which ‘forgets the extra details’. This map should be a ‘homomorphism’ of algebras, but I’ll postpone the definition of that concept.

Let me give some examples. I’ll take the operad that I described last time, and describe some of its algebras, and homomorphisms between these.

]]>In this post I’m using simple graphs as an example of how operads can be used to assemble networks. Simple graphs have undirected edges. But they’re just one example of our approach. For networks where communication channels are directed, we use graphs that take this into account: for example, ‘directed graphs’.

Here’s what I said about this issue… with some emphasis added at points where I address the issue you raise:

There are some restrictions on what counts as a simple graph. If the vertices are agents of some sort and the edges are communication channels, these restrictions imply:

• We allow at most one channel between any pair of agents, since there’s at most one edge between any two vertices of our graph.

• The channels do not have a favored direction, since there are no arrows on the edges of our graph.

• We don’t allow a channel from an agent to itself, since an edge can’t start and end at the same vertex.

For other purposes we may want to drop some or all of these restrictions.There is an appalling diversity of options! We might want to allow multiple channels between a pair of agents. For this we could use multigraphs.We might want to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. For this we could use directed graphs.And so on.To avoid sinking into a mire of special cases, we need the full power of modern mathematics. Instead of separately studying all these various kinds of networks, we need a unified notion that subsumes all of them.

To do this, the Metron team came up with something called a ‘network model’. There is a network model for simple graphs, a network model for multigraphs,

a network model for directed graphs, a network model for directed graphs with 3 colors of vertex and 15 colors of edge, and more.

I’ll explain the general concept of ‘network model’ later. First I want to illustrate some things you can do with network models, and I’ll do that next time using simple graphs. But fear not—our setup can handle a wide variety of networks.

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(3) 3

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(5) 3 ]]>

arch1 wrote:

At worst I’ll be a straight man.

Like a comedian, every mathematician seems more funny with a straight man.

1) empty set (since sticking graphs together doesn’t create or destroy vertices).

Right!

2)-4) 2^C(t,2) (since each pair of distinguishable vertices can independently be joined by an arc, or not)

If C(t,2) means the binomial coefficient $\binom{t}{2}$, then you’re right! If

then is the set of simple graphs with vertices, so its cardinality is

5) 2^C(3,2)=8 (same reason)

Right again! In this example is *also* the set of simple graphs wiht vertices, so its cardinality is *also*

**Moral:** In this particular example, the algebra is very similar to the operad it’s an algebra of. That’s not always true, but every typed operad has an algebra of this kind, with

Replace “P” with “C” in my answers (the vertices are distinguishable but their order within the pair doesn’t matter)

]]>1) empty set (since sticking graphs together doesn’t create or destroy vertices)

2-4) 2^P(t,2) (since each pair of distinguishable vertices can independently be joined by an arc, or not)

5) 2^P(3,2)=8 (same reason) ]]>