## Categorical Foundations of Network Theory

Jacob Biamonte got a grant from the Foundational Questions Institute to run a small meeting on network theory:

It’s being held 25-28 May 2015 in Turin, Italy, at the ISI Foundation. We’ll make slides and/or videos available, but the main goal is to bring a few people together, exchange ideas, and push the subject forward.

### The idea

Network theory is a diverse subject which developed independently in several disciplines. It uses graphs with additional structure to model everything from complex systems to theories of fundamental physics.

This event aims to further our understanding of the mathematical theory underlying the relations between seemingly different networked systems. It’s part of the Azimuth network theory project.

### Timetable

With the exception of the first day (Monday May 25th) we will kick things off with a morning talk, with plenty of time for questions and interaction. We will then break for lunch at 1:00 p.m. and return for an afternoon work session. People are encouraged to give informal talks and to present their ideas in the afternoon sessions.

#### Monday May 25th, 10:30 a.m.

Jacob Biamonte: opening remarks.

For Jacob’s work on quantum networks visit www.thequantumnetwork.org.

John Baez: network theory.

For my stuff see the Azimuth Project network theory page.

#### Tuesday May 26th, 10:30 a.m.

Operads are a formalism for sticking small networks together to form bigger ones. David has a 3-part series of articles sketching his ideas on networks.

#### Wednesday May 27th, 10:30 a.m.

Eugene Lerman: continuous time open systems and monoidal double categories.

Eugene is especially interested in classical mechanics and networked dynamical systems, and he wrote an introductory article about them here on the Azimuth blog.

#### Thursday May 28th, 10:30 a.m.

Tobias Fritz: ordered commutative monoids and theories of resource convertibility.

Tobias has a new paper on this subject, and a 3-part expository series here on the Azimuth blog!

### Location and contact

ISI Foundation
Via Alassio 11/c
10126 Torino — Italy

Phone: +39 011 6603090
Email: isi@isi.it
Theory group details: www.TheQuantumNetwork.org

### 23 Responses to Categorical Foundations of Network Theory

1. Eugene Lerman says:

It feels a bit funny to read that I do/am interested in classical mechanics. I have done a bit of symplectic geometry and Hamiltonian systems. I suppose one can argue these are the same things as classical mechanics… :)

What are hot topics in classical mechanics these days? Non-holonomic systems?

• John Baez says:

Isn’t “Hamiltonian systems” just the way people with a Ph.D. say “classical mechanics”?

What are hot topics in classical mechanics these days?

Network theory!

• Eugene Lerman says:

John wrote

Network theory!

in response to:

What are hot topics in classical mechanics these days?

But we have no idea even where to start to build a theory of networks of Hamiltonian systems!

• Eugene Lerman says:

Isn’t “Hamiltonian systems” just the way people with a Ph.D. say “classical mechanics”?

Not really. Some classical systems are not conservative. I know a few that voted for Obama.

• John Baez says:

Eugene wrote:

But we have no idea even where to start to build a theory of networks of Hamiltonian systems!

No idea even where to start? I don’t think it’s that bad. Port-Hamiltonian systems are the way people usually tackle this issue—and if people are doing things suboptimally, we can improve it. We really should tackle this issue soon, before everyone and his brother jumps aboard the network theory bandwagon.

• Arjan van der Schaft, Port-Hamiltonian systems: an introductory survey, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006.

Abstract. Abstract. The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models.

If they’re talking about it at the ICM it must be hot, right?

And “any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system” suggests there’s a compact closed category with port-Hamiltonian systems as morphisms, and/or an operad describing how to connect them.

• Eugene Lerman says:

I have looked at Port-Hamiltonian systems. Let’s just say that I am very sceptical that they do what the say and say what they do, rather than publicly badmouth an ICM speaker.

• John Baez says:

Well, if something is not right we can fix it, so I don’t think “we don’t even know where to start”: we can start with what people are doing, fix mistakes, and make everything elegant. And the good thing is, there are lots of well-known examples of port-Hamiltonian systems, which can guide us.

• Eugene Lerman says:

How do you fix a formalism that have variables dual to position variables? Position variables live in a manifold. I never heard of manifolds having vector space duals.

How do you fix a formalism for which Dirac structures in local coordinates are not skew-symmetric? When people don’t distinguish between a Riemannian metric and a symplectic form, I get very confused.

• John Baez says:

I never heard of manifolds having vector space duals.

But they have contangent bundles.

Having spent a lot of time talking to physicists, I find they’re often on the right track even if they use words differently (i.e., wrong) or screw up some stuff. Anyway, you’ve gotten me eager to straighten out this subject. The more problems there are, the more fun it’ll be! The numerous examples will help us figure out what to do.

2. Eugene Lerman says:

Having spent a lot of time talking to physicists, I find they’re often on the right track even if they use words differently (i.e., wrong) or screw up some stuff. Anyway, you’ve gotten me eager to straighten out this subject.

I am glad you want to sort this out.

Here is an example I have no idea how to formulate in terms of port-Hamiltonian systems/bond graphs/what not. Take two particles on a 2-sphere interacting by way of a potential (say they are connected by a spring). It’s a network consisting of two nodes. For each node you have the cotangent bundle of a sphere. Now how do you tear it apart and and put back together? What are efforts/flows here?

• John Baez says:

Here you are ‘coupling’ two systems by taking the product of their phase spaces and then adding a term to the Hamiltonian that depends on variables from each system.

I think electrical engineers are more used to ‘composing’ two systems, by identifying some variables of one system with those of the other. In composition, you create a new phase space that’s a pushout of two other phase spaces. For example, this is what happens when you attach two electrical circuits by connecting some wires from one to wires from another: you identify some currents and potentials in the first circuit with some currents and potentials in the second.

So, I think these are different operations—and I’ve spent a lot more time thinking about composition than about coupling. I can understand some forms of composition, at least, as composition of morphisms in cospan categories. Composing cospans is done by pushout.

But composition may be a limiting case of coupling! If in your example, if we take a limit where the potential between two particles becomes huge except when they have the same position, and zero when they’re at the same position, this will force their positions and momenta to be equal.

Of course, coupling is a special case of ‘changing the Hamiltonian by adding an extra term’. After we’ve built up a complicated networked system by composing subsystems, we can study perturbations of Hamiltonians where we add extra terms that only depend jointly on two subsystems when those subsystems ‘touch’. That’s what I’d be inclined to do, anyway.

• Eugene Lerman says:

I don’t disagree with you. And it looks like you don’t disagree with me either — there is no theory of networks of Hamiltonian systems yet and it’s not clear how to start building it.

At some point a year or two ago I toyed with the idea of coupling mechanical systems by using (generalized) forces. The geometry of such a set up looked like a fun thing to play with. Unfortunately I didn’t (and don’t) have enough physical examples to check if this exercise was going to be on the right track; if it it was going to make physical sense.

• John Baez says:

John wrote:

I think electrical engineers are more used to ‘composing’ two systems, by identifying some variables of one system with those of the other. In composition, you create a new phase space that’s a pushout of two other phase spaces. For example, this is what happens when you attach two electrical circuits by connecting some wires from one to wires from another: you identify some currents and potentials in the first circuit with some currents and potentials in the second.

I should have said “pullback”, not “pushout”. We’re taking a product of two phase spaces and then taking an “equalizer” where we demand that some functions on the first equal some functions on the other—for example, some currents and voltages in one circuit equal currents and voltages in another. Doing a product and then an equalizer in this way is doing a pullback.

I slipped because I often think of electrical circuits as graphs with extra structure, and to compose these we do a pushout. But the functor from graphs to phase spaces is contravariant, and it carries pushouts to pullbacks.

Eugene wrote:

I don’t disagree with you. And it looks like you don’t disagree with me either — there is no theory of networks of Hamiltonian systems yet and it’s not clear how to start building it.

I’m afraid disagreement isn’t a symmetric relation. Well, okay: it’s not exactly “clear” how to build the theory of networks of Hamiltonian systems—but it doesn’t seem hard, either.

• Eugene Lerman says:

The category of manifolds in general and symplectic manifolds in particular don’t have pullbacks. Do you want to look at derived pullbacks?

3. lee bloomquist says:

Tangentially perhaps, is there some possibility of discussing the Chu space category?

4. Eugene Lerman says:

John wrote:

I’m thinking I can get away with using any convenient category of smooth spaces.

I think we’d be better off with $C^\infty$ schemes. $C^\infty$ schemes have a good notion of a Poisson algebra. But nothing is worked out.

• John Baez says:

I’ve made a bunch of progress, and my ideas have changed a bit, but I’ll wait and explain this at the workshop. I’m sure there’s a lot of fun stuff to do here.

5. This May, a small group of mathematicians is going to a workshop on the categorical foundations of network theory, or Jacob Biamonte. I’m trying to get us mentally prepared for this. We all have different ideas, yet they should fit together somehow.

Tobias Fritz, Eugene Lerman and David Spivak have all written articles here about their work, though I suspect Eugene will have a lot of completely new things to say, too. Now it’s time for me to say what my students and I have doing.

6. Here’s a new paper on network theory:

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

While my paper with Jason Erbele, Categories in control, studies signal flow diagrams, this one focuses on circuit diagrams. The two are different, but closely related.

7. We’re getting ready for the Turin workshop on the Categorical Foundations of Network Theory. So, we’re trying to get our thoughts in order. A bunch of blog articles seems like a good way to go about it.

8. John Baez says:

Someone asked me what the workshop achieved. I replied:

First, we figured out a lot about how the approach Brendan Fong and I are using to describe networks (“decorated cospans”) is related to the approach David Spivak uses (“the operad of wiring diagrams”). David and a student and Brendan will try to prove some theorems about this and write this up.

Second, Eugene Lerman and I made some progress understanding each other’s thoughts on networks of nonlinear classical mechanical systems. There’s a short paper I should write on this, but I’m being distracted…

… by my new work with Brendan and Blake Pollard on networks in stochastic thermodynamics! This seems more urgent to me, since I believe it will help us understand living systems.

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