<![CDATA[Network Theory (Part 27)]]>

<![CDATA[Network Theory (Part 27)]]>This quarter my graduate seminar at UCR will be about network theory. I have a few students starting work on this, so it seems like a good chance to think harder about the foundations of the subject. I’ve decided that bicategories of spans play a basic role, so I want to talk about those.

If you haven’t read the series up to now, don’t worry! Nothing I do for a while will rely on that earlier stuff. I want a fresh start. But just for a minute, I want to talk about the big picture: how the new stuff will relate to the old stuff.

So far this series has been talking about three closely related kinds of networks:

• Markov processes
• stochastic Petri nets
• stochastic reaction networks

but there are many other kinds of networks, and I want to bring some more into play:

• circuit diagrams
• bond graphs
• signal-flow graphs

These come from the world of control theory and engineering—especially electrical engineering, but also mechanical, hydraulic and other kinds of engineering.

My goal is not to tour different formalisms, but to integrate them into a single framework, so we can easily take ideas and theorems from one discipline and apply them to another.

For example, in Part 16 we saw that a special class of Markov processes can also be seen as a special class of circuit diagrams: namely, electrical circuits made of resistors. Also, in Part 17 we saw that stochastic Petri nets and stochastic reaction networks are just two different ways of talking about the same thing. This allows us to take results from chemistry—where they like stochastic reaction networks, which they call ‘chemical reaction networks’—and apply them to epidemiology, where they like stochastic Petri nets, which they call ‘compartmental models’.

As you can see, fighting through the thicket of terminology is half the battle here! The problem is that people in different applied subjects keep reinventing the same mathematics, using terminologies specific to their own interests… making it harder to see how generally applicable their work actually is. But we can’t blame them for doing this. It’s the job of mathematicians to step in, learn all this stuff, and extract the general ideas.

We can see a similar thing happening when writing was invented in ancient Mesopotamia, around 3000 BC. Different trades invented their own numbering systems! A base-60 system, the S system, was used to count most discrete objects, such as sheep or people. But for ‘rations’ such as cheese or fish, they used a base 120 system, the B system. Another system, the ŠE system, was used to measure quantities of grain. There were about a dozen such systems! Only later did they get standardized.

Circuit diagrams

But enough chit-chat; let’s get to work. I want to talk about circuit diagrams—diagrams of electrical circuits. They can get really complicated:

This is a 10-watt audio amplifier with bass boost. It looks quite intimidating. But I’ll start with a simple class of circuit diagrams, made of just a few kinds of parts:

• resistors,
• inductors,
• capacitors,
• voltage sources

and maybe some others later on. I’ll explain how you can translate any such diagram into a system of differential equations that describes how the voltages and currents along the wires change with time.

This is something you’d learn in a basic course on electrical engineering, at least back in the old days before analogue circuits had been largely replaced by digital ones. But my goal is different. I’m not mainly interested in electrical circuits per se: to me the important thing is how circuit diagrams provide a pictorial way of reasoning about differential equations… and how we can use the differential equations to describe many kinds of systems, not just electrical circuits.

So, I won’t spend much time explaining why electrical circuits do what they do—see the links for that. I’ll focus on the math of circuit diagrams, and how they apply to many different subjects, not just electrical circuits.

Let’s start with an example:

This describes a current flowing around a loop of wire with 4 elements on it: a resistor, an inductor, a capacitor, and a voltage source—for example, a battery. Each of these elements is designated by a cute symbol, and each has a real number associated to it:

• This is a resistor:

and it comes with a number $R,$ called its resistance.

• This is an inductor:

and it comes with a number $L,$ called its inductance.

• This is a capacitor:

and it comes with a number $C,$ called its capacitance.

• This is a voltage source:

and it comes with a number $V,$ called its voltage.

You may wonder why inductance got called $L$ instead of $I.$ Well, it’s probably because $I$ stands for ‘current’. And then you’ll ask why current is called $I$ instead of $C.$ I don’t know: maybe because $C$ stands for ‘capacitance’. If every word started with its own unique letter, we wouldn’t have these problems. But then we wouldn’t need words.

Here’s another example:

This example has two new features. First, it has places where wires meet, drawn as black dots. These dots are often called nodes, or sometimes vertices. Since ‘vertex’ starts with V and so does ‘voltage’, let’s call the dots ‘nodes’. Roughly speaking, a graph is a thing with nodes and edges, like this:

This suggests that in our circuit, the wires with elements on them should be seen as edges of a graph. Or perhaps just the wires should be seen as edges, and the elements should be seen as nodes! This is an example of a ‘design decision’ we have to make when formalizing the theory of circuit diagrams. There are also various different precise definitions of ‘graph’, and we need to try to choose the best one.

A second new feature of this example is that it has some white dots called terminals, where wires end. Mathematically these terminals are also nodes in our graph, but they play a special role: they are places where we are allowed to connect this circuit to another circuit. You’ll notice this circuit doesn’t have a voltage source. So, it’s like piece of electrical equipment without its own battery. We need to plug it in for it to do anything interesting!

This is very important. Big complicated electrical circuits are often made by hooking together smaller ones. The pieces are best thought of as ‘open systems’: that is, physical systems that interact with the outside world. Traditionally, a lot of physics focuses on ‘closed systems’, which don’t interact with the outside the world—the part of the world we aren’t modeling. But network theory is all about how we can connect open systems together to form larger open systems (or closed systems). And this is one place where category shows up. As we’ll see, we can think of an open system as a ‘morphism’ going from some inputs to some outputs, and we can ‘compose’ morphisms to get new morphisms by hooking them together.

Differential equations from circuit diagrams

Let me sketch how to get a bunch of ordinary differential equations from a circuit diagram. These equations will say what the circuit does.

We start with a graph having some set $N$ of nodes and some set $E$ of edges. To say how much current is flowing along each edge it will be helpful to give each edge a direction, like this:

So, define a graph to consist of two functions

$s,t : E \to N$

Then each edge $e$ will have some node $s(e)$ as its source, or starting-point, and some node $t(e)$ as its target, or endpoint:

(This kind of graph is often called a directed multigraph or quiver, to distinguish it from other kinds, but I’ll just say ‘graph’.)

Next, each edge is labelled by one of four elements: resistor, capacitor, inductor or voltage source. It’s also labelled by a real number, which we call the resistance, capacitance, inductance or voltage of that element. We will make this part prettier later on, so we can easily introduce more kinds of elements without any trouble.

Finally, we specify a subset $T \subseteq N$ and call these nodes terminals.

Our goal now is to write down some ordinary differential equations that say how a bunch of variables change with time. These variables come in two kinds:

• Each edge $e$ has a current running along it, which is a function of time denoted $I_e$ . So, for each $e \in E$ we have a function

$I_e : \mathbb{R} \to \mathbb{R}$

• Each edge $e$ also has a voltage across it, which is a function of time denoted $V_e$ . So, for each $e \in E$ we have a function

$V_e : \mathbb{R} \to \mathbb{R}$

We now write down a bunch of equations obeyed by these currents and voltages. First there are some equations called Kirchhoff’s laws:

• Kirchhoff’s current law says that for each node that is not a terminal, the total current flowing into that node equals the total current flowing out. In other words:

$\displaystyle{ \sum_{e: t(e) = n} I_e = \sum_{e: s(e) = n} I_e }$

for each node $n \in N - T.$ We don’t impose Kirchhoff’s current law at terminals, because we want to allow current to flow in or out there!

• Kirchhoff’s voltage law says that we can choose for each node a potential $\phi_n,$ which is a function of time:

$\phi_n : \mathbb{R} \to \mathbb{R}$

such that

$V_e = \phi_{s(e)} - \phi_{t(e)}$

for each $e \in E.$ In other words, the voltage across each edge is the difference of potentials at the two ends of this edge. This is a slightly nonstandard way to state Kirchhoff’s voltage law, but it’s equivalent to the usual one.

In addition to Kirchhoff’s laws, there’s an equation for each edge, relating the current and voltage on that edge. The details of this equation depends on the element labelling that edge, so we consider the four cases in turn:

• If our edge $e$ is labelled by a resistor of resistance $R$ :

we write the equation

$V_e = R I_e$

This is called Ohm’s law.

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

we write the equation

$\displaystyle{ V_e = L \frac{d I_e}{d t} }$

I don’t know a name for this equation, but you can read about it here.

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

we write the equation

$\displaystyle{ I_e = C \frac{d V_e}{d t} }$

I don’t know a name for this equation, but you can read about it here.

• If our edge $e$ is labelled by a voltage source of voltage $V$ :

we write the equation

$V_e = V$

This explains the term ‘voltage source’.

Puzzles

Next time we’ll look at some examples and see how we can start polishing up this formalism into something more pretty. But you can get to work now:

Puzzle 1. Starting from the rules above, write down and simplify the equations for this circuit:

Puzzle 2. Do the same for this circuit:

Puzzle 3. If we added a fifth kind of element, our rules for getting equations from circuit diagrams would have more symmetry between voltages and currents. What is this extra element?

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