Complex networks with tuneable dimensions as a universality playground

Ana P. Millán, Giacomo Gori, Federico Battiston, Tilman Enss, Nicolò Defenu

Published 2020-06-18Version 1

Allowing to relate exactly the behaviour of a wide range of real interacting systems with abstract mathematical models, the theory of universality is one of the core successes of modern physics. Over the years, many of such interacting systems have been conveniently mapped into networks, physical architectures on top of which collective and in particular critical behavior may emerge. Despite a few insights, a clear understanding of the relevant parameters for universality on network structures is still missing. The comprehension of such phenomena needs the identification of a class of inhomogeneous structures, whose connectivity and spectral properties may be simply varied, allowing to test their influence on critical phenomena. Here, we construct a complex network model where the probability for the existence of a bond between two nodes is proportional to a power law of the nodes' distance $1/r^{1+\sigma}$ with $\sigma\in\mathbb{R}$. By an explicit numerical computation we prove that the spectral dimensions for such model can be continuously tuned in the interval $d_{s}\in[1,\infty)$. We discuss this feature in relation to other structural properties, such as the Hausdorff dimension and local connectivity measures. Offering fully tuneable spectral properties governing universality in interacting systems, we propose our model as a tool to probe universal behaviour on inhomogeneous structures. We suggest that similar structures could be engineered in atomic, molecular and optical devices in order to tune universal properties to a desired value.