# A Spectral Density Approach in Research of Internet Topology Properties.

**ABSTRACT** Spectral density approach for distinguishing graphs was studied in this paper. Firstly, spectral density approach was testified for being effective in distinguishing different graphs by making comparisons among the spectrums of three different kind of graphs, the ER random graph, BA scale-free graph and the Internet topology graph. Secondly, we focused our studies on the properties of Internet graph that its spectrum could represent, and found that in standard spectral density analysis part, we found that the spectral density plot of Internet graph has a feature of having a maximum when ¿=0 and the second maximum when ¿=0.5 around. In SLS analysis part, we found the SLS spectrum had a set of highest tuples when SLS=1 and second highest tuples when SLS>2. Besides, a relationship of the power law distribution was observed when SLS>2, but there is no power-law relationship found when SLS. What was found here could be used to identify an Internet topology graph properties.

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**ABSTRACT:**We study the spectra and eigenvectors of the adjacency matrices of scale-free networks when bidirectional interaction is allowed, so that the adjacency matrix is real and symmetric. The spectral density shows an exponential decay around the center, followed by power-law long tails at both spectrum edges. The largest eigenvalue lambda1 depends on system size N as lambda1 approximately N1/4 for large N, and the corresponding eigenfunction is strongly localized at the hub, the vertex with largest degree. The component of the normalized eigenfunction at the hub is of order unity. We also find that the mass gap scales as N(-0.68).Physical Review E 12/2001; 64(5 Pt 1):051903. DOI:10.1103/PhysRevE.64.051903 · 2.33 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We perform spectral analysis of the Internet topology at the AS level, by adapting the standard spectral filtering method of examining the eigenvectors corresponding to the largest eigenvalues of matrices related to the adjacency matrix of the topology. We observe that the method suggests clusters of ASes with natural semantic proximity, such as geography or business interests. We examine how these clustering properties vary in the core and in the edge of the network, as well as across geographic areas, over time, and between real and synthetic data. We observe that these clustering properties may be suggestive of traffic patterns and thus have direct impact on the link stress of the network. Finally, we use the weights of the eigenvector corresponding to the first eigenvalue to obtain an alternative hierarchical ranking of the ASes.Proceedings - IEEE INFOCOM 03/2003; DOI:10.1109/INFCOM.2003.1208688 -
##### Article: Scale-free networks

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**ABSTRACT:**The study of network topologies provides interesting insights into the way in which the principles on which interconnected systems are constructed influence the dynamics of diffusion and communication processes in many kinds of socio-technical systems. Empirical research has shown that there are principles of construction similar to those of the laws of nature for social networks and their technical derivatives, like E-mail networks, the internet, publication co-authoring, or business collaboration. For decades, the paradigm of a randomly connected network has been used as a model for real world networks, in ignorance of the fact that they are only a poor fit for such networks. Apparently, all the above-mentioned networks share the same building blocks. They attach new members over time and the attachment prefers existing members that are already well connected. This principle of “preferential attachment” leads to interesting properties that have to be taken into consideration when analyzing and designing systems with some kind of network background. What are called “scale-free” networks seems to be a better fit for the description of real world networks. They use preferential attachment as a construction principle to resample real world networks. Their behavior in terms of diffusion and communication processes is fundamentally different from that of random networks. To illustrate the potential value of the discovery of scale-free networks for applications in information systems related research, an example will be used in this article to illustrate their usefulness for realistic network modeling. A scale-free communication network of security traders will show what impact network topology has on the dynamics of complex socio-technical systems.Business and Information Systems Engineering the international journal of Wirtschaftsinformatik 08/2006; 48(4):267-275. DOI:10.1007/s11576-006-0058-2 · 0.54 Impact Factor