Communicability betweenness in complex networks

Abstract

Betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks. They allow us to study the extent to which a node takes part when information is passed around the network. Nodes with high betweenness may be regarded as key players that have a highly active role. At one extreme, betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes. At the other extreme, an alternative type of betweenness has been defined by considering all possible walks of any length. In this work, we propose a betweenness measure that lies between these two opposing viewpoints. We allow information to pass through all possible routes, but introduce a scaling so that longer walks carry less importance. This new definition shares a similar philosophy to that of communicability for pairs of nodes in a network, which was introduced by Estrada and Hatano (Phys. Rev. E 77 (2008) 036111). Having defined this new communicability betweenness measure, we show that it can be characterized neatly in terms of the exponential of the adjacency matrix. We also show that this measure is closely related to a Fréchet derivative of the matrix exponential. This allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations. Using illustrative synthetic and real life networks, we show that the new betweenness measure behaves differently to existing versions, and in particular we show that it recovers meaningful biological information from a protein-protein interaction network.3

Communicability across evolving networks

“…The walk viewpoint is also in line with the influential work of Katz [17] for the study of static, undirected social networks. The explicit use of a walk based measure of centrality was proposed for the static case in [9], and the idea has been shown to lead to very powerful measures that are useful across a range of application areas [6,10,11]. A further benefit of the walk counting approach is that the combinatorics can be conveniently described and implemented in terms of operations in linear algebra, and we will show that this feature can be carried through to the dynamic case.…”

A Block Krylov Method to Compute the Action of the Fréchet Derivative of a Matrix Function on a Vector with Applications to Condition Number Estimation

“…Given a matrix function f : C n×n → C n×n , one commonly needs to compute f (A)b for a large, sparse matrix A and a vector b ∈ C n . For example, when f (A) = e A is the matrix exponential, this computation arises when solving evolution equations with exponential integrators (e.g., [12], [13], [15]) or in network analysis (e.g., [6], [7]). In particular, [7] also uses the Fréchet derivative of the matrix exponential, which we introduce shortly.…”

Milestones in graphical bioinformatics

“…Those interested in complex networks, including protein interaction networks, and their analysis should consult several papers of Estrada and colleagues as a good introduction in this topic. [125,[153][154][155][156][157][158][159][160][161][162][163][164][165][166][167] Table 25 shows that research in China in graphical bioinformatics has deep roots, and it appears that China will soon, if not already, be the leading country in the development of graphical bioinformatics. In China, the dominant groups of researchers come from mathematical institutions, and they are interested in discrete mathematics and graph theory.…”