Disordered Complex Networks: Energy Optimal Lattices and Persistent Homology

Abstract

Disordered complex networks are of fundamental interest in statistical physics, and they have attracted recent interest as stochastic models for information transmission over wireless networks. While mathematically tractable, a network based on the regulation Poisson point process model offers challenges vis-a-vis network efficiency. Strongly correlated alternatives, such as networks based on random matrix spectra (the Ginibre network), on the other hand offer formidable challenges in terms of tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean lattices <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interpolate</i> between Poisson and rigidly structured networks, and allow us to achieve the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">best of both worlds</i> : significantly improve upon the Poisson model in terms of network efficacy measured by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Signal to Interference plus Noise Ratio</i> (abbrv. SINR) and the related concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">coverage probabilities</i> , at the same time retaining a considerable measure of mathematical and computational simplicity and robustness to erasure and noise. We investigate the optimal choice of the base lattice in this model, connecting it to the celebrated problem optimality of Euclidean lattices with respect to the Epstein Zeta function, which is in turn related to notions of lattice energy. This leads us to the choice of the triangular lattice in 2D and face centered cubic lattice in 3D, whose Gaussian perturbations we consider. We provide theoretical analysis and empirical investigations to demonstrate that the coverage probability decreases with increasing strength of perturbation, eventually converging to that of the Poisson network. In the regime of low disorder, our studies suggest an approximate statistical behaviour of the coverage function near a base station as a log-normal distribution with parameters depending on the Epstein Zeta function of the lattice, and related approximate dependencies for a power-law constant that governs the network coverage probability at large thresholds. In 2D, we determine the disorder strength at which the perturbed triangular lattice (abbrv. PTL) and the Ginibre networks are the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">closest</i> measured by comparing their network topologies via a comparison of their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Persistence Diagrams</i> in the total variation as well as the symmetrized nearest neighbour distances. We demonstrate that, at this very same disorder, the PTL and the Ginibre networks exhibit very similar coverage probability distributions, with the PTL performing at least as well as the Ginibre. Thus, the PTL network at this disorder strength can be taken to be an effective substitute for the Ginibre network model, while at the same time offering the advantages of greater tractability both from theoretical and empirical perspectives.