What links Alzheimer’s disease, the bridges of Königsberg and Twitter?

A mathematical puzzle originating in 18th century Prussia has led to insights in fields as diverse as banking, social networking, epidemiology – and now Alzheimer’s disease

Everything is a form of communication – the colours of flowers, dollar bills, bird song, the synapses of brains, and of course words. And with the necessity for communication comes the need for connectivity, so it’s little wonder that one of the fastest growing areas of research in the life sciences is “graph theory”, which attempts to describe the connections and communication between objects in a network – be it a network of banks, internet sites, ants or neurons.

The progression of Alzheimer’s is accompanied by a buildup of amyloid plaque (which we learn today may be halted by a new drug) and the breakdown of communication between nerve cells. It turns out that graph theory can provide fascinating insights into the faulty wiring behind the progressive memory loss of Alzheimer’s. But what exactly is graph theory?

To discover the origins of the theory we have to go back to the 18th century and the ancient Prussian city of Königsberg, now Kaliningrad – that tiny city state wedged between Poland and Lithuania. It was here that Leonard Euler solved the long-standing Bridges of Königsberg Problem, which has had a profound effect on the development of network theory.

A recent wave of expansion in network theory was spurred on by a seminal paper by Watts and Strogatz published in Nature in 1998. In that work the authors developed a set of mathematical parameters to characterise three particular networks – the brain of the earthworm, the power grid of the US and the web of collaboration between actors listed in the Internet Movie Database. They found that all three networks had what they termed “small world” properties, meaning that the nodes of a system are linked through relatively few intermediate steps.

The idea of six degrees of separation highlights some of the basic concepts of small world networks by stating that within the network of six billion people on the planet just six links separate us from any other person.

It turns out that nearly all networks found in nature have small world architectures, perhaps because small worlds maximise the efficiency of information transfer at a low wiring cost. The human brain is a small-world network and it has proved useful to characterise its structure and function in terms of the clustering coefficients and path lengths of graph theory. In Alzheimer’s disease and schizophrenia, brains tend to move towards a more random architecture. Graph theory has identified hubs that are vital for maintaining communication between anatomically distant locations. For example, the temporal lobe is crucial for memory formation, and graph theory has helped to pinpoint a hub in the parietal lobe that contributes to sustaining communication between the temporal lobe and areas of the frontal lobe that are quite far away from it.

In parallel with developments in the neurosciences, small world theory continues to be used to study connectivity in social networks like Facebook. Research has also started in the field of epidemiology with early work suggesting that analysing Twitter networks may help to predict the outbreak of a flu epidemic sooner.

Laurence O’Dwyer is a research fellow in the department of psychiatry, Goethe University, Germany

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