How size of networks skyrocket with a single new link

A single new connection can dramatically enhance the size of a network – no matter whether this connection represents an additional link in the internet, a new acquaintance within a circle of friends or a connection between two nerve cells in the brain.

The results, which are published in Nature Physics, were part of a theoretical study carried out by researchers from the Max Planck Institute for Dynamics and Self-Organisation, the Bernstein Center for Computational Neuroscience Göttingen and the University Göttingen. This study mathematically describes for the first time the influence of single additional links in a network. (Nature Physics, published online on January 16th, 2011).

Imagine the following scenario: In your sports team you get to know a new player and arrange to go out and see a movie on the next weekend. The new team member brings along three friends - and suddenly by adding one new contact, your own circle of friends has grown by four people.

Growth processes of this sort occur in many networks: Neurons in the brain constantly establish new connections, websites link to each other and a person travelling infected with influenza creates a network of infected places with each intermediate stop. From a scientist's point of view, such growth processes are still poorly understood: How does a network change when single links are added? How quickly does a network grow in this way?

To answer these questions, scientists from Göttingen tracked the growth of networks link by link. A new connection, however, can not only add one new element. It can also merge two networks (as in the example in the sports team above). The researchers focused on a special form of network growth that introduces a form of competition between possible links: If several new connections are possible, only the one connection is created: the one that results in the smallest new network.

"There is evidence, that growing networks of neurons at first prefer forming small groups and thus roughly follow the growth process we discuss", says Jan Nagler, staff researcher at the University of Göttingen and the Max Planck Institute for Dynamics and Self-Organization.