A wavelet view of small-world networks

The human eye is a powerful tool to gain an understanding of the structure of small networks of tens of vertices. However, direct analysis by the eye is hopeless for a network of millions of vertices. The theory of wavelets provides a powerful microscopy to look at large complex networks to answer specific questions about their structure. Wavelet multiresolution representations of networks provide a coarse-to-fine strategy for characterizing and classifying networks by processing the minimum amount of information. In particular, we show that the small-world property of a class of networks can easily be derived from its coarse description in the lowest resolution subspace of the wavelet decomposition.