Modeling spatial networks by contact graphs of disk packings

Spatial networks are ubiquitous in the real world, with examples including Internet, power grids, neural networks, and so on. In this paper, based on a proposed disk packing, we present a contact graph as an exactly solvable model for spatial networks. We then derive analytically some critical structural properties of the model, including degree distribution, clustering coefficient, and diameter, and show that it displays simultaneously the striking scale-free small-world phenomena observed in most real networks. We also determine all the eigenvalues and their multiplicities of the transition probability matrix for the contact graph, and exploit the eigenvalues to derive closed-form expressions for Kemeny constant of random walks and the number of spanning trees on the contact graph. Finally, we derive closed-form expressions for some interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance. As an application of resistance distances, we study the coherence of the first-order consensus dynamics, which tends to a small constant as the graph grows, implying that the graph is resistant to noise.