Complex networks with tuneable dimensions as a universality playground

Abstract

Universality is one of the key concepts in understanding critical phenomena. However, for interacting inhomogeneous systems described by complex networks a clear understanding of the relevant parameters for universality is still missing. Here we discuss the role of a fundamental network parameter, the spectral dimension, neglected in previous investigations. For this purpose, we construct a complex network model where the probability of a bond between two nodes is proportional to a power law of the nodes' distances. By explicit computation we prove that the spectral dimension for this model can be tuned continuously from 1 to infinity, and we discuss related network connectivity measures. We propose our model as a tool to probe universal behaviour on inhomogeneous structures and comment on the possibility that the universal behaviour of correlated models on such networks mimics the one of continuous field theories in fractional euclidean dimensions. We suggest that similar structures could be engineered in atomic, molecular and optical devices in order to tune universal properties to a desired value.

Local topological moves determine global diffusion properties of hyperbolic higher-order networks

“…In particular, the spectral dimension constitutes a fundamental quantity to capture how geometry affects dynamics. For instance, it is is known to characterize the return time of random walkers [51,52], the stability of synchronised states [25,26], universal critical phenomena [53][54][55][56][57][58][59], quantum diffusion [60,61] or the universal properties of different quantum gravity approaches [62][63][64]. Remarkably, the value of the spectral dimension is not universal in networks but can vary significantly from network to network.…”