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 
then Ohm’s law says:
where 
• If our edge 
we have
Here we need to think of the voltage and current as functions of time.
• If our edge 
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 
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 
Thus each edge 
Definition. Given a set 
We use the letter 
the positive real numbers, and 
Now we want to give our
To handle this nicely, we need to make a category of
Definition. Given 
such that the following diagrams commute:
There is a category 
Warning: the morphisms in
They are just a step along the way. A morphism of the kind we really want is a diagram like this in
where 
You see, if 
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 
• an object
• a morphism from
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 
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.
Posted by John Baez 
:
is negative, and this sends 
‘downhill’ towards the minimum. 
yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The latter separates the basins of attraction for the stable equilibria.
is a constant and 
is restricted to multiples of 
states, the unitary time evolution of a state 
is generated by a self-adjoint 
matrix 
called the Hamiltonian. In other words, 
and evolving for a time 
the expected value of the observable associated to a self-adjoint operator 
![[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t.](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0+%5Cquad+%5Ciff+%5Cquad+%5Cdisplaystyle%7B%5Cfrac%7Bd%7D%7Bd+t%7D%7D+%5Clangle+O+%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+0+%5Cquad+%5Cforall+%5C%3A+%7C%5Cpsi_0+%5Crangle%2C+%5Cforall+t.+&bg=ffffff&fg=333333&s=0 "[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t.")
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. 
is the vector built from the diagonal entries of 
More concretely, 
is the 
th component of the vector 
![[O,H] =0](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D+%3D0+&bg=ffffff&fg=333333&s=0 "[O,H] =0")
and all 
obeying the master equation. █![[O,H]=0](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D%3D0&bg=ffffff&fg=333333&s=0 "[O,H]=0")
the expected value of 
and all 
However, John and Brendan saw that—unlike in quantum mechanics—you need more than just the expectation value of the observable ![[O,H]=0.](https://s0.wp.com/latex.php?latex=%5BO%2CH%5D%3D0.&bg=ffffff&fg=333333&s=0 "[O,H]=0.")
You really need both
? 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:![[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0+%5Cquad+%5Ciff+%5Cquad+%5Cdisplaystyle%7B%5Cfrac%7Bd%7D%7Bd+t%7D%7D+%5Clangle+O+%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+0+%5Cquad+%5Cforall+%5C%3A+%7C%5Cpsi_0+%5Crangle%2C+%5Cforall+t+%5Cge+0+&bg=ffffff&fg=333333&s=0 "[O, H] = 0 \quad \iff \quad \displaystyle{\frac{d}{d t}} \langle O \rangle_{\psi(t)} = 0 \quad \forall \: |\psi_0 \rangle, \forall t \ge 0")
direction is easy to show, and it follows from John and Brendan’s theorem. The point is to show the 
direction. Since 
are an orthonormal basis of eigenvectors, and 
are the corresponding eigenvalues. We then have:
being linearly independent. For any infinitesimal stochastic operator 
connected components iff we can reorder (permute) the states of the system such that 
which implies![[O, H] = 0](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0&bg=ffffff&fg=333333&s=0 "[O, H] = 0")
![[O, H] = 0.](https://s0.wp.com/latex.php?latex=%5BO%2C+H%5D+%3D+0.&bg=ffffff&fg=333333&s=0 "[O, H] = 0.")
█
so that 
generates a 1-parameter unitary group. Show that the 
is 
sums to 1: 
be a possibly time-dependent observable, and write 
for its expected value with respect to some initial state ![\displaystyle{ \frac{d}{d t}\langle O(t)\rangle_{\psi(t)} = \left\langle [O(t), H] \right\rangle_{\psi(t)} + \left\langle \frac{\partial O(t)}{\partial t}\right\rangle_{\psi(t)} }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%7B+%5Cfrac%7Bd%7D%7Bd+t%7D%5Clangle+O%28t%29%5Crangle_%7B%5Cpsi%28t%29%7D+%3D+%5Cleft%5Clangle+%5BO%28t%29%2C+H%5D+%5Cright%5Crangle_%7B%5Cpsi%28t%29%7D+%2B+%5Cleft%5Clangle+%5Cfrac%7B%5Cpartial+O%28t%29%7D%7B%5Cpartial+t%7D%5Cright%5Crangle_%7B%5Cpsi%28t%29%7D+%7D&bg=ffffff&fg=333333&s=0 "\displaystyle{ \frac{d}{d t}\langle O(t)\rangle_{\psi(t)} = \left\langle [O(t), H] \right\rangle_{\psi(t)} + \left\langle \frac{\partial O(t)}{\partial t}\right\rangle_{\psi(t)} }")
It is easy to see that a function 
solving
that preserves the differential equations and fixes the diagonal 
and thus could account for this invariant subsystem. It turns out that what matters is the structure of the mutual dependencies of the three subsystems making up the big system. That is, the evolution of 
depends only on 
the evolution of 
depends only on 
and the evolution of 
depends only on 
has no symmetries. Nonetheless, the existence of the invariant subsystem living on the diagonal 
can be deduced from certain properties of the graph 
The key is the existence of a surjective map of graphs 
with exactly one node, call it 
and one arrow. It is also crucial that 
has the following lifting property: there is a unique way to lift the one and only arrow of 
is the set of edges, 
is the set of nodes, and the two arrows assign to an edge its source and target, respectively. To each node we attach a phase space (or more formally a manifold, perhaps with boundary or corners). Here ‘attach’ means that we choose a function 
it assigns to each node 
a phase space 
(If we think of the set 
as a category of manifolds and smooth maps, then 
is simply a functor.) 
where 
as the collection of phase spaces of the subsystems constituting the network 
We refer to 
is 
while the phase space of the network 
is the real line 
in a graph 
depends on the states of the subsystem associated to the node 
To make this precise requires the notion of an ‘open system’, or ‘control system’, which we define below. It also requires a way to associate an open system to the set of arrows coming into a node/vertex 
controlled by the the phase space 
is a map 
is the tangent bundle of the space 
so that for all 
is a vector tangent to 
Given phase spaces 
For us, submersions of the form 
are general enough.)
(this is a very short tree, a corolla if you like). The input tree of a node 
In the example the input tree 
For instance for the input tree in the example we just discussed, 
is the map
then 
is also a network. This allows us to talk about the phase space 
of an input tree. In our running example,
of open systems associated with every node 
is
to each one of them.
by 
and the space of vector fields on 
). In our running example the interconnection map for the network 
where 
is a phase space function and 
is a collection of open systems compatible with the graph and the phase space function. The corresponding vector field on the total space of the network is obtained by interconnecting the open systems.
In this category, by definition, a morphism from a network 
which is compatible with the phase space functions:
defines a map 
of total phase spaces in the opposite direction:
The objects of this category are pairs consisting of a phase space and a vector field on this phase space. A morphism in this category is a smooth map of phase spaces that intertwines the two vector fields. That is:
in 
then a morphism from 
to some dynamical system 
is simply a piece of an integral curve of the vector field 
defined on an interval 
And if 
is the constant vector field on the circle then a morphism from 
Proving that a given dynamical system has a periodic orbit is usually hard.
so that 
and the two interconnection maps 
and 
are compatible:
of open systems on the network 
