k-dense Communities in the Internet AS-Level Topology

Extracting a set of well connected subgraphs as com- munities from the Internet AS-level topology graph is crucially important for assessing the performance of protocols and routing algorithms, for designing ecient networks, and for evaluating the impact of failures. A huge number of community extraction methods have been proposed in the literature, among which the k-core decomposition and the k-clique community extraction methods. The former method is computationally e- cient, but it only discovers coarse-grained and loosely connected communities. On the other hand, k-clique can extract ne-grained and tightly connected communities, but is NP hard and therefore useless for analyzing the Internet AS-level topology graph. In the paper we inves- tigate the Internet structure by exploiting an ecient algorithm for extracting k-dense communities, where a k-clique community implies a k-dense community, which in turn implies a k-core community. The paper provides two innovative contributions. The rst is the application of the k-dense method to the Internet AS-level topology graph - obtained from the CAIDA, DIMES and IRL datasets - to identify well- connected communities and to analyze how these are connected to the rest of the graph. The second contribution relates to the study of the most well-connected communities with the support of two additional datasets: a geographical dataset (which lists, for each AS, the countries in which it has at least one geographical location) and the IXP dataset (which maintains, for each IXP, its geographical position and the list of its participants). We found that the k-max- dense community holds a central position in the Internet AS-level topology graph structure since its 101 ASs (less than the 0.3% of Internet ASs) are involved in more than 39% of all Internet connections. We also found that those ASs are connected to at least one IXP and have at least one geographical location in Europe (only 70.3% of them have at least one additional geographical location outside Europe)

k-Dense communities in the Internet AS-level topology graph

In this paper we investigate the structure of the Internet by exploiting an efficient algorithm for extracting k-dense communities from the Internet AS-level topology graph. The analyses showed that the most well-connected communities consist of a small number of ASs characterized by a high level of clusterization, although they tend to direct a lot of their connections to ASs outside the community. In addition these communities are mainly composed of ASs that participate at the Internet Exchange Points (IXPs) and have a worldwide geographical scope. Regarding k-max-dense ASs we found that they play a primary role in the Internet connectivity since they are involved in a huge number of Internet connections (42% of Internet connections). We also investigated the properties of three classes of k-max-dense ASs: Content Delivery Networks, Internet Backbone Providers and Tier-1s. Specifically, we showed that CDNs and IBPs heavily exploit IXPs by participating in many of them and connecting to many IXP participant ASs. On the other hand, we found that a high percentage of connections originated by Tier-1 ASs are likely to involve national ASs which do not participate at IXPs

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

Parallel (k)(k)-Clique Community Detection on Large-Scale Networks

The analysis of real-world complex networks has been the focus of recent research. Detecting communities helps in uncovering their structural and functional organization. Valuable insight can be obtained by analyzing the dense, overlapping, and highly interwoven k-clique communities. However, their detection is challenging due to extensive memory requirements and execution time. In this paper, we present a novel, parallel k-clique community detection method, based on an innovative technique which enables connected components of a network to be obtained from those of its subnetworks. The novel method has an unbounded, user-configurable, and input-independent maximum degree of parallelism, and hence is able to make full use of computational resources. Theoretical tight upper bounds on its worst case time and space complexities are given as well. Experiments on real-world networks such as the Internet and the World Wide Web confirmed the almost optimal use of parallelism (i.e., a linear speedup). Comparisons with other state-of-the-art k-clique community detection methods show dramatic reductions in execution time and memory footprint. An open-source implementation of the method is also made publicly available

A Structural Analysis of the Internet AS-level topology

The study of the structural characteristics of the Internet topology at the Autonomous System (AS) level of abstraction is an important and interesting subject that has attracted significant interest over the last few years. Above all, a deep knowledge of the Internet underlying structure helps researchers in designing a more accurate model of the network; as a result, engineers can design applications and protocols that can take into account the underlying structure and test their projects on synthetic graphs, thereby developing more efficient algorithms. A significant challenge for researchers analyzing the Internet is how to interpret the global organization of the graph as the coexistence of its structural blocks associated with more highly interconnected parts, namely communities. While a huge number of papers have already been published on the issue of community detection, very little attention has so far been devoted to the discovery and interpretation of Internet communities. The contribution of this work is twofold. First, we study the evolution of the Internet AS-level topology over the last 9 years by means of two innovative approaches: the k-dense method and the dK-analysis. Second, we focus on substructures that play a key role in the Internet connectivity, and we investigate the classes of the ASes and the nature of the connections that create such communities. We find that as the Internet grows over time, some of its structural properties remain unchanged. Although the size of the network, as well as the kMAX -dense index (an index of the maximum level of density reached in a network), has doubled over the last 9 years, we show that after proper normalizations the k-dense decomposition has remained stable. Besides, we provided a clear evidence that the formation of denser and denser sub-graphs over time has been triggered by the proliferation of Internet eXchange Points (IXP) and public peering connections. We found that ASes within most densely-connected substructures are usually Network Service Providers, Content Providers, or Content Delivery Networks; in addition, all of them participate to at least one IXP