Evaluating links through spectral decomposition

Spectral decomposition has been rarely used to investigate complex networks. In this work we apply this concept in order to define two types of linkdirected attacks while quantifying their respective effects on the topology. Several other types of more traditional attacks are also adopted and compared. These attacks had substantially diverse effects, depending on each specific network (models and realworld structures). It is also showed that the spectral-based attacks have special effect in affecting the transitivity of the networks. PACS numbers: 89.75.-k, 89.75.Hc, 89.20.-a Evaluating links through spectral decomposition 2 The theory of graphs (e.g. [1]) and networks (e.g. [2, 3]) represents one of the most multidisciplinary, integrated and applicable areas of theoretical mathematics and computing. Although its origin is often traced back to Euler’s solution of the Königsberg bridge problem, graphs have been around for much longer, at least since the first map was draw on sand ‡. Because graphs and networks can represent most discrete structures possibly underlying dynamical systems [3, 4], they are particularly useful for modeling a vast range of problems. The identification of structured connections in growing graphs, especially the existence of hubs and their fundamental importance (e.g. [2]), helped to catalyse a surge of interest which had already been sparkled by random graphs and small world networks studies, giving rise to the new theory of complex networks. A good deal of the investigations in complex networks have focused on relatively simple properties such as the node degree (i.e. the number of connections established by a node), clustering coefficient (i.e. the degree of interconnectivity among the immediate neighbors of a node) and the shortest path length between two nodes. These measurements[5] are particularly important because they correspond to the distinguishing features of the main complex network models. For instance, small world networks are characterized by low mean shortest path length together with high clustering coefficient, and scale free networks exhibit power law degree distributions. However, as these measurements are not enough to provide a complete, invertible, representation of the complex network of interest, they will not be enough to directly express many important connectivity properties. There are so many possible measurements of complex networks that it is useful to organize them into categories (e.g. [5]). A particular interesting and useful category of measurements are those called spectral (e.g. [6, 7, 8]), in the sense of involving the eigenvalues of the adjacency matrices of the analyzed graphs. Spectral approaches to graphs and networks are particularly important because of many reasons including the relationship between eigenvalues and the dynamics of the network, connectedness, cuts, modularity and cycles, among others. Such concepts and methods have progressively attracted the attention from the complex networks community, to the extent that some of the best community finding algorithms in this area are now based on spectral methods (e.g. [9]). Spectral approaches in graphs, have many relationships with theoretical and applied physics, they may consider the adjacency (e.g. [6, 8]) or Laplacian matrices (e.g. [7]) of graphs. In this work we concentrate attention in the former type of approaches. More specifically, because the spectrum of a graph does not provide a complete representation, we focus our attention on the possibility to use the eigenspaces of graphs [8] in order to derived more powerful features for characterizing the graph connectivity. Complete representations are important in graph studies because they allow a one to one mapping between any graph (including its isomorphisms) into a feature space which can be used ‡ Maps are a special kind of graph called geographical, which is characterized by the fact that the edges have well-defined positions in an embedding space. Evaluating links through spectral decomposition 3 for unambiguous graph classification (e.g. [5]), avoiding degenerate mappings §. While any graph can be precise and completely represented in terms of its spectrum and eigenspaces, such formulation ultimately depends on the node labeling for the correct identification of the eigenspaces. Therefore, such a representation is not invariant to node label permutations and graph isomorphisms. While the existence of a complete and invariant representation of graphs does not seem to be likely (e.g. [8]), it is still interesting to consider additional features rather than just the graph spectrum. One of the most natural such a complementation can be achieved by considering also the eigenspaces of the graphs. We consider here the problem of quantifying the importance of links in networks using the spectral decomposition of the adjacency matrix. Based on link spectral measurements that are described below, a fraction of the links is removed and the effect of this removal on network topology is quantified for some specific model or real networks. For comparison’s sake, the same procedure is also applied using other, nonspectral, link measurements. There are many works dealing with vulnerability of model and real networks to attacks on nodes or links [10, 11, 12, 13, 14, 15, 16]. Those works do not use spectral measurements. Spectral techniques are often used to express centrality measures of nodes [17] and for community detection [9, 18, 19], but were also used for the network vulnerability problem [20, 21, 22]. None of these works used spectral decomposition. Spectral decomposition is used in Ref. [23], where the authors consider the problem of reconstructing a network after perturbation. This article is organized as follows. First, the basic concepts from complex network (e.g. [2, 3, 4, 5]) and eingenspace (e.g. [6, 8]) theories are presented in an introductory and self-contained way. Then the experimental methodology is explained regarding the generation of the synthetic complex network models and measurements used for the evaluation, which is followed by the presentation and discussion of the results.