Non-Homogeneous Fractal Hierarchical Weighted Networks

Abstract

A model of fractal hierarchical structures that share the property of non-homogeneous weighted networks is introduced. These networks can be completely and analytically characterized in terms of the involved parameters, i.e., the size of the original graph Nk and the non-homogeneous weight scaling factors r 1, r 2, · · · rM. We also study the average weighted shortest path (AWSP), the average degree and the average node strength, taking place on the non-homogeneous hierarchical weighted networks. Moreover the AWSP is scrupulously calculated. We show that the AWSP depends on the number of copies and the sum of all non-homogeneous weight scaling factors in the infinite network order limit.

Dynamics with Chaos and Fractals

“…Owing to the important roles of Sierpinski fractals in several applications like weighted networks, trapping problems, antenna engineering, city planning, and urban growth [135,136,137,138,139], we expect that the results will be helpful in further fields of applications. One of the relevant applications involves optimization theory.…”

First-Order Network Coherence and Eigentime Identity on the Weighted Cayley Networks

“…Each node of B (1) 1 , B (2) 1 , B (3) 1 links the attaching node A 1 , i.e. A 0 in G 0 , with unitary weight.…”

“…For convenience, the central node A n−1 in G n−1 is labeled A n in G n . The three nodes B (1) n−1 , B (2) n−1 , B (3) n−1 in G n−1 are labeled as B (1) n , B (2) n , B (3) n in G n , sequentially. The weighted Cayley networks are set up.…”

“…From the iterative construction method of G n and for the sake of convenience, the node set of G n can be regarded as merging four groups, sequentially denoted by {A n }, G (1) n , G (2) n , G (3) n . The four groups are obtained as follows.…”

“…Finally, the last two terms take into account all the possible paths between each nodes of subgraph G (1) n , G (2) n , G (3) n and the central node A n . Using the scaling mechanism for the edge weights, the first term in Eq.…”