Complexity

Volume 2017 (2017), Article ID 6210878, 7 pages

https://doi.org/10.1155/2017/6210878

## Structure Properties of Koch Networks Based on Networks Dynamical Systems

^{1}School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China^{2}School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China^{3}Department of Mathematics and Computer Science, Adelphi University, Garden City, NY 11530, USA

Correspondence should be addressed to Jia-Bao Liu and Shaohui Wang

Received 8 November 2016; Revised 20 January 2017; Accepted 12 February 2017; Published 6 March 2017

Academic Editor: Pietro De Lellis

Copyright © 2017 Yinhu Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce an informative labeling algorithm for the vertices of a family of Koch networks. Each label consists of two parts: the precise position and the time adding to Koch networks. The shortest path routing between any two vertices is determined only on the basis of their labels, and the routing is calculated only by few computations. The rigorous solutions of betweenness centrality for every node and edge are also derived by the help of their labels. Furthermore, the community structure in Koch networks is studied by the current and voltage characterizations of its resistor networks.

#### 1. Introduction

The WS small-world models [1] and BA scale-free networks [2] are two famous random networks which stimulated an in-depth understanding of various physical mechanisms in empirical complex networks. The two main shortcomings are the uncertain creating mechanism and the huge computation in analysis. Deterministic models always have important properties similar to random models, such as being scale-free, having small-world behavior, and being highly clustered, and thus they could be used to imitate empirical networks appropriately. Hence, the study of the deterministic models of a complex network is increasing recently.

Inspired by the simple recursive operation and techniques of plane filling and generating processes of fractals, several deterministic models [3–18] have been created imaginatively and studied carefully. The lines in the famous Koch fractals [19] are mapped into vertices, and there is an edge between two vertices if two lines are connected; the generated novel networks were named Koch networks [20]. This novel class of networks incorporates some key properties which characterize the majority of real-life networked systems: a power-law distribution with exponent in the range between 2 and 3, a high clustering coefficient, a small diameter, and average path length and degree correlations. Besides, the exact numbers of spanning trees, spanning forests, and connected spanning subgraphs in the networks are enumerated by Zhang et al. in [20]. All these features are obtained exactly according to the proposed generation algorithm of the networks considered [21–29]. The deterministic models of the complex network have a fixed shortest path, but how to mark it only by their labels is rarely researched [30–34].

However, some important properties in Koch networks, such as vertex labeling, the shortest path routing algorithm, the length of the shortest path between arbitrary two vertices, the betweenness centrality, and the current and voltage properties of Koch resistor networks have not yet been researched. In this paper, we introduce an informative labeling and routing algorithm for Koch networks. By the intrinsic advantages of the labels, we calculate the shortest path distances between two arbitrary vertices in a couple of computations. We derive the rigorous solutions of betweenness centrality of every node and edge, and we also research the current and voltage characteristics of Koch resistor networks.

#### 2. Koch Networks

The Koch networks are constructed in an iterative way. Let be the Koch networks after iterations, where is a structural parameter.

*Definition 1. *The Koch networks are generated as follows: initially , is a triangle. For , is obtained from by adding groups of vertices to each of the three vertices of every existing triangle in .

*Remark 2. *Each group consists of two new vertices, called son vertices. Both sons are connected to one another and to their father vertices; thus, the three vertices shape a new triangle.

That is to say, we can get from just by replacing each existing triangle in with the connected clusters on the right-hand side of Figure 1.