Extended corona product as an exactly tractable model for weighted heterogeneous networks

TL;DR

This paper introduces an exactly solvable model for weighted heterogeneous networks using an extended corona product, enabling analytical study of their structural and dynamical properties relevant to real-world systems.

Contribution

It develops a novel weighted corona product model and derives analytical expressions for key network properties and dynamics, bridging a gap in modeling weighted complex networks.

Findings

Model accurately reproduces properties of real weighted networks

Derived spectra enable analysis of random walk dynamics

Analytical formulas for hitting times and spanning trees

Abstract

Various graph products and operations have been widely used to construct complex networks with common properties of real-life systems. However, current works mainly focus on designing models of binary networks, in spite of the fact that many real networks can be better mimicked by heterogeneous weighted networks. In this paper, we develop a corona product of two weighted graphs, based on which and an observed updating mechanism of edge weight in real networks, we propose a minimal generative model for inhomogeneous weighted networks. We derive analytically relevant properties of the weighted network model, including strength, weight and degree distributions, clustering coefficient, degree correlations and diameter. These properties are in good agreement with those observed in diverse real-world weighted networks. We then determine all the eigenvalues and their corresponding multiplicities of the transition probability matrix for random walks on the weighted networks. Finally, we apply the obtained spectra to derive explicit expressions for mean hitting time of random walks and weighted counting of spanning trees on the weighted networks. Our model is an exactly solvable one, allowing to analytically treat its structural and dynamical properties, which is thus a good test-bed and an ideal substrate network for studying different dynamical processes, in order to explore the impacts of heterogeneous weight distribution on these processes.