# Dense Networks With Mixture Degree Distribution

^{1}Key Laboratory of High-Confidence Software Technology, Peking University, Beijing, China^{2}School of Electronics Engineering and Computer Science, Peking University, Beijing, China^{3}College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

Complex networks have become a powerful tool to describe the structure and evolution in a large quantity of real networks in the past few years, such as friendship networks, metabolic networks, protein–protein interaction networks, and software networks. While a variety of complex networks have been published, dense networks sharing remarkable structural properties, such as the scale-free feature, are seldom reported. Here, our goal is to construct a class of dense networks. Then, we discover that our networks follow the mixture degree distribution; that is, there is a critical point above which the cumulative degree distribution has a power-law form and below which the exponential distribution is observed. Next, we also prove the networks proposed to show the small-world property. Finally, we study random walks on our networks with a trap fixed at a vertex with the highest degree and find that the closed form for the mean first-passage time increases logarithmically with the number of vertices of our networks.

## 1 Introduction

The exploding interest in complex networks during the several decades of the 21st century is rooted in the discovery that despite the diversity of complex networks, the structure and the evolution of each network are driven by a common set of basic laws and principles. Examples include the Internet and the World Wide Web [1], biological networks [2], social networks [3], and communication networks [4, 5], to mention but a few. There are two common considerable properties, the small-world effect and the scale-free feature. The well-known WS-network (Watts and Strogatz, WS) was proposed by Watts and Strogatz in Nature to explain small-world phenomena in diverse networks by two indices, diameter and clustering coefficient [6]. The most pioneering of generally studied networks is the BA-network (Barabási and Albert, BA) built by Barabási and Albert in Science using two rules, namely, growth and preferential attachment [7].

In currently existing studies, the main concentration is to create networks which have scale-free and small-world properties as mentioned above. However, most networks are sparse, which means that the average degree of networks is asymptotically equal to a positive constant under the limitation of a large number of vertices. The principal reason is that numbers of real-world networks have been found to indicate sparsity. On the contrary, a few scholars have discovered the existence of dense networks in [8, 9]. For probing such networks, several available networks have been presented and analytically explored with primary methods, including mean-field theory [10, 11], generate function [12], and rate equation [13, 14]. In particular, the vast majority of these networks are proved to have no scale-free feature. To put this in another way, the degree distribution of these networks does not obey the power-law distribution. Therefore, it is of interest to develop new theoretical frameworks for producing networks with both the density feature and the scale-free feature. Here, we propose a class of networks with the appropriate structural properties mentioned above. Specifically speaking, these networks are precisely proved to be not only scale-free and small-world but also dense. Throughout this article, because graphs are abstract representations of networks, there is no need to distinguish between graph and network, which indicates that these two terms are considered to be the same.

The remainder of this article can be organized into the following sections. In Section 2, our task is to introduce network construction and discuss some extensively reported structural properties, including the average degree, mixture degree distribution, small-world property, and mean first-passage time. Among them, our networks are analytically proved to show both the density and scale-free features since the power-law exponent of cumulative degree distribution is equal to constant 2. Afterward, all the networks have a smaller diameter and a higher clustering coefficient, suggesting that they share the small-world property. Then, we derive an explicit expression of the mean first-passage time on our networks associated with the trapping problem. Finally, we close this article with a Conclusion and Discussions in the last section.

## 2 Network Construction and Topological Properties

In this section, our intention is to generate a class of networks: let *m* and step

### A Network Construction

Here, we will present our network,

Step 0: We start from a single vertex, we designate it as the active vertex of the network, and the active vertex is placed in the layer zero, denoted by

Step 1: We add a cycle *m* vertices and *m* edges, each vertex in the cycle is labeled as 0, and we connect each vertex labeled as 0 to the active vertex; then, we obtain

Step 2: We generate *m* replicas of *m* replicas to the new active vertex, and attach each of the non-bottom vertices of the *m* replicas to form an *m* cycle, so we acquire

**FIGURE 1**. Illustration of iterative construction leads to proposed network

Step 3: We add *m* replicas of *m* replicas with the active vertex, and connect each of the non-bottom vertices of these *m* replicas to form

These steps can be easily generalized. In fact, step *t* will involve the following operation:

Step *t*: We add *m* replicas of *m* replicas with the active vertex; and connect each of the non-bottom vertices of these *m* replicas to produce

Indefinitely repeating the steps of replication and connection, apparently, we obtain a class of networks. As an illustrative example, a network

Now, we compute some related basic parameters of our networks by construction. Let *t* as follows:

At step

Until now, we have finished the construction of our networks and computed certain fundamental quantities. It is an extremely evident fact to discover the hierarchy of our network related to the step. It is worth noticing that the involved network construction with hierarchy has been discussed in [1]. In [1], Ravasz *et al.* constructed a hierarchical network that combines the scale-free property with a high clustering. However, it should be noted that the approach adopted in this article is slightly different from the approach in [1]. While Ravasz *etal.* applied an iterative process for all vertices, our method is based on an idea that the next network *m*, our networks are different due to the number of vertices and edges being different, but the topological structure of our networks is the same. Additionally, as will be shown later, our network

### B Topological Properties

As has been mentioned above, we mainly focus on the investigation of certain topological properties related to the potential structure of the proposed network. In this section, we will calculate several topological properties, including the density feature, mixture degree distribution, small-world property, and mean first-passage time. What we present here will be significant constituents in the coming subsections.

**Density feature:** A key topological structure of a network is its average degree. Average degree

We give a detailed calculation process of the average degree in the Appendix. From Eq. 3, it goes without saying that the average degree *t* and does not tend to a real positive constant. In the large number of vertices, the average degree

**FIGURE 2**. Diagram of the average degree

Mixture degree distribution: The concept of degree is the most fundamental character and measure of a vertex in a network. Since in a network every vertex has a degree value, some large and some small, the distribution of vertex in the network is a key topological feature, which may be of great concern in application. The degree distribution is one of the most important topological features of a network. Degree distribution can be applied to determine whether a given network is scalefree or not. Combined with the process of network construction, it is straightforward to find that the degree spectrum of network

where notation *k* in network

For a network *t* and has degree *m* vertices, that is, the active vertex of the *m* units added to the network in the last layer (namely,

On the basis of the aforementioned Table 1 and Eq. 4, the dependence of the cumulative distribution *t* is captured by the following equation:

where power-law exponent *k* in Eq. 5 yields

The detailed calculation process of cumulative degree distribution is presented in Appendix. The cumulative degree distribution of our networks is composed of two parts, exponential distribution and power-law distribution. Besides, this result does not match the statement in the previous work. Genio believes that there is no network whose power-law exponent belongs to *et al.* explore two models of scale-free networks that have the degree distribution exponent *et al.* present a modeling framework which produces networks that are both dense and have scale-free properties in [9]. In addition, in [19], Ma *et al.* propose a framework for producing scale-free networks with the dense feature using two operations, that is, first-order subdivision and line operation. However, our approach is different from those in the above methods for constructing scale-free networks with dense properties. Meanwhile, the lights shed by them may be helpful to construct some novel networks with certain constraints. Next, we will discuss the small-world property of our networks.

Small-world property: Watts and Strogatz proposed the small-world property of complex networks by using two features: a relative smaller diameter and a higher clustering coefficient [6]. The small-world property of our networks is still not well discussed; despite that, they well describe these two important parameters of topological structures. In the coming discussion, we focus on the diameter and clustering coefficient of our networks.

Diameter: The distance between two vertices is the smallest number of edges to get from *u* to *v*. The longest shortest path between all pairs of vertices is called the diameter. The diameter is itself a feature of network structure and can be applied to characterize a communication delay over a network. In general, the larger the diameter is, the lower the communication efficiency is. The diameter of our network is denoted as

We give an exhaustive calculation process of diameter in Appendix. The reason comprises two main cases: 1) all vertices of the layer *t* and *m*. Note that this

Clustering coefficient: The clustering coefficient is another vital property of a network, which provides the measure of the local structure within the network. The most immediate measure of clustering is the clustering coefficient *i*. By definition, the clustering coefficient of a vertex *i* is the ratio of the total number *C*’s as follows:

The clustering coefficient of a vertex *i* in our networks is as follows:

The detailed process of the clustering coefficient will be given in Appendix. The degree of the clustering coefficient of a whole network is captured by the average value of the clustering coefficient,

Taking the limit on the number of vertices, the clustering coefficient

**FIGURE 3**. Diagram of the clustering coefficient

Consequently, our networks, with a smaller diameter and a higher clustering coefficient, can be considered as small-world networks.

Mean first-passage time: In this subsection, we have an attempt to study the trapping problem on our network *t* has the largest degree, referred to as *v* moves to one of its neighbors *v* in network

Here we think a walker starts from the vertex *v* at the initial time. Obviously, we can obtain the fact that the transition probability *v* to reach *u* holds the following equation:

where *i* and *u*, and

Armed with the rule mentioned above, we want to discuss the most important quantity for the trapping problem, generally named the first-passage time *v* to first arrive at the trap in the trapping problem. For our network *v* by *v* to first drop into the trap, namely, the active vertex

The classical method to solve the aforementioned equation is the generating function. Without loss of generality, we can write down the corresponding generating function of

A trial yet helpful fact associated with

Before proceeding further, we first define two notations, *s* steps and *w* at the layer *w*, after *s* steps. Combined with the structure of network

where *s* is equal to one and

Then, together with Eq. 13, the generating function

in which we apply a result

Meanwhile, let

On the basis of Eq. 16, doing the derivative of both sides of Eq. 15 evolves the exact solution of

For each vertex at the layer *L* can be expressed in terms of

Then, the first-passage time

Here, we again apply the hierarchical structure of network

Plugging Eqs 17 and 18 and the value of

Moreover, we think the logarithm of the number of vertices of network

Thus, in the large limit of the number of vertices of network*t*.

**FIGURE 4**. Diagram of the mean first-passage time

This is a bit different from some previous results in the existing literature; for instance, for a complete graph with *N* vertices, the mean first-passage time is exactly equal to

## 3 Conclusion and Discussion

In our article, we present a class of scale-free networks with the dense feature. On the basis of our analysis, we deduce some striking results.1) The average degree of our networks is not approximately a fixed constant value, and its value diverges with the step *t*, and then we reveal the fact that our networks are dense. 2) The cumulative degree distribution of our networks

## Data Availability Statement

The original contributions presented in the study are included in the article/supplementary Material; further inquiries can be directed to the corresponding author.

## Author Contributions

XW provided this topic and wrote the paper. FM discussed and modified it, and BY guided the manuscript. All authors contributed to manuscript and approved the submitted version.

## Funding

This research was supported in part by the National Key Research and Development Program of China under Grant No. 2019YFA0706401; in part by the National Natural Science Foundation of China under Grant No. 61632002; in part by the General Program of the National Natural Science Foundation of China under Grant No. 61872399, Grant No. 61872166, and Grant No. 61672264; in part by the Young Scientists Fund of the National Natural Science Foundation of China under Grant No. 61802009, Grant No. 61902005, and Grant No. 62002002; and in part by the National Natural Science Foundation of China under Grant No. 61662066.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: dense feature, scale-free property, small-world property, mean first-passage time, mixture degree distribution

Citation: Wang X, Ma F and Yao B (2021) Dense Networks With Mixture Degree Distribution. *Front. Phys.* 9:647346. doi: 10.3389/fphy.2021.647346

Received: 29 December 2020; Accepted: 15 February 2021;

Published: 26 April 2021.

Edited by:

Nuno A. M. Araújo, University of Lisbon, PortugalReviewed by:

Haroldo V. Ribeiro, State University of Maringá, BrazilRainer Klages, Queen Mary University of London, United Kingdom

Copyright © 2021 Wang, Ma and Yao. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Xiaomin Wang, wmxwm0616@163.com