Hongzhong Deng

Spectral Measure of Structural Robustness in Complex Networks

We introduce the concept of natural connectivity as a measure of structural robustness in complex networks. The natural connectivity characterizes the redundancy of alternative routes in a network by quantifying the weighted number of closed walks of all lengths. This definition leads to a simple mathematical formulation that links the natural connectivity to the spectrum of a network. The natural connectivity can be regarded as an average eigenvalue that changes strictly monotonically with the addition or deletion of edges. We calculate both analytically and numerically the natural connectivity of three typical networks: regular ring lattices, random graphs, and random scale-free networks. We also compare the proposed natural connectivity to other structural robustness measures within a scenario of edge elimination and demonstrate that the natural connectivity provides sensitive discrimination of structural robustness that agrees with our intuition.

Robustness of random graphs based on graph spectra.

It has been recently proposed that the robustness of complex networks can be efficiently characterized through the natural connectivity, a spectral property of the graph which corresponds to the average Estrada index. The natural connectivity corresponds to an average eigenvalue calculated from the graph spectrum and can also be interpreted as the Helmholtz free energy of the network. In this article, we explore the use of this index to characterize the robustness of Erdős-Rényi (ER) random graphs, random regular graphs, and regular ring lattices. We show both analytically and numerically that the natural connectivity of ER random graphs increases linearly with the average degree. It is also shown that ER random graphs are more robust than the corresponding random regular graphs with the same number of vertices and edges. However, the relative robustness of ER random graphs and regular ring lattices depends on the average degree and graph size: there is a critical graph size above which regular ring lattices are more robust than random graphs. We use our analytical results to derive this critical graph size as a function of the average degree.

Heterogeneity of Scale-free Networks

The heterogeneity of scale-free networks is studied using the network structure entropy (NSE). The NSE of scale-free networks is presented analytically by introducing the degree-rank function. It is shown that the normalized NSE of scale-free networks is only dependent on the scaling exponent and is independent of the size or the minimum degree of networks when scaling exponent is greater than 2. Given the size and the minimum degree of scale-free networks, it is shown that the NSE reached a minimum value when scaling exponent is about 1.7 and then the scale-free networks become more homogeneous as scaling exponent increases after the minimum value.

A new measure of heterogeneity of complex networks based on degree sequence

Many unique properties of complex networks are due to the heterogeneity. The measure and analysis of heterogeneity is important and desirable to research the behaviours and functions of complex networks. In this paper,entropy of degree sequence (EDS) as a new measure of the heterogeneity of complex networks is proposed and normalized entropy of degree sequence (NEDS) is defined. EDS is agreement with the normal meaning of heterogeneity within the context of complex networks compared with conventional measures. The heterogeneity of scale-free networks is studied using EDS. The analytical expression of EDS of scale-free networks is presented by introducing degree-rank function. It is demonstrated that scale-free networks become more heterogeneous as scaling exponent decreases. It is also demonstrated that NEDS of scale-free networks is independent of the size of networks which indicates that NEDS is a suitable and effective measure of heterogeneity.

Spectral measure of robustness for Internet topology

The natural connectivity as a novel robustness measure of complex networks is proposed. The natural connectivity has a clear physical meaning and a simple mathematical formulation. It is shown that the natural connectivity can be derived mathematically from the graph spectrum as an average eigenvalue and that it changes strictly monotonically with the addition or deletion of edges. By comparing the natural connectivity with other typical robustness measures within a scenario of edge elimination, it is demonstrated that the natural connectivity has an acute discrimination which agrees with our intuition. The robustness of global Internet AS-level topology and Chinese Internet AS-level topology is studied using natural connectivity.

ATTACK VULNERABILITY OF COMPLEX NETWORKS BASED ON LOCAL INFORMATION

We introduce a novel model for attack vulnerability of complex networks with a tunable attack information parameter. Based on the model, we study the attack vulnerability of complex networks based on local information. We employ the generating function formalism to derive the exact value of the critical removal fraction of nodes for the disintegration of networks and the size of the giant component based on local information. We show that hiding just a small fraction of nodes can prevent the breakdown of a network and that it is a cost-efficient strategy for enhancing the robustness of complex networks to hide the information of networks.

Relationship between degree-rank function and degree distribution of protein-protein interaction networks

It is argued that both the degree-rank function r=f(d), which describes the relationship between the degree d and the rank r of a degree sequence, and the degree distribution P(k), which describes the probability that a randomly chosen vertex has degree k, are important statistical properties to characterize protein-protein interaction (PPI) networks, both rank-degree plot and frequency-degree plot are reliable tools to analyze PPI networks. An exact mathematical relationship between degree-rank functions and degree distributions of PPI networks is derived. It is demonstrated that a power law degree distribution is equivalent to a power law degree-rank function only if scaling exponent is greater than 2. The puzzle that the degree distributions of some PPI networks follow a power law using frequency-degree plots, whereas the degree sequences do not follow a power law using rank-degree plots is explained using the mathematical relationship.