Protocol Duffers

A graph, yesterday
A graph, yesterday

What can we tell by both the order and size of a graph? One of the basic theorems of graph theory states that for any graph G, the sum of the degrees of the nodes equals twice the number of edges of G. That is, if the degree of any node is the number of edges connected to it (for node n1 with two edges connected to it, its degree = 2), the sum of all the degrees of the graph will be double the size of the graph (the number of edges). In other words, a network is not simply made up of a certain number of elements connected to one another, but is constituted by, qualified by, the connectivity of the nodes. How connected are you? What type of connection do you have? For a square, the sum of the degrees is 8 (the nodes [the square’s corners] each have two edges [the square’s lines] connected to them), while the sum of the edges is 4. In the IT industries connectivity is purely a quantitative measure (bandwidth, number of simultaneous connections, download capacity). Yet, in a different vein, Deleuze and Guattari describe network forms such as the rhizome as, in effect, edges that contain nodes (rather than vice versa), or even, paradoxically, as edges without nodes. In graph theory we see that the connectivity of a graph or network is a value different from a mere count of the number of edges. A graph not only has edges between nodes but edges connecting nodes.

This paragraph (from Galloway and Thacker on protocols) is typical of the faults of this kind of writing. Nothing that it says is entirely incorrect; and yet it confuses and misleads where it ought to clarify.

It is certainly true that there is a relationship between the ratio between the order and size of a graph, and the degree of its nodes. This can be stated precisely: given that the sum of all the degrees of the graph will be double the size of the graph, and the average degree of nodes in the graph will be that sum divided by the number of nodes, then the average degree of nodes in the graph will be twice the number of edges divided by the number of nodes. OK, so what? “A network is not simply made up of a certain number of elements connected to one another” – except that it still is. No extra information has been introduced by observing these ratios. There isn’t an additional property of “connectivity” (in the sense meant here, but see below) that is not inferrable from what we already know about size, order and the degree of each node. Saying “a graph not only has edges between nodes but edges connecting nodes” is a little like saying “the sun not only warms sunbathers, but also increases their temperature”.

The reference to “connectivity” as the term is used informally “in the IT industries” is largely a red herring here. The size, order and degrees of a graph are also “purely…quantitative” – what else would they be? As for Deleuze and Guattari, who can say? “Edges that contain nodes (rather than vice versa)” – who says that nodes “contain” edges? What could it possibly mean for either to contain the other? “Edges without nodes” do not exist in standard graph theory – there are no edges-to-nowhere or edges-from-nowhere. A rhizome’s structure is graph-like, in that nodes (in the botanical sense) put out multiple roots and shoots which connect to other nodes, but to map a rhizome as a graph we must introduce abstract “nodes” to represent the ends of shoots; only then can segments of the rhizome be considered “edges” between nodes (in the graph theoretical sense). None of this is particularly helpful in this context.

When we talk about “connectivity” in graph theory, we are typically talking about paths (traceable along one or more edges, e.g. from A to B and then from B to C) between nodes; the question that interests us is whether there are any nodes that are unreachable along any path from any other nodes, whether there are any disconnected subgraphs, how redundant the connections between nodes are, and so on. “Connectivity” in this sense is indeed not a function of the counts of nodes and edges (although if the number of edges is fewer than the number of nodes minus one, your graph cannot be fully connected…). But it is also not a matter of the degrees of nodes. A graph may be separable into multiple disconnected subgraphs, and yet every node may have a high degree, having multiple edges going out to other nodes within the subgraph to which it belongs. In this sense, it is indeed true that “the connectivity of a graph is…different from a mere count of the number of edges” (it is in fact the k-vertex-connectedness of the graph, a precise notion quite separate from that of degree). But the way in which it is really true is quite different from – and much more meaningful than – the way in which the above paragraph tries to suggest it is true.

What has happened here? The authors have clearly done their reading, but they have not synthesized their knowledge at the technical level: they move from learned fact to learned fact without understanding the logical infrastructure that connects them, being content instead to associate at the level of figurative resemblance. If pressed, writers in this style will often claim that they are identifying “homologies” (abusing that word also in the process) between things, and that one thing’s having a similar sort of conceptual shape to another is sufficient reason to associate them. But the available connectives in that case are weak (“it is surely no coincidence that…”, and other rhetorical substitutes for being able to demonstrate a reliably traversable connection), and it is often impossible to move from the resulting abstract quasi-structure back to the level of the explanandum without falling into total incoherence. The required “aboutness” just isn’t there: there is no negotiable passage back from the talk-about-talk to the talk-about-the-things-the-original-talk-was-about.

In the analysis of literary texts (and other cultural artifacts) we often are looking for structures of similar-patterning: for things which “look like” one another, which share a field of associations or a way of relating elements within that field. It is usually quite legitimate to compare two poems and to say that both have a common “logic” in the way they relate temporality and subjective identity-formation, or something like that. But it is foolish to apply the tools of literary analysis to objects whose primary mode of organisation is not figurative. Skimming along the surface of the language used by technicians in the description of their tasks, one may well discover patterns of association that are “telling”, that reveal something at the level of ideology. I am not proposing that cultural studies give up the jouissance of unmasking – without it, the discipline would lose its entire raison d’etre. But I would like to put in a plea for technical focus, of a kind appropriate to the domain, when dealing with technical subjects. You don’t have to ignore the things you’ve been trained to recognise, but you do need to be able to be undistracted by them. Get it right, then be clever. The payoffs may take longer in coming, but they’re so much realer.