Journal of Applied Mathematics

Volume 2018, Article ID 1017308, 7 pages

https://doi.org/10.1155/2018/1017308

## The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

^{1}LRIT-CNRST URAC 29, Rabat IT Center, Faculty of Sciences, Mohammed V University in Rabat, BP 1014, Rabat, Morocco^{2}Winona State University, Winona, MN 55987, USA^{3}Laboratory of Conception and Systems, Faculty of Sciences, Mohammed V University in Rabat, BP 1014, Rabat, Morocco

Correspondence should be addressed to Raihana Mokhlissi; moc.liamg@anahiarissilhkom

Received 19 April 2018; Accepted 8 July 2018; Published 3 September 2018

Academic Editor: Heping Zhang

Copyright © 2018 Raihana Mokhlissi 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

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

#### 1. Introduction

Recently, the analysis of complex networks has received a major boost caused by the huge network data resources and many systems in the real world can be described and characterized by complex networks [1]. Some scientific studies have inspired researchers to construct network models to explain the existing common characteristics in real-life systems. Among the well-known models of the complex networks, there is a small-world network. It displays rich behavior as observed in a large variety of real systems including Internet (websites with navigation menus), electric power grids, networks of brain neurons, telephone call graphs, and social networks. It is characterized by specific structural features: large clustering coefficient and small average distance. To analyze this class of complex networks, theories are needed to explain their inherent and emergent properties. New formal models of these networks are needed to predict accurately their performance, assert the guarantees of their reliability, and quantify their robustness. The graph theory has a powerful tool to simplify this theoretical study by enumerating** the spanning trees** of a network [2]. The latter are defined as a connected and acyclic subgraph of having all vertices (nodes) of and some or all its edges. The goal of this paper is to know how many spanning trees can have a network. The enumeration of these spanning trees tends to be one of the most important parameters that characterizes the network reliability [3]. We denote the number of spanning trees by , also known as the complexity of a network. In general, it can be obtained by calculating the determinant or the eigenvalues of the Laplacian matrix corresponding to the network [4]. However, this general method is not acceptable for large and complex networks due to its high computing time complexity. Therefore, it is interesting to develop techniques and methods to facilitate the calculation of the number of spanning trees and find its exact formula for special classes of networks. In this context, our work proposes a combinatorial method for determining the spanning trees number for some complex networks, which is the decomposition method [5]. It relies on the principle of a process of “Divide and Conquer" by dividing a problem in subproblems, solving each of these subproblems and then incorporating the partial results for a general solution.

As an application of the number of spanning trees of a network, we use** the entropy of spanning trees** or what is called the asymptotic complexity (see, e.g., Dehmer, Emmert-Streib, Chen, Li, and Shi [2, 6]). By calculating this entropy, we can estimate how the network will evolve to infinity. This parameter permits us to quantify the robustness of complex networks and to characterize their structures [7]. It is related to the ability of the network to resist random changes in its structures. Many researchers have used this measure to estimate the robustness of some complex networks and the heterogeneity of their structures such as the small-world Farey graph [8], the two-tree network [9], the planar unclustered networks [10], the prism and antiprism graphs [11], and the lattices [12].

The novelty of our work is to analytically investigate two generalized families of small-world networks, called the Small-World Exponential network. See, e.g., Mokhlissi, Lotfi, Debnath and El Marraki [13] and Liu, Dolgushev, Qi and Zhang [14], and the Koch network. See, e.g., Zhang, Zhou, Xie, Chen, Lin and Guan [15] and Zhang, Gao, Chen, Zhou, Zhang, and Guan [16]. The first network is based on complete graphs and the second network is based on the classical fractal Koch curve [17], which has many important properties observed in real networks. To generalize these two networks, we add two important parameters related to the size of the cyclic subgraphs and the dimension of the cyclic subgraphs (the number of the cyclic subgraphs added). We suggest two iterative algorithms generating their structures, we determine their topological properties, and we calculate their complexities. In the end, we evaluate and compare their spanning trees entropy with other networks having the same average degree as the Hanoi network, the Flower network, the Honeycomb lattice. As a result, we conclude that the generalized Small-World Exponential network and the generalized Koch network have the same spanning tree entropy, so the same robustness although their structures and properties are totally different, and this entropy depends just on the size of the cyclic subgraphs, which means the articulation nodes degree of the first iteration increases according to the dimension of the cyclic subgraphs; it does not influence the spanning tee entropy. The scope of this study is that the generalization of these two small-world networks does not affect the concept of the small-world networks (large clustering coefficient and small average distance). The work of this paper presents an alternative perspective in the analysis of small-world networks that exhibit typical features of real-world systems.

The outline of this paper is organized as follows. In Section 2, we present the preliminaries and the used methodology. The construction, the properties, and the complexity of the generalized Small-World Exponential network and the generalized Koch network are provided in Sections 3 and 4. Then, the spanning trees entropy of these small-world networks are presented in Section 5. Finally, the conclusion is included in Section 6.

#### 2. Preliminaries

In this section, we introduce some notations and the method used to facilitate the calculation of the complexity of a complex network. Let be a connected planar graph with being its number of vertices, being its number of edges, and being its number of faces; it has no loops and no parallel edges. The number of vertices of a graph refers to its order and its number of edges refers to its size. The terms graph and network are used indistinctly. A network is said to be a small-world network if the distance between two random nodes grows proportionally to the logarithm of the number of nodes in the network, that is, , while the clustering coefficient (measure of the degree to which nodes in a network tend to cluster together) is not small.

**Euler’s formula [22]:** Euler’s formula is a topological invariant that characterized the topological properties related to the number of vertices, edges, and faces.

Theorem 1. *Let be a connected planar graph with vertices, edges, and faces. These numbers are connected by the well-known Euler’s relation; then*

The selection of the appropriate method for calculating the spanning trees number is a key factor in a given network. For this work, we put forward a decomposition method to make the number of spanning trees easy for computation. This method relies on the principle of Divide and Conquer; we decompose the graph into different subgraphs according certain constraints: by following one node, two nodes, an edge, and a path. In this work, we study the case where subgraphs are connected by one vertex (see Figure 1). To apply this method, we follow this algorithm:(1)We decompose the original graph into different subgraphs that are connected to one vertex.(2)We calculate the number of spanning trees for each of subgraph.(3)We collect the results to obtain the complexity of the original graph.