Hierarchy measure for complex networks

Nature, technology and society are full of complexity arising from the intricate web of the interactions among the units of the related systems (e.g., proteins, computers, people). Consequently, one of the most successful recent approaches to capturing the fundamental features of the structure and dynamics of complex systems has been the investigation of the networks associated with the above units (nodes) together with their relations (edges). Most complex systems have an inherently hierarchical organization and, correspondingly, the networks behind them also exhibit hierarchical features. Indeed, several papers have been devoted to describing this essential aspect of networks, however, without resulting in a widely accepted, converging concept concerning the quantitative characterization of the level of their hierarchy. Here we develop an approach and propose a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and, as we demonstrate, is capable of capturing the essential features of the structure and the degree of hierarchy in a complex network. The measure we introduce is based on a generalization of the m-reach centrality, which we first extend to directed/partially directed graphs. Then, we define the global reaching centrality (GRC), which is the difference between the maximum and the average value of the generalized reach centralities over the network. We investigate the behavior of the GRC considering both a synthetic model with an adjustable level of hierarchy and real networks. Results for real networks show that our hierarchy measure is related to the controllability of the given system. We also propose a visualization procedure for large complex networks that can be used to obtain an overall qualitative picture about the nature of their hierarchical structure.Comment: 29 pages, 9 figures, 4 table

Link prediction in very large directed graphs: Exploiting hierarchical properties in parallel

Link prediction is a link mining task that tries to find new edges within a given graph. Among the targets of link prediction there is large directed graphs, which are frequent structures nowadays. The typical sparsity of large graphs demands of high precision predictions in order to obtain usable results. However, the size of those graphs only permits the execution of scalable algorithms. As a trade-off between those two problems we recently proposed a link prediction algorithm for directed graphs that exploits hierarchical properties. The algorithm can be classified as a local score, which entails scalability. Unlike the rest of local scores, our proposal assumes the existence of an underlying model for the data which allows it to produce predictions with a higher precision. We test the validity of its hierarchical assumptions on two clearly hierarchical data sets, one of them based on RDF. Then we test it on a non-hierarchical data set based on Wikipedia to demonstrate its broad applicability. Given the computational complexity of link prediction in very large graphs we also introduce some general recommendations useful to make of link prediction an efficiently parallelized problem.Peer ReviewedPostprint (published version