recent push towards studying increasingly larger quantum mechanical systems, where the analysis is only beginning to undergo a shift towards embracing the concepts of complex networks.

For example, theoretical and experimental attention has turned to explaining transport in photosynthetic complexes comprising tens to hundreds of molecules and thousands of atoms using quantum mechanics. Likewise, in condensed matter physics using the language of “chiral quantum walks”, the topological structure of the interconnections comprising complex materials strongly affects their transport properties.

An ultimate goal is a mathematical theory and formal description which pinpoints the similarities and differences between the use of networks throughout the quantum sciences. This would give rise to a theory of networks augmenting the current statistical mechanics approach to complex network structure, evolution, and process with a new theory based on quantum mechanics.

Topics of special interest to the satellite include

• Quantum transport and chiral quantum walks on complex networks
• Detecting community structure in quantum systems
• Tensor algebra and multiplex networks
• Quantum information measures (such as entropy) applied to complex networks
• Quantum critical phenomena in complex networks
• Quantum models of network growth
• Quantum techniques for reaction networks
• Quantum algorithms for problems in complex network science
• Foundations of quantum theory in relation to complex networks and processes thereon
• Quantum inspired mathematics as a foundation for network science

Info

QNET will be held at the NetSci Conference venue at the Clark Kerr Campus of the University of California, on June 2nd in the morning (8am-1pm).

Links

• Main conference page: NetSci2014
• Call for abstracts and the program

It sounds interesting! You’ll notice that the list of topics seems reminiscent of some of the things we’ve been talking about right here on Azimuth! A general theme of the Network Theory Series has been geared towards developing frameworks to describe networked systems through a common language and then to map the use of tools and results across disciplines. It seems like a great place to talk about these ideas. Oh, and here’s a current list of the speakers:

• Leonardo Banchi (UCL, London)
• Ginestra Bianconi (London)
• Silvano Garnerone (IQC, Waterloo)
• Laetitia Gauvin (ISI Foundation)
• Marco Javarone (Sassari)
• Tomi Johnson (Oxford)

and again, the organizers are

• Mauro Faccin (ISI Foundation)
• Filippo Radicchi (Indiana University)
• Zoltán Zimborás (UCL)

From the call, we can notice that a central discussion topic at QNET will be about contrasting stochastic and quantum mechanics. Here on Azimuth we like this stuff. You might remember that stochastic mechanics was formulated in the network theory series to mathematically resemble quantum theory (see e.g. Part 12). This formalism was then employed to produce several results, including a stochastic version of Noether’s theorem by John and Brendan in Parts 11 and 13—recently Ville has also written Noether’s Theorem: Quantum vs Stochastic. Several other results were produced by relating quantum field theory to Petri nets from population biology and to chemical reaction networks in chemistry (see the Network Theory homepage). It seems to me that people attending QNET will be interested in these sorts of things, as well as other related topics.

One of the features of complex network science is that it is often numerically based and geared directly towards interesting real-world applications. I suspect some interesting results should stem from the discussions that will take place at this workshop.

By the way, here’s a view of downtown San Francisco at dusk from Berkeley Hills California from the NetSci homepage:

Noether’s Theorem: Quantum vs Stochastic

guest post by Ville Bergholm

In 1915 Emmy Noether discovered an important connection between the symmetries of a system and its conserved quantities. Her result has become a staple of modern physics and is known as Noether’s theorem.

The theorem and its generalizations have found particularly wide use in quantum theory. Those of you following the Network Theory series here on Azimuth might recall Part 11 where John Baez and Brendan Fong proved a version of Noether’s theorem for stochastic systems. Their result is now published here:

• John Baez and Brendan Fong, A Noether theorem for stochastic mechanics, J. Math. Phys. 54:013301 (2013).

One goal of the network theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. John and Brendan proved their stochastic version of Noether’s theorem by exploiting ‘stochastic mechanics’ which was formulated in the network theory series to mathematically resemble quantum theory. Their result, which we will outline below, was different than what would be expected in quantum theory, so it is interesting to try to figure out why.

Recently Jacob Biamonte, Mauro Faccin and myself have been working to try to get to the bottom of these differences. What we’ve done is prove a version of Noether’s theorem for Dirichlet operators. As you may recall from Parts 16 and 20 of the network theory series, these are the operators that generate both stochastic and quantum processes. In the language of the series, they lie in the intersection of stochastic and quantum mechanics. So, they are a subclass of the infinitesimal stochastic operators considered in John and Brendan’s work.

The extra structure of Dirichlet operators—compared with the wider class of infinitesimal stochastic operators—provided a handle for us to dig a little deeper into understanding the intersection of these two theories. By the end of this article, astute readers will be able to prove that Dirichlet operators generate doubly stochastic processes.

Before we get into the details of our proof, let’s recall first how conservation laws work in quantum mechanics, and then contrast this with what John and Brendan discovered for stochastic systems. (For a more detailed comparison between the stochastic and quantum versions of the theorem, see Part 13 of the network theory series.)

The quantum case

I’ll assume you’re familiar with quantum theory, but let’s start with a few reminders.

In standard quantum theory, when we have a closed system with states, the unitary time evolution of a state is generated by a self-adjoint matrix called the Hamiltonian. In other words, satisfies Schrödinger’s equation:

The state of a system starting off at time zero in the state and evolving for a time is then given by

The observable properties of a quantum system are associated with self-adjoint operators. In the state the expected value of the observable associated to a self-adjoint operator is

This expected value is constant in time for all states if and only if commutes with the Hamiltonian :

In this case we say is a ‘conserved quantity’. The fact that we have two equivalent conditions for this is a quantum version of Noether’s theorem!

The stochastic case

In stochastic mechanics, the story changes a bit. Now a state is a probability distribution: a vector with nonnegative components that sum to 1. Schrödinger’s equation gets replaced by the master equation:

If we start with a probability distribution at time zero and evolve it according to this equation, at any later time have

We want this always be a probability distribution. To ensure that this is so, the Hamiltonian must be infinitesimal stochastic: that is, a real-valued matrix where the off-diagonal entries are nonnegative and the entries of each column sum to zero. It no longer needs to be self-adjoint!

When is infinitesimal stochastic, the operators map the set of probability distributions to itself whenever and we call this family of operators a continuous-time Markov process, or more precisely a Markov semigroup.

In stochastic mechanics, we say an observable is a real diagonal matrix, and its expected value is given by

where is the vector built from the diagonal entries of More concretely,

where is the th component of the vector

Here is a version of Noether’s theorem for stochastic mechanics:

Noether’s Theorem for Markov Processes (Baez–Fong). Suppose is an infinitesimal stochastic operator and is an observable. Then

if and only if

and

for all and all obeying the master equation.   █

So, just as in quantum mechanics, whenever the expected value of will be conserved:

for any and all However, John and Brendan saw that—unlike in quantum mechanics—you need more than just the expectation value of the observable to be constant to obtain the equation You really need both

together with

for all initial data to be sure that

So it’s a bit subtle, but symmetries and conserved quantities have a rather different relationship than they do in quantum theory.

A Noether theorem for Dirichlet operators

But what if the infinitesimal generator of our Markov semigroup is also self-adjoint? In other words, what if is both an infinitesimal stochastic matrix but also its own transpose: ? Then it’s called a Dirichlet operator… and we found that in this case, we get a stochastic version of Noether’s theorem that more closely resembles the usual quantum one:

Noether’s Theorem for Dirichlet Operators. If is a Dirichlet operator and is an observable, then

Proof. The direction is easy to show, and it follows from John and Brendan’s theorem. The point is to show the direction. Since is self-adjoint, we may use a spectral decomposition:

where are an orthonormal basis of eigenvectors, and are the corresponding eigenvalues. We then have:

where the third equivalence is due to the vectors being linearly independent. For any infinitesimal stochastic operator the corresponding transition graph consists of connected components iff we can reorder (permute) the states of the system such that becomes block-diagonal with blocks. Now it is easy to see that the kernel of is spanned by eigenvectors, one for each block. Since is also symmetric, the elements of each such vector can be chosen to be ones within the block and zeros outside it. Consequently

implies that we can choose the basis of eigenvectors of to be the vectors which implies

Alternatively,

implies that

where we have used the above sequence of equivalences backwards. Now, using John and Brendan’s original proof, we can obtain   █

In summary, by restricting ourselves to the intersection of quantum and stochastic generators, we have found a version of Noether’s theorem for stochastic mechanics that looks formally just like the quantum version! However, this simplification comes at a cost. We find that the only observables whose expected value remains constant with time are those of the very restricted type described above, where the observable has the same value in every state in a connected component.

Puzzles

Suppose we have a graph whose graph Laplacian matrix generates a Markov semigroup as follows:

Puzzle 1. Suppose that also so that is a Dirichlet operator and hence generates a 1-parameter unitary group. Show that the indegree and outdegree of any node of our graph must be equal. Graphs with this property are called balanced.

Puzzle 2. Suppose that is doubly stochastic Markov semigroup, meaning that for all each row and each column of sums to 1:

and all the matrix entries are nonnegative. Show that the Hamiltonian obeys

and all the off-diagonal entries of are nonnegative. Show the converse is also true.

Puzzle 3. Prove that any doubly stochastic Markov semigroup is of the form where is the graph Laplacian of a balanced graph.

Puzzle 4. Let be a possibly time-dependent observable, and write for its expected value with respect to some initial state evolving according to the master equation. Show that

This is a stochastic version of the Ehrenfest theorem.