Extended Corona Product as an Exactly Tractable Model for Weighted Heterogeneous Networks

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.

Exploring Trapping Problem on Weighted Heterogeneous Networks Induced by Extended Corona Product

“…The family of weighted heterogeneous network employed as test-bed here was proposed by Qi et al in ref [18]. It was produced iteratively by integrating extended corona product with weight reinforcement mechanism and parameterized by δ(δ is a positive integer).…”

“…To this end, considerable deterministic network models that can reproduce common structural properties of real networks had been put forward to serve as test-bed [12]- [15]. Given the observation that massive networks are always composed of small pieces such as communities, modules and motifs [16]- [18], quite a few deterministic networks had been built out of smaller networks by employing graph products such as hierarchical product [19], [20], corona product [21], [22].…”

“…In this paper, we study two types of random walks on a family of weighted heterogeneous networks induced by extended corona product [18], [21], [22] with different numbers of traps. In more detail, we introduce the weighted heterogeneous networks proposed by Qi et al in Ref [18] in section 2. We study standard random walk and delayed random walk on the networks with one trap fixed at a initial node and derive the analytical expressions of AT T in section 3 and section 4 respectively.…”

Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

“…designing models for complex networks. Frequently used graph operations and products include edge iteration [8,9], planar triangulation [10,11,12], Kronecker product [13,14,15], hierarchical product [16,17,18], and corona product [19,20,21].…”

Spectra of some new extended corona

“…Recently, Adiga et al [1,2] have introduced some new extension of corona, neighborhood corona and edge corona of two graphs and computed their A-spectrum, L-spectrum and Q-spectrum. In [23], an extended corona product for weighted graphs is defined and then applying this generalized corona product and the reinforcement mechanism of edge weight in realistic networks, the authors introduced a simple generative model for heterogeneous weighted networks, which leads to rich topological and weighted properties.…”