An energy-based heterogeneity measure for quantifying structural irregularity in complex networks

https://doi.org/10.1016/j.jocs.2019.06.008Get rights and content

Highlights

  • We propose a novel yet efficient normalized measure for quantifying the structural heterogeneity characteristics of graphs.

  • The proposed measure is based on the difference between energy and Laplacian energy.

  • The simulation results imply that the proposed measure has lower bound 0 and upper bound 1, for regular graphs and star-like graphs, respectively.

  • The proposed measure can identify the structural topological differences of synthetic and real-world complex networks.

Abstract

In this article, we propose a novel and efficient measure to quantify the structural heterogeneity characteristics of a graph. This measure is also used for comparing and classifying the complex networks based on the spectral graph theory, by quantifying the difference between the energy and Laplacian energy extracted from the underlying networks. Experimental results captured from simulations on the synthetic and real-world networks imply that for regular graphs and star-like graphs, the proposed measure has lower and upper bound 0 and 1, respectively and in cases in which the graphs are non-isomorphic, the measure returns non-zero values.

Introduction

Data characterization for classifying and clustering purposes is one of the main challenges in analyzing the non-vectoral pattern data such as strings, trees and graphs. Unlike pattern vectors, analysis of tree or graph data provides labeling scheme or structural ordering of nodes. Identification and quantification of similarities and differences of various graph models are among the fundamental and challenging issues in different fields of science and engineering. Our main aim is to find an efficient method for characterization of the structure of a graph, which provides an indication of its inherent complexity. Therefore, in this article, we propose a novel measure for comparing and classifying the networks on the basis of the spectral graph theory, which quantitatively addresses the differences between the energy and Laplacian energy extracted from underlying networks. The experimental results from simulations on the syntactic and real-world networks indicate that this measure can return zero for regular graphs and 1 for the star-like graphs; and even when the graphs are non-isomorphic, non-zero values are returned. Moreover, the proposed measure can identify the topological differences with functional impact on information flow passing the network, such as absence/presence of crucial links which connect or separate the components accurately with simple calculations.

The paper is structured in five sections. Section 2 is devoted to the literature review and related work including analysis and definition of the main heterogeneity measures. Section 3 introduces the proposed heterogeneity measure in addition to required theorems for extracting its characteristics. Section 4 provides evidence of the proposed measure by utilizing synthetic and real datasets, and explains the details of the numerical results captured through the simulation experiments. Finally, Section 5 presents concluding remarks and suggestions for future contributions.

Section snippets

Irregularity measures and related literature

In the recent years, many studies have focused on graph analysis and complex networks to define a new measures that properly indicate the irregularity, and in particular is applicable to the extremal graphs (i.e., graphs that expose the maximum value for a given irregular measure (IM)) relevant to available measures.

Definition 1 [1]: An IM graph like G is a real function F:I(G)R of a G invariant set I such that F(G) = 0 if G is regular.

We assume that G is a simple and undirected graph with a

The proposed measure

In this section, we present a novel heterogeneity measure based on the graph spectral energy in order to evaluate the structural complexity of the graphs. Despite the fact that there are several alternative measures for quantifying the structural characteristics of the graphs, there are few measures with low computational complexity and high classification power. In this paper, an easy and efficient technique is explored. This metric can be applied to all kinds of graphs. Then, we will apply

The experiments

The empirical evaluation of different graph characteristics can be divided into two parts. After introducing the data sets used in this paper, we will analyze and assess the proposed heterogeneity measure along with the other measures through different simulation tests.

Concluding remarks and future work

The main aim of the present paper is to provide and develop a novel and efficient measure for quantifying the structural complexity of graphs. We defined a novel measure to evaluate the heterogeneity of the complex networks based on difference between energy and Laplacian energy of graphs. In order to evaluate the complexity measures and analyze their characteristics, we conducted the empirical results by using synthetic and real-world datasets. The simulation results have shown that the

Declaration of Competing Interest

Please check the following as appropriate:

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject

Farshad Safaei received the B.Sc., M.Sc., and Ph.D. degrees in Computer Engineering from Iran University of Science and Technology (IUST) in 1994, 1997 and 2007, respectively. He is currently an assistant professor in the Department of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran. His research interests are Performance Evaluation of Computer Systems, Networks-on-Chips, and Complex Networks.

References (35)

  • P.O. Boaventura-Netto

    Graph irregularity: discussion, graph extensions and new proposals

    Rev. Matemã¡tica Teorã­a Y Apl.

    (2015)
  • L.V. Collatz et al.
    (1957)
  • M.O. Albertson

    The irregularity of a graph

    Ars Combinatoria

    (1997)
  • P. Hansen et al.

    Graphs and Discovery

  • H. Abdo et al.

    The total irregularity of a graph

    arXiv preprint arXiv

    (2012)
  • E. Estrada

    Quantifying network heterogeneity

    Phys. Rev. E

    (2010)
  • C.S. Edwards

    The largest vertex degree sum for a triangle in a graph

    Bull. London Math. Soc.

    (1977)
  • Farshad Safaei received the B.Sc., M.Sc., and Ph.D. degrees in Computer Engineering from Iran University of Science and Technology (IUST) in 1994, 1997 and 2007, respectively. He is currently an assistant professor in the Department of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran. His research interests are Performance Evaluation of Computer Systems, Networks-on-Chips, and Complex Networks.

    Sepehr Tabrizchi received the B.Sc. degree in Computer Engineering from Azad university branch of Najaf Abad (IAUN) in 2014, and the M.Sc. degree in Computer System Architecture from Azad university branch of science and research Tehran (SRBIAU), Tehran, Iran, in 2017. He is currently pursuing Ph.D. degree in Computer Engineering at the Computer Engineering & Science Department, Shahid Beheshti University (National University of Iran), Tehran, Iran. His research activities include Complex and Social Networks.

    Amir Hosein Hadian Rasanan is a bachelor student of Computer sciences in Shahid Beheshti University. He is currently working on numerical algorithms to simulate the biological systems.

    Marzieh Zare received her PhD from University of North Texas in 2013. She is now a postdoctoral fellow in the school of computer science, the Institute for Research in Fundamental Sciences (IPM), Tehran, Iran. Her research interests are complex networks, neuroscience, and complexity/physics- inspired artificial Intelligence.

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