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Could there be a ‘universal law’ for network growth?

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Correlation does not imply causation. However, some correlations are at least fascinating, and here’s one that’s getting a lot of attention: the apparent structural similarity between the growth of the universe, that of the human brain, and complex artificial networks like the Internet or Twitter.

In fact, according to the research presented by Dmitri Krioukov of the University of California in San Diego, equations similar to Einstein’s descriptions of the universe might apply to things like the Internet.

First, let’s dispose of any cosmological theology (“Wow! The universe is a giant brain!”) that might arise from what follows: Krioukov emphasises in this media announcement:

“By no means do we claim that the universe is a global brain or a computer. But the discovered equivalence between the growth of the universe and complex networks strongly suggests that unexpectedly similar laws govern the dynamics of these very different complex systems.”

So let’s take a look at the paper itself, published in full (hooray for open science!) here.

What Krioukov and his collaborators have turned up is that the universe’s development seems to follow a power-law structure: “the causal network representing the large-scale structure of spacetime in our accelerating universe is a power-law graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks,” they state in the abstract (Wikipedia’s description of the power-law will suffice for background).

In other words, Krioukov is suggesting that the growth of the universe seems to observe a mathematical relationship that’s also been documented in such diverse Earth-bound fields as the human brain, the Internet and social networks.

How can this be?

If The Register understands Krioukov’s paper correctly, it partly follows from the constraint that the speed of light places on causality: macro events in spacetime cannot be linked if the speed of light prevents it (I’m ignoring micro phenomena such as entanglement, for the moment). Any “causet” – causal set – has to exist within a single “time cone”.

That means clustering is almost inevitable.

The conclusion that the universe follows a power law seems to be simpler: if we take the spacetime we observe now, and what’s called “de Sitter spacetime” (the eventual universe that is, depressingly, cold and empty), the large-scale graph for de Sitter spacetime produced a power law.

So what’s this got to do with the Internet?

The Krioukov paper points out that the Internet “looks” like the universe in this: at the large scale, it looks homogeneous (think “the cloud”!), but at the small scale, the Internet is lumpy. To put it in the more academic terms of the paper:

“De Sitter spacetime is homogeneous and isotropic, as is the hyperbolic space, but if we take a real network, e.g. the Internet, and map it to this homogeneous space, then after the mapping, the node density in the space is non-uniform”.

And this is where the paper offers up something tangible and useful: to those who have lived it, the growth of the Internet might seem random and unpredictable. However, it may have been governed by an “invisible hand” – equations “similar to Einstein’s” – which could “be used to predict and possibly control the fine-grained dynamics of links and nodes in networks”.

Predicting how a network might grow and behave is, as anybody who watched last week’s “Click Frenzy” disaster in Australia will attest, a discipline with real economic value.

And the connection with the brain? It's also a network that arises in conditions that challenge simple explanation - but you start with one cell, and end up with sufficient billions to comprehend articles like this one. ®

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