I’m currently reading Small World, Mark Buchanan’s account of ‘small world’ networks – the concept behind the old idea of six degrees of separation (which I thought I’d seen fairly rigorously disproved somewhere).
What strikes me so far is the vagueness of it all. Perhaps it will become clearer – I’m only on page 68 – but there seems to be an imprecision that’s most unusual for a mathematical discipline. This could be down to the way Buchanan is presenting things of course – his style is very readable but this does sometimes (not always!) bring a degree of smoothing over.
Just as an example, we are told about Erdos in 1959 solving the puzzle of how many roads are required, placed randomly, to join 50 towns. Buchanan tells us ‘It turns out, the random placement of about 98 roads is adequate to ensure that the great majority of towns are linked.’
I’m sorry? What does about 98 mean? How about ensuring the vast majority are linked? That’s small consolation if you live in one of the towns that is isolated.
The other vagueness, in the ‘six degrees of separation’ model is what we really count as an acquaintance. It’s such a fuzzy concept, it’s hard to see just how it can be made to operate with the precision required by mathematics. I have nearly 1,000 people in my email address book. Are they all acquaintances? How about those lovely people on the Nature Network with whom I often exchange comments about blog entries, but none of whom have I ever met or spoken to, and only two have I ever emailed?
For that matter, what about my ‘harvest’ emails? Is somebody an acquaintance because I’ve seen their email address? Probably not. How about when someone sends me an email and copies in lots of other people. Are those email addresses part of my contact circle? Dunno – and I doubt if the people who play around with this interesting, but in some senses rather futile feeling, research do either.
The transition between where a network is unconnected and where it is connected is called the percolation threshold. It is defined for an infinite network as the degree of connectivity where there is the first infinite cluster.
The percolation threshold is a true critical point, the properties of the network diverge as a function of connectivity as the percolation threshold is approached. That is they change exponentially as the critical point is approached.
The point isn’t that there is a specific number of nodes between any two arbitrary nodes, but rather that as you increase the connectivity there is a point where a differential change (i.e. a single contact) makes a gigantic difference by connecting two very large but previously unconnected clusters.
David – I think I can see what you’re saying (dare I say it, that was a bit tech-speak), but I don’t think it really deals with my two issues which are the lack of connection to the real world, and the inability to be precise about definitions.