With apologies to René Magritte and
@paulhollins (who gave me the idea) I think it is useful to be reminded of the distinction between reality and pictures. We are currently investing certain pictures (not unlike the one below) with mystical properties which we believe grant privileged access to an 'objective reality'. This was Magritte's target when he painted his pipe: but for him it was a clever joke about mimesis and representation. For us, and for the pictures we now confuse with reality, the consequences are more serious: policies are made by looking at such pictures, people will get hurt, others will get rich and some will get elected.
Having said this, it is not quite the same joke as Magritte's, because the pictures of social networks can lead to actions whose consequences appear to be consistent with the logic of the picture. This is to say that a social network picture is a 'logical representation': one whose formal characteristics are isomorphic to lived experience. The question about the nature of a social network picture is a question about the relationship between reality and its logical representation. It is analogous to the relationship between geometry and the natural world.
Descartes saw geometry precisely in this way: that activity of the mind which connected with the body to produce relations which were isomorphic to transcendental reality. I think it is awareness of the connection to the transcendental which is missing in the contemporary obsession with social network graphs. We have forgotten about our bodies and we have forgotten about what we cannot know.
The connection between the transcendent and measurement is beautifully expressed by Blake's "The Ancient of Days". I would suggest that instead of measuring the heavens, Blake's Urizen (the bearded man) is now measuring pictorial depictions of reality. That's what the social network analysts are doing. The irony of this is that Urizen represents rational man. Instead of rational man encountering the heavens and constraining them with his compass, he now encounters the artefacts of his own technological making, similarly and vainly trying to constrain them too. We appear to have gone up a level of recursion!
But geometry and its relation to the transcendental haven't gone away. When we look at a picture - particularly a picture of labelled nodes and arcs - we are examining a logical structure and piecing together its inherent logic. In so doing, we re-encounter something fundamental about the relation between logic and the natural world, and in this case about the relation between geometry and the natural world. Geometry is literally "earth-measurement": that means it is precisely at the point of encounter between reason and experience. Whatever reason can bring to bear on the geometer's constructions, experience and the body moves in harmony with it. Geometry, fundamentally, is an art of harmony. Is it so fanciful to suppose that the earliest geometers (and the ones whose work most closely resembles our social network analysts) were the astrologers for whom the 'music of the spheres' played such an important role as they examined the starry heavens?
When we take those social network geometries and we apply the logical reasoning of formal techniques like category theory, we may be able to uncover something of the territory of our experience of those graphs and pretty pictures. At a time when we are so confused about the import of the miraculous and enchanting pretty pictures that our technology presents to us, revealing the deeper logic behind those pictures as the logic of our experience might be extremely valuable.