Hidden in full view - the secret life of networks

You know that odd feeling you get when you start to understand something you kinda knew about for a while, and sorta knew was important, but never really took the trouble to push through the mental mill? And you know the other odd feeling, when something suddenly appears really rather simple yet full of implications that point somewhere important -- possibly very important?

Good. So you might know how it feels when both happens at once. That's roughly where I'm at, now I've taken the trouble to go and look at scale free networks.

It turns out, of course, that plenty of people far brighter than me have been going on about these for ages. But I've never seen them discussed outside the (many) fields where they've been popping up as significant: this could easily be due to my dilettantism and congenital laziness, and I'm making a bit of an arse of myself in blethering on about something everyone else knew about already.

(But I don't care. Oh, the deadly rush of a new enthusiasm combined with a blog.)

Networks. Points -- nodes -- connected to other points. That's all they are. A random network is one where the connections are, well, random. Each node can be connected to a random number of other nodes, and there's no way of guessing what that number is. You might think that if you looked at lots of networks - friends, sexual partners, organisations, collections of genes, the Intarwebs - then you'd find a lot of randomness - or, where there was some pattern, it'd be unique to a particular network.

Not at all. In all of those examples, and in plenty of others, networks turn out to be scale free. That's jargon for saying that the number of nodes with lots of connections are rarer than those with few (as you might expect, correctly this time), but that once you've counted all the nodes with certain numbers of connections , there's a very strong and very simple mathematical way to guess the probability of any node having any number of connections.

It doesn't matter how big the network is: if it's scale free, once you've counted the proportion of nodes with five connections to the nodes with ten connections, you can predict how many will have a hundred or a thousand.

That follows a power law - it's roughly exponential. That means that nodes with really high numbers of connections will be vanishingly rare - but the formula that says how rare follows the same rules as, say, the signal strength you pick up at a certain distance from a radio transmitter, or the force of gravity you feel as you move away from a body. Or the likelihood of earthquakes of a particular strength, or the distribution of aspects of linguistics, or... you get the picture. It appears to be a fundamental aspect of the reality of interconnected systems, and it's popping up everywhere.

That's very cool. But so what?

Well, scale free networks have certain very interesting characteristics. They are extremely resistant to random damage: the chances of a random error hitting a very important node are inversely proportional to the importance of that node - yet for most nodes, even very important ones, there will be enough others to help the network survive until it re-establishes its normal form. Conversely, if you have a directed attack that knows about important nodes, then a few well-aimed hits can utterly ruin the network in very short order.

You can see why that matters for such interesting areas such as terrorism networks - how they can be disrupted, and how they can be disruptive - or disease spread, or the Internet itself.

Indeed, scale free network theory started around eight years ago with a study of the way the Internet was structured: we're just beginning to join the dots.

But for me, the money shot is the realisation that biological evolution can be considered in those terms. The structures within cells - and the way DNA builds genes - have many characteristics of scale free networks, and when you model the way scale free networks react to random changes, you get systems that evolve much more efficiently towards a particular target than if you start with a random network.

This is extremely interesting: there are many aspects of biological evolution that remain mysterious (but don't tell the creationists - they haven't spotted the good stuff). We may be on the brink of a revolution in our understanding of how complex systems follow simple rules - and in science, these are the times worth staying alive for.

Or it could be just another excitement, like chaos theory, that turns into something suggestive but not that useful.

I don't think so. Let's watch and see what happens.