In the last decade or so, the study of networks has had a profound effect on the way we understand the spread of everything from fashion and ideas to forest fires and disease. 

But this better understanding of individual networks has revealed a gaping hole in our knowledge of how networks interact with each other. That looks to be hugely important. Many systems, rather than being individual networks, are actually networks of networks: the financial system, the economy, our brain and our genetic control system to name just a few.

What’s puzzling about all these systems is that they demonstrate emergent behaviour that single networks alone cannot reproduce. 

So it’s no surprise that with the triumphs in understanding single networks under their belts, complexity scientists have set their sights on the more ambitious goal of understanding ‘networks of networks’. Consequently, this area is set to become one of the fastest growing in science.  

Today, Anna Saumell-Mendiola, Angeles Serrano and Marian Boguna at the University of Barcelona in Spain reveal their approach to the problem. These guys have studied the way that disease-spreading networks interact when they become linked. 

To keep things simple they consider two identical networks that share a few common links.  Each network models the spread of disease using the so-called susceptible-infected-susceptible or SIS model. The idea is that each individual in the network can either be susceptible to disease or infected by the disease. In other words, they can be either a receiver or a transmitter of the disease. 

Susceptible individuals become infected if they are near infected individuals and infected individuals become susceptible again after a short, predefined period of time.

The SIS model is well understood and accurately reproduces many of the features of disease transmission that epidemiologists observe in the real world, such as the transition from healthy to epidemic states and back again.

In their new work, Saumell-Mendiola and co ask what happens when the same disease is circulating in two different networks that are only loosely linked. 

The answer is somewhat counter-intuitive. These guys calculate that the global epidemic threshold of the disease can be smaller than the epidemic thresholds in the two networks separately.

That’s significant. It implies that a disease can be globally endemic–that it can be self-sustaining without any external source of infection–even if it is not spreading in either network separately. And all this can happen even if the two networks share only a few connections.

That’s exactly the kind of subtle, emergent behaviour that network scientists have found it hard to understand and model with models of single networks.   

Incidentally, this model has important real world applications. There are plenty of examples of coupled networks, such as the spread of sexually transmitted disease in heterosexual and homosexual populations. These populations are largely separate but linked by a few bisexual individuals. 

The work of Saumell-Mendiola and co could provide some important insights into how best to stop the spread of disease in this kind of coupled network.

Clearly there’s more to be gleaned from this kind of study. The bigger question, of course, is whether there is a more general  understanding we can gain  from the study of networks of networks, whether a kind of statistical mechanics of networks will eventually emerge. 

There’s no shortage of interest and funding in this area. So time will surely tell.

Ref: arxiv.org/abs/1202.4087: Epidemic Spreading On Interconnected Networks