Imagine a network with a random walker on it. The random walker is able to move from node to node, but only if a link connects these nodes. This network also grows. When a growth event occurs, a new node is attached to the current location of the random walker. Depending on the speed the walker moves, different networks will be built. If the walker moves slowly star-like networks will emerge. Alternatively, if the walker moves very fast (infinitely), we build a type of scale-free network. Below are some videos showing this process in which the walker is moving at different rates. In the first video the walker moves slowly, and in the second video the walker moves quickly.

This model is a simple example of a dynamical network, that is, a network that changes with time. However, our interest lies in trying to understand the uncertainty associated with the walker's position as the network grows infinitely large. A more technical way to ask this is how does the positional entropy of the walker evolve as the network grows? Below are two videos that show the expected position of the walker as a function of the network size (these two videos correspond to the two videos above). It can be seen that although the walker is intrinsically diffusive, it behaves like a wave when network growth is coupled to its position.

This process acts as a minimal model to demonstrate how growth can funnel dynamics into certain outcomes. In the language of information theory the behaviour of the random walker is called compressible. More information regarding this work can be found here, here and here.