On the distributions of Laplacian eigenvalues versus node degrees in complex networks

https://doi.org/10.1016/j.physa.2009.12.005Get rights and content

Abstract

In this paper, the important issue of Laplacian eigenvalue distributions is investigated through theory-guided extensive numerical simulations, for four typical complex network models, namely, the ER random-graph networks, WS and NW small-world networks, and BA scale-free networks. It is found that these four types of complex networks share some common features, particularly similarities between the Laplacian eigenvalue distributions and the node degree distributions.

Introduction

Recently, the notion of complex networks has become more and more popular for research in physics, engineering, biology and even social sciences [1], [2], [3], [4], [5], [6], [7].

A network can be viewed as a graph consisting of nodes connected by edges according to certain rules, in which the nodes and edges typically represent some physical, biological or social entities and their relationships, respectively [8]. One of the greatest challenges in understanding the nature and essence of various complex networks today is how the network structures affect their dynamical behaviors [4], [5], [9]. To better understand and to deal with various complex networks, it is important and even necessary to study and comprehend the structural characteristics of such networks. From a graph-theoretic perspective, the Laplacian matrix of a network is essential, because it contains most information about the underlying network, such as the node-degree distribution and edge connectivity, among others. On the other hand, for a constant matrix, the spectrum of its eigenvalues plays a fundamental role in its behavior and functioning. Based on the study and comparison of the eigenvalues of the Laplacian matrices of different networks, called Laplacian eigenvalues hereafter, one can gain a basic understanding of the structural properties of such networks, as well as their similarities and differences, which are very useful for many relevant applications such as network synchronization [10], [9].

Among the many representative network models, the classical Erdös–Rényi (ER) random-graph networks [11] and the relatively new Watts–Strogatz (WS) and Newman–Watts (NW) small-world networks [12], [13] and Barabási–Albert (BA) scale-free networks [14] are particularly significant and important. The question to be addressed in this paper is: “Are there any similar statistical properties of the Laplacian eigenvalues with respect to the topological features in ER, WS, NW and BA network models?” Noticeably, this question has already been addressed from different points of view in the literature. For example, it was shown in Ref. [15] that the largest eigenvalues of the Laplacian and adjacency matrices of a scale-free network are determined mainly by the largest-degree node of the network, while the smallest nonzero eigenvalue of the Laplacian matrix depends on the way the nodes are connected to each other. Also, based on the random matrix theory it was shown in Ref. [16] that despite the differences between the spectral densities of some different types of networks, their eigenvalue fluctuations are about the same and they all follow the Gaussian orthogonal ensemble distribution of the random matrices. From a somewhat different standpoint and approach, this paper performs a careful numerical analysis based on intensive computer simulations, to study the eigenvalue distributions of the unweighted symmetrical Laplacian matrices of four representative complex network models: the ER random-graph networks, WS small-world networks, NW small-world networks, and BA scale-free networks. It is well known that for large-sized networks, their possible ensembles or realizations are very large. For instance, the set of all possible ER random-graph networks with 2500 nodes has roughly O(22500×2499/2) different network realizations, implying that there are totally about O(22500×2499/2) different Laplacian matrices, which is literally intractable given the currently available computing power. Therefore, following the common practice this paper carries out a theory-guided numerical study of the subject in interest.

The main finding of this paper is that the Laplacian eigenvalues of the aforementioned four types of complex networks have very different properties in general and yet meanwhile they also share some common features. In particular, based on the spectral theory of graphs [17], [18], the results obtained in this paper reveal that the distributions of the Laplacian eigenvalues are very similar to the distributions of the node-degree distributions, which further verify some similar findings in Refs. [15], [16]. More precisely, by plotting the Laplacian eigenvalues of the four network models against the node-degree indices in increasing order, we obtain various “eigenvalue curves” and found that in different realizations of the same type of network topology of the same size, the relative deviations of their Laplacian eigenvalue curves are very small—with only up to ±5% errors. This phenomenon shows that, for a certain network with fixed size and given node-degree distribution, the eigenvalue distributions of these four very different types of complex networks are all insensitive to the network connectivity characteristics, namely the locations of the 1’s and 0’s in their coupling matrices, where 1 means connected and 0 means unconnected.

For an unweighted symmetrical Laplacian matrix, the smallest eigenvalue is always zero and the others are strictly positive, denoted as 0=λ1<λ2λ3λN [2], [3], [4], [5], [6], [7], [8], [10]. In many applications, the largest and the smallest nonzero Laplacian eigenvalues play a very important role, therefore they have attracted more attention than the other eigenvalues [19], [20]. One contribution of this paper is to further reveal and confirm, from a somewhat different viewpoint, the similarities between the Laplacian eigenvalue distributions and the node-degree distributions of the aforementioned four types of complex networks, supported by theory-guided numerical simulations. By plotting the node degrees against the node-degree indices in increasing order, various “node-degree curves” are obtained, showing that (i) for the ER random-graph networks, the eigenvalue curves are similar to the node-degree curves, and all these curves are located nearby, mostly with up to ±5% errors; (ii) for the BA scale-free networks, the eigenvalue curves are also similar to the node-degree curves and their differences are also located nearby, mostly with up to ±7.5% errors; (iii) for the WS and NW small-world networks, however, the eigenvalue curves are quite different from their corresponding node-degree distributions in some small-index regions, although they are still similar in other regions.

The rest of this paper is organized as follows. Section 2 provides a brief summary of some notations and generating algorithms for the ER random-graph networks, WS and NW small-world networks and BA scale-free networks, useful throughout the paper. Section 3 presents some theory-guided numerical analysis of the Laplacian eigenvalue and node-degree distributions, reporting the main results of the paper. Section 4 gives more detailed analysis about the Laplacian eigenvalue properties of the four complex network models, respectively. Finally, conclusions are drawn in Section 5.

Section snippets

Notations

N: the number of nodes in a random graph model;

p: the probability parameter;

ki: the degree of node i, defined to be the number of edges directly connecting node i;

k={k1,k2,kN}: the set of node degrees arranged in the non-decreasing order: k1k2kN;

IN:N×N identity matrix;

A:N×N adjacency matrix, defined by aij={1if ij,i adjacent j0otherwise ;L:N×N Laplacian matrix, defined by lij={kiif i=j1if ij,i adjacent j0otherwise ;λ(L)={λ1,λ2,λN}: the set of eigenvalues of L arranged in the

Distributions of Laplacian eigenvalues

To start with, some numerical results of extensive simulations are shown, to have a sense of different distributions of Laplacian eigenvalues for the above-reviewed four typical network models, namely the ER random-graph model, the WS small-world network model, the NW small-world model, and the BA scale-free network model, respectively.

The results are summarized in Fig. 1(a), where the number of simulation runs on each network model is 1000, therefore totally 4000 network realizations were

Four typical network models

In this section, the aforementioned four different types of networks are discussed, respectively.

Conclusions

In this paper, the distributions of Laplacian eigenvalues of four representative complex network models have been carefully investigated through extensive theory-guided numerical simulations, including the ER random-graph networks, WS small-world networks, NW small-world networks, and BA scale-free networks. The results show that although the Laplacian eigenvalues of these four types of complex networks have many different properties in general, they nevertheless share some common features,

Acknowledgements

This research was supported by the Hong Kong Research Grants Council under the GRF Grants CityU 111708 and CityU 122906.

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