Distances in random graphs with finite variance degrees

In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with \prob(D_j\leq x)=F(x). We assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_{\nu}N$, when the base of the logarithm equals \nu=\expec[D_j(D_j -1)]/\expec[D_j]>1. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.Comment: 40 pages, 2 figure

Router-level community structure of the Internet Autonomous Systems

The Internet is composed of routing devices connected between them and organized into independent administrative entities: the Autonomous Systems. The existence of different types of Autonomous Systems (like large connectivity providers, Internet Service Providers or universities) together with geographical and economical constraints, turns the Internet into a complex modular and hierarchical network. This organization is reflected in many properties of the Internet topology, like its high degree of clustering and its robustness. In this work, we study the modular structure of the Internet router-level graph in order to assess to what extent the Autonomous Systems satisfy some of the known notions of community structure. We show that the modular structure of the Internet is much richer than what can be captured by the current community detection methods, which are severely affected by resolution limits and by the heterogeneity of the Autonomous Systems. Here we overcome this issue by using a multiresolution detection algorithm combined with a small sample of nodes. We also discuss recent work on community structure in the light of our results

On Characterizing Network Hierarchy

Our previous work in topology characterization and hierarchy [1] introduced a hierarchy metric to explore the hierarchical structure in various networks. This metric is non-intuitive and complicated. In this paper, we propose a simpler and more natural metric for measuring network hierarchy. This simpler metric uses slightly different criteria in selecting backbone links than the more complicated one. Nevertheless, the network classifications according to both metrics agree with each other. Furthermore, we have extended the hierarchy analysis to examine path characteristics and found that the hierarchical nature of degree-based networks better resembles the hierarchy of the Internet at the AS level than at the routerlevel

Performance Analysis, Data Sharing and Tools Integration in Grids: New Approach based on Ontology

In this paper, we propose a new approach to performance analysis, data sharing and tools integration in Grids that is based on ontology. We devise a novel ontology for describing the semantics of monitoring and performance data that can be used by performance monitoring and measurement tools. We introduce an architecture for an ontology-based model for performance analysis, data sharing and tools integration. At the core of this architecture is a Grid service which offers facilities for other services to archive and access ontology models along with collected performance data, and to conduct searches and perform reasoning on that data. Using an approach based on ontology, performance data will be easily shared and processed by automated tools, services and human users, thus helping to leverage the data sharing and tools integration, and increasing the degree of automation of performance analysis