Almerima Jamakovic

On the relationship between the algebraic connectivity and graph's robustness to node and link failures

We study the algebraic connectivity in relation to the graphs robustness to node and link failures. Graphs robustness is quantified with the node and the link connectivity, two topological metrics that give the number of nodes and links that have to be removed in order to disconnect a graph. The algebraic connectivity, i.e. the second smallest eigenvalue of the Laplacian matrix, is a spectral property of a graph, which is an important parameter in the analysis of various robustness-related problems. In this paper we study the relationship between the proposed metrics in three well-known complex network models: the random graph of Erdos-Renyi, the small-world graph of Watts-Strogatz and the scale-free graph of Barabasi-Albert. From (Fielder, 1973) it is known that the algebraic connectivity is a lower bound on both the node and the link connectivity. Through extensive simulations with the three complex network models, we show that the algebraic connectivity is not trivially connected to graphs robustness to node and link failures. Furthermore, we show that the tightness of this lower bound is very dependent on the considered complex network model.

Weighted spectral distribution for internet topology analysis: theory and applications

Comparing graphs to determine the level of underlying structural similarity between them is a widely encountered problem in computer science. It is particularly relevant to the study of Internet topologies, such as the generation of synthetic topologies to represent the Internets AS topology. We derive a new metric that enables exactly such a structural comparison: the weighted spectral distribution. We then apply this metric to three aspects of the study of the Internets AS topology. i) We use it to quantify the effect of changing the mixing properties of a simple synthetic network generator. ii) We use this quantitative understanding to examine the evolution of the Internets AS topology over approximately seven years, finding that the distinction between the Internet core and periphery has blurred over time. iii) We use the metric to derive optimal parameterizations of several widely used AS topology generators with respect to a large-scale measurement of the real AS topology.

On the relationships between topological measures in real-world networks

Over the past several years, a number of measures have been introduced to characterize the topology of complex networks. We perform a statistical analysis of real data sets, representing the topology of different real-world networks. First, we show that some measures are either fully related to other topological measures or that they are significantly limited in the range of their possible values. Second, we observe that subsets of measures are highly correlated, indicating redundancy among them. Our study thus suggests that the set of commonly used measures is too extensive to concisely characterize the topology of complex networks. It also provides an important basis for classification and unification of a definite set of measures that would serve in future topological studies of complex networks.

Influence of the network structure on robustness

The classical connectivity is typically used to capture the robustness of networks. Robustness, however, encompasses more than this simple definition of being connected. A spectral metric, referred to as the algebraic connectivity, plays a special role for the robustness since it measures the extent to which it is difficult to cut the network into independent components. We rely on the algebraic connectivity to study the robustness to random node and link failures in three important network models: the random graph of Erdos-Renyi, the small-world graph of Watts and Strogatz and the scale-free graph of Barabasi-Albert. We show that the robustness to random node and link failures significantly differs between the three models. This points to explicit influence of the network structure on the robustness. The homogeneous structure of the random graph of Erdos-Renyi implies an invariant robustness under random node failures. The heterogeneous structure of the small-world graph of Watts and Strogatz and scale-free graph of Barabasi-Albert, on the other hand, implies a non-trivial robustness to random node and link failures.

Tuning Topology Generators Using Spectral Distributions

An increasing number of synthetic topology generators are available, each claiming to produce representative Internet topologies. Every generator has its own parameters, allowing the user to generate topologies with different characteristics. However, there exist no clear guidelines on tuning the value of these parameters in order to obtain a topology with specific characteristics. In this paper we optimize the parameters of several topology generators to match a given Internet topology. The optimization is performed either with respect to the link density, or to the spectrum of the normalized Laplacian matrix. Contrary to approaches in the literature that rely only on the largest eigenvalues, we take into account the set of all eigenvalues. However, we show that on their own the eigenvalues cannot be used to construct a metric for optimizing parameters. Instead we present a weighted spectral method which simultaneously takes into account all the properties of the graph.

Mixing biases: structural changes in the AS topology evolution

In this paper we study the structural evolution of the AS topology as inferred from two different datasets over a period of seven years. We use a variety of topological metrics to analyze the structural differences revealed in the AS topologies inferred from the two different datasets. In particular, to focus on the evolution of the relationship between the core and the periphery, we make use of a recently introduced topological metric, the weighted spectral distribution. We find that the traceroute dataset has increasing difficulty in sampling the periphery of the AS topology, largely due to limitations inherent to active probing. Such a dataset has too limited a view to properly observe topological changes at the AS-level compared to a dataset largely based on BGP data. We also highlight limitations in current measurements that require a better sampling of particular topological properties of the Internet. Our results indicate that the Internet is changing from a core-centered, strongly customer-provider oriented, disassortative network, to a soft-hierarchical, peering-oriented, assortative network.

A weighted spectrum metric for comparison of internet topologies

Comparison of graph structures is a frequently encountered problem across a number of problem domains. Comparing graphs requires a metric to discriminate which features of the graphs are considered important. The spectrum of a graph is often claimed to contain all the information within a graph, but the raw spectrum contains too much information to be directly used as a useful metric. In this paper we introduce a metric, the weighted spectral distribution, that improves on the raw spectrum by discounting those eigenvalues believed to be unimportant and emphasizing the contribution of those believed to be important. We use this metric to optimize the selection of parameter values for generating Internet topologies. Our metric leads to parameter choices that appear sensible given prior knowledge of the problem domain: the resulting choices are close to the default values of the topology generators and, in the case of some generators, fall within the expected region. This metric provides a means for meaningfully optimizing parameter selection when generating topologies intended to share structure with, but not match exactly, measured graphs.

Capturing Internet Traffic Dynamics through Graph Distances

Studies of the Internet have typically focused either on the routing system, i.e. the paths chosen to reach a given destination, or on the evolution of traffic on a physical link. In this paper, we combine routing and traffic, and study for the first time the evolution of the traffic on the Internet topology. We rely on the traffic and routing data of a large transit provider, spanning almost a month.