Analysis of directed networks via the matrix exponential

Abstract The matrix exponential has been identified as a useful tool for the analysis of undirected networks, with sound theoretical justifications for its ability to model important aspects of a given network. Its use for directed networks, however, is less developed and has been less successful so far. In this article we discuss some methods to identify important nodes in a directed network using the matrix exponential, taking into account that the notion of importance changes whether we consider the influence of a given node along the edge directions (downstream influence) or how it is influenced by directed paths that point to it (upstream influence). In addition, we introduce a family of importance measures based on counting walks that are allowed to reverse their direction a limited number of times, thus capturing relationships arising from influencing the same nodes, or being influenced by the same nodes, without sacrificing information about edge direction. These measures provide information about branch points.

[1]  Ayman Farahat,et al.  Authority Rankings from HITS, PageRank, and SALSA: Existence, Uniqueness, and Effect of Initialization , 2005, SIAM J. Sci. Comput..

[2]  J. A. Rodríguez-Velázquez,et al.  Subgraph centrality in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Lothar Reichel,et al.  Error Estimates and Evaluation of Matrix Functions via the Faber Transform , 2009, SIAM J. Numer. Anal..

[4]  Desmond J. Higham,et al.  Network Properties Revealed through Matrix Functions , 2010, SIAM Rev..

[5]  David F. Gleich,et al.  PageRank beyond the Web , 2014, SIAM Rev..

[6]  Lothar Reichel,et al.  Analysis of directed networks via partial singular value decomposition and Gauss quadrature , 2014 .

[7]  Michele Benzi,et al.  Total communicability as a centrality measure , 2013, J. Complex Networks.

[8]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[9]  Mike Tyers,et al.  BioGRID: a general repository for interaction datasets , 2005, Nucleic Acids Res..

[10]  Michele Benzi,et al.  The Physics of Communicability in Complex Networks , 2011, ArXiv.

[11]  Michele Benzi,et al.  MATRIX FUNCTIONS , 2006 .

[12]  Reinhard Schneider,et al.  Using graph theory to analyze biological networks , 2011, BioData Mining.

[13]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[14]  Michele Benzi,et al.  On the Limiting Behavior of Parameter-Dependent Network Centrality Measures , 2013, SIAM J. Matrix Anal. Appl..

[15]  Ernesto Estrada,et al.  The Structure of Complex Networks: Theory and Applications , 2011 .

[16]  Leo Katz,et al.  A new status index derived from sociometric analysis , 1953 .

[17]  M. Benzi,et al.  Quadrature rule-based bounds for functions of adjacency matrices , 2010 .

[18]  B. Hall.  Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , 2004 .

[19]  Tsippi Iny Stein,et al.  The GeneCards Suite: From Gene Data Mining to Disease Genome Sequence Analyses , 2016, Current protocols in bioinformatics.

[20]  Desmond J. Higham,et al.  Googling the Brain: Discovering Hierarchical and Asymmetric Network Structures, with Applications in Neuroscience , 2011, Internet Math..

[21]  L. Knizhnerman.  Calculation of functions of unsymmetric matrices using Arnoldi's method , 1991 .