| Complex systems are ubiquitous throughout the world, both in nature and within man-made structures. Over the past decade, large amounts of network data have become available and, correspondingly, the analysis of complex networks has become increasingly important. One of the fundamental questions in this analysis is to determine the most important elements in a given network. Measures of node importance are usually referred to as node centrality, and many centrality measures have been proposed over the years. Here, we focus on the analysis and computation of centrality measures based on matrix functions.;First, we examine a node centrality measure based on the notion of total communicability, defined in terms of the row sums of the exponential of the adjacency matrix of the network. We argue that this is a natural metric for ranking nodes in a network, and we point out that it can be computed very rapidly even in the case of large networks. Furthermore, we propose the total sum of node communicabilities as a useful measure of network connectivity.;Next, we compare various parameterized centrality rankings based on the matrix expo- nential and matrix resolvent with degree and eigenvector centrality. We demonstrate an analytical relationship between these rankings which helps to explain explain the observed robustness of these rankings on many real world networks, even though the scores produced by the centrality measures are not stable.;Finally, we propose an extension of these measures to directed networks, and we apply them to the problem of ranking hubs and authorities. The extension is achieved by bipartization, i.e., the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix. We explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original, directed network, and we give an interpretation of centrality and communicability in this new context, leading to a technique for ranking hubs and authorities. |
