The more complex a system is, the more its seemingly local properties are given by context.

I’ve discussed this in terms of functional properties. That is properties of a system which seem to do something for a purpose. The hand grasps that it may lift the coffee cup to the mouth. A crucial point here is that the hand grasps before the coffee reaches the mouth. The grasp is caused by the end towards which it is oriented, which is in the future. This makes a mess of our theories in which the future is strictly caused by the past, and forces us to offer a complicated explanation (and a lot of handwaving) to recover such a strictly past-to-future model out of a seemingly ends-oriented phenomenon. It is however no coincidence that much more parsimonious language exists. By simply admitting ends proper into the discourse things become comprehensible again.

A deeper dive into this discussion must wait for another time. The crucial thing to note in the above is that the function of the hand as coffee-getter is not something that is intrinsic to the hand, but only how it fits into the larger situation of the person’s (me) behavior in getting another sip of coffee. The hand is not a coffee-getter in itself, but rather functions as one in a context. The functional property is contextually inherited.

But contextual inheritance can be much more mundane than this. It enters the picture as soon as we start dealing with systems qua systems. Consider a network, say, a social network. Networks are represented by nodes and edges — things and their connections to one another.

In a social network we might consider those connections to represent friendships. A common question we ask of networks is how “central” is a given node in to the network? Typically we really mean something vague here, like “how important is this node?”. It turns out there is not one way to answer this question.

A common first answer is to look at the so-called degree of each node. The degree is just how many connections a node has to other nodes. In the sketch below we see six nodes, each with their degree marked in blue.

The chap in the center of the network has five friends, and the other individuals each have one friend, namely the chap in the center. We say that the center node is of degree 5, the other nodes are each of degree 1.

Notice what has happened to our language: we say the node has a degree, but degree is a property of how a node is connected to other nodes. If you take the node out of the network, what is its degree?

Contextual inheritance has slipped in. A node “has” degree based on how it is connected to other nodes. The degree is not “in” the node, but it’s not not in the node either. But this context seems pretty close by, if it is indeed contextual inheritance, it is still quite local context that is generating the property “degree of the node”. But this is not always the case.

Returning to the specific issue of the “importance” of a node in a network, intuitively, in the above the node degree seems to do a decent job. The center node in the network that connects all the others has a degree 5 while the others are degree 1. It is clearly, in some sense, the most important node.

Let’s look at the node degrees for a different network.

Something intuitively off with this one. The central node that binds together what would otherwise be three separate networks has the lowest degree of any node in the network! So is our intuition off, or is our metric bunk? All things being equal, I tend to side with the intuition. We know things when we see them. Formalisms are clunky.

As it happens, for this very reason, there a ton of ways to measure the “importance” of a node on a network. Another way to measure is the “betweenness” of a node. Betweenness is calculated by, first, establishing the shortest path between every pair of nodes on a network (in terms of number of hops it takes to get to one node from the other), and then counting the number of times such a path routes through a given node.

Here is the second network with the betweenness counts for each node.

Aha! So despite that the central node is the lowest in terms of degree, it is the highest in terms of betweenness. You can see why intuitively: every path that connects those node clusters must pass through.

Now, again, the central node “has” a betweenness of 75. But where is that value generated? Certainly not in the node. And not even in the way that node is connected to its neighbors. It has to do with the structure of the entire network. The betweenness is something that is about both the node and the network. It is both local and non-local.

The more complex a system is, the more its seemingly local properties are contextually inherited.