Popular models in network neuroscience model network growth as a function of spatial proximity and connectional similarity. These models typically start with a sparse “seed” network of connections, and then proceed to simulate growth by adding connections to this network in discrete time steps. Here, we show that the assumption of network growth in a variant of these models can be subsumed by a type of shrinkage, or the weakening of dominant structural patterns in the network. We illustrate this relationship in structural and proximity networks on our example brain-imaging data.
We now compute and visualize proximity networks and show that these networks accurately approximate structural networks. Here, we define proximity as distance(-α) and simply set α = 1. In practice, we usually fit α to the data, although this has no qualitative effect on our results.
Scatter plots of connectivity and proximity networks
We now sparsify proximity matrices by weakening the contribution of their first several components. This weakening is formally known as shrinkage, and is commonly used to clean covariance or correlation matrices. In our analyses, we find that shrunken proximity networks approximate structural networks with considerably higher accuracy.