Monday, 16 June 2014

More on Social Networks and Category Theory

Recently I've been thinking a lot about learning analytics, big data and all those pretty pictures of social networks that people are producing in tools like Gephi and R. There's something important about this stuff. Maybe the most important thing is it is something which fascinates us, but which is also very difficult to pin down. Is this just a variety of visualised statistics? But then, statistics has the hold it does over the social sciences because it produces something like 'empirical regularity' by averaging events, and then inferences can be made from the statistical regularities (we have understood for a long time a statistical correlation between smoking and cancer, although the causal connection has only very recently been discovered). The issue for social network analysis is that there doesn't tend to be an emphasis on establishing regularities. In which case, the visualisations do not seem to be part of an empirical effort. That means it's about something else.

In place of regularity, we have visualised 'patterns', 'clusters', etc. The identification of these leads to questions about similarities between different situations which pertain to each cluster and pattern. The Triple Helix people, for example, on looking at clusters of industry and universities, will point to the discourse dynamics in those regions and the intermixing of academic labour with industrial innovation. But I wonder what Victorian biologists would make of this as scientific procedure. They too looked at pattern and cluster (for example, on a butterfly's wing), but their efforts were to establish the regularity of occurrence of pattern, identify its genus, species, etc. Unfortunately social networks are much bigger than butterflies, and the pattern of a network tends to be a one-off. However, this is not to rule out the possibility of a typology of social networks - but I don't see it happening anywhere - certainly not in the learning analytics world (and maybe it's not a good idea!). Typologies would produce a different kind of regularity.

Category theory tells us about a typology of connection - and maybe that's a starting point. My understanding of it is only beginning to emerge (I may have to revisit what I say now, but I have to start somewhere!) To begin with, we have the difference between a monomorphism and an epimorphism as the differences between types of connection. I see these two mappings as being essentially about the differences between the contexts within which a person might say there is a connection. There are some connections where there is a mapping to a single point from a number of points: it is a point of 'focus' - perhaps it's like a 'resolution' - for example, the resolution of a musical dissonance. Mapping onto, often termed 'surjective', is related to the concept of epimorphism (although not equivalent to it). A 'monomorphic' mapping is different from epimorphism in the sense that monomorphism maps points uniquely onto other points (for example, mapping chairs onto students). Monomorphism and Epimorphism can be expressed using an arrow notation which in turn reveals the algebra. Monomorphism is written:

Essentially, this means that the map i¡f = i¡g --> f = g. Badiou argues that a monomorphism preserves difference. That is to say that if f and g are different, this difference is preserved in i. Monomorphic, or one-to-one mappings are related to what set theory calls 'injective mappings'.

The dual of the monomorphism is the epimorphism. In an Epimorphism, we start with an identity (a single arrow) and it is followed by the possibility of difference immanent in that identity (2 arrows).
It seems that the possibility of introducing difference and preserving identity in transformations between objects begins to open out an expressive power in the category theory notation that goes far beyond the simple 'connecting' of points. Thinking in this way introduces its own algebra, but an algebra which embraces the degrees of uncertainty and absence which get lost in our fascination with social networks. 

The algebra begins to appear when we start to think about how the mappings between objects might be viewed from other objects. For example, it is a common situation in a social network for a node to have a number of points pointing at it. In category theory, we can speculate on the objects which sees all these other objects. Such a situation where there is a central object which can see the other objects and which presents the internal logic between those objects is a kind of universe called a 'Topos'.

If we look at a social network graph as a Topos, then we examine its topology asking "who can see what?", "what is it like to be an inhabitant of this topos?". But more importantly, we cannot ask those questions without asking of ourselves, "where am I in this topos?". That, to me, is the really important question - and the reason why category theory may provide a solution to the mystery of why we are so fascinated by these topologies.

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