A framework for second-order eigenvector centralities and clustering coefficients

We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron–Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks

Complex networks and public funding: the case of the 2007-2013 Italian program

In this paper we apply techniques of complex network analysis to data sources representing public funding programs and discuss the importance of the considered indicators for program evaluation. Starting from the Open Data repository of the 2007-2013 Italian Program Programma Operativo Nazionale 'Ricerca e Competitivit\`a' (PON R&C), we build a set of data models and perform network analysis over them. We discuss the obtained experimental results outlining interesting new perspectives that emerge from the application of the proposed methods to the socio-economical evaluation of funded programs.Comment: 22 pages, 9 figure

Mathematical Formulation of Multi-Layer Networks

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich. Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multi-layer complex systems. In this paper, we introduce a tensorial framework to study multi-layer networks, and we discuss the generalization of several important network descriptors and dynamical processes --including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion-- for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multi-layer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.Comment: 15 pages, 5 figure

On the limiting behavior of parameter-dependent network centrality measures

We consider a broad class of walk-based, parameterized node centrality measures for network analysis. These measures are expressed in terms of functions of the adjacency matrix and generalize various well-known centrality indices, including Katz and subgraph centrality. We show that the parameter can be "tuned" to interpolate between degree and eigenvector centrality, which appear as limiting cases. Our analysis helps explain certain correlations often observed between the rankings obtained using different centrality measures, and provides some guidance for the tuning of parameters. We also highlight the roles played by the spectral gap of the adjacency matrix and by the number of triangles in the network. Our analysis covers both undirected and directed networks, including weighted ones. A brief discussion of PageRank is also given.Comment: First 22 pages are the paper, pages 22-38 are the supplementary material

Brain structural covariance networks in obsessive-compulsive disorder: a graph analysis from the ENIGMA Consortium.

Brain structural covariance networks reflect covariation in morphology of different brain areas and are thought to reflect common trajectories in brain development and maturation. Large-scale investigation of structural covariance networks in obsessive-compulsive disorder (OCD) may provide clues to the pathophysiology of this neurodevelopmental disorder. Using T1-weighted MRI scans acquired from 1616 individuals with OCD and 1463 healthy controls across 37 datasets participating in the ENIGMA-OCD Working Group, we calculated intra-individual brain structural covariance networks (using the bilaterally-averaged values of 33 cortical surface areas, 33 cortical thickness values, and six subcortical volumes), in which edge weights were proportional to the similarity between two brain morphological features in terms of deviation from healthy controls (i.e. z-score transformed). Global networks were characterized using measures of network segregation (clustering and modularity), network integration (global efficiency), and their balance (small-worldness), and their community membership was assessed. Hub profiling of regional networks was undertaken using measures of betweenness, closeness, and eigenvector centrality. Individually calculated network measures were integrated across the 37 datasets using a meta-analytical approach. These network measures were summated across the network density range of K = 0.10-0.25 per participant, and were integrated across the 37 datasets using a meta-analytical approach. Compared with healthy controls, at a global level, the structural covariance networks of OCD showed lower clustering (P < 0.0001), lower modularity (P < 0.0001), and lower small-worldness (P = 0.017). Detection of community membership emphasized lower network segregation in OCD compared to healthy controls. At the regional level, there were lower (rank-transformed) centrality values in OCD for volume of caudate nucleus and thalamus, and surface area of paracentral cortex, indicative of altered distribution of brain hubs. Centrality of cingulate and orbito-frontal as well as other brain areas was associated with OCD illness duration, suggesting greater involvement of these brain areas with illness chronicity. In summary, the findings of this study, the largest brain structural covariance study of OCD to date, point to a less segregated organization of structural covariance networks in OCD, and reorganization of brain hubs. The segregation findings suggest a possible signature of altered brain morphometry in OCD, while the hub findings point to OCD-related alterations in trajectories of brain development and maturation, particularly in cingulate and orbitofrontal regions