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One way to study a network is to break it down into into its simplest pattern of links. These simple patterns are called motifs and their numbers usually depend on the type of network.

One of the big puzzles of network science is that some motifs crop up much more often than others. These motifs are clearly important. Remove them (or change their distribution) and the behaviour of the network changes too. But nobody knows why.

Today, Xiao-Ke Xu at Hong Kong Polytechnic University and friends say they know why and the answer is intimately linked to the existence of rich clubs within a network.

Let’s step back for a bit of background. In many networks, a small number of nodes are well connected to large number of others. The group of all well-connected nodes is known as the “rich club” and it is known to play an important role in the network of which it is part.

Rich clubs are particularly influential in a specific class of network in which the number of links between nodes varies in a way that is scale free (ie follows a power law).

This is an important class. It includes the internet, social networks, airline networks and many naturally occurring networks such as gene regulatory networks.

(However, it doesn’t apply to networks in which the links form at random or networks in which the links are highly uniform, like power grids.)

Network theorists have long known that because rich club members can influence large numbers of other nodes, it’s possible to manipulate the entire network, simply by targetting the small number of rich club members.

But a crucial factor is the network of links between rich club members themselves. If the rich club is poorly connected it will have a different influence on the the network as a whole than if it is well connected.

So Xu and co began playing with a network in which they could change the connectedness of the rich club. At the same time, they measured the number of motifs in this network and how this changed.

It turns out that the shape of motifs that link rich club members is different to those that appear elsewhere in the network. So the ratio of these different types of motifs changes in a predictable way as the connectedness of the rich club varies.

That immediately suggests that if you measure the distribution of motifs, you can determine the nature of the rich club. In other words, you can determine the existence and nature of a rich club without measuring it directly.

But more importantly, it also explains why certain motifs are more important than others and why they crop up more often than expected: they are merely a reflection of the nature of the network’s rich club.

Fascinating stuff. It’s looking increasingly clear that these rich clubs are even more important than network scientists had imagined.

Ref: arxiv.org/abs/1106.5301: Optimizing And Controlling Functions Of Complex Networks By Manipulating Rich-Club Connections

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