Complex networks with tuneable dimensions as a universality playground

TL;DR

This paper introduces a tunable complex network model where the spectral dimension can be continuously adjusted, providing a new platform to explore universality in inhomogeneous systems and their relation to fractional-dimensional field theories.

Contribution

The authors construct a novel complex network model with a controllable spectral dimension, enabling systematic study of universality in inhomogeneous and fractional-dimensional structures.

Findings

Spectral dimension can be tuned from 1 to infinity in the model.

The model serves as a tool to investigate universal behavior in complex networks.

Discussion on the mimicry of continuous field theories in fractional dimensions.

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 for universality, the spectral dimension. 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.