Estimating the Manifold Dimension of a Complex Network Using Weyl’s Law

The dimension of the space underlying real-world networks has been shown to strongly influence the networks structural properties, from the degree distribution to the way the networks respond to diffusion and percolation processes. In this paper we propose a way to estimate the dimension of the manifold underlying a network that is based on Weyl’s law, a mathematical result that describes the asymptotic behaviour of the eigenvalues of the graph Laplacian. For the case of manifold graphs, the dimension we estimate is equivalent to the fractal dimension of the network, a measure of structural self-similarity. Through an extensive set of experiments on both synthetic and real-world networks we show that our approach is able to correctly estimate the manifold dimension. We compare this with alternative methods to compute the fractal dimension and we show that our approach yields a better estimate on both synthetic and real-world examples.