Computation of generalized matrix functions

We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants are also investigated. Numerical experiments are performed to illustrate the effectiveness and efficiency of our techniques in computing generalized matrix functions arising in the analysis of networks.Comment: 25 paged, 2 figure

Updating and downdating techniques for optimizing network communicability

The total communicability of a network (or graph) is defined as the sum of the entries in the exponential of the adjacency matrix of the network, possibly normalized by the number of nodes. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. The total communicability can be computed quickly even for large networks using techniques based on the Lanczos algorithm. In this work we introduce some heuristics that can be used to add, delete, or rewire a limited number of edges in a given sparse network so that the modified network has a large total communicability. To this end, we introduce new edge centrality measures which can be used to guide in the selection of edges to be added or removed. Moreover, we show experimentally that the total communicability provides an effective and easily computable measure of how "well-connected" a sparse network is.Comment: 20 pages, 9 pages Supplementary Materia

Non-backtracking alternating walks

The combinatorics of walks on a graph is a key topic in network science. Here we study a special class of walks on directed graphs. We combine two features that have previously been considered in isolation. We consider alternating walks, which form the basis of algorithms for hub/authority detection and for discovering directed bipartite substructure. Within this class, we restrict to non-backtracking walks, since this constraint has been seen to offer advantages in related contexts. We derive a recursive formula for counting the total number of non-backtracking alternating walks of a given length, leading to an expression for any associated power series expansion. We discuss computational issues for the widely used cases of resolvent and exponential series, showing that non-backtracking can be incorporated at very little extra cost. We also derive an appropriate asymptotic limit which gives a parameter-free, spectral analogue. We perform tests on an artificial data set in order to quantify the advantages of the new methodology. We also show that the removal of backtracking allows us to identify larger bipartite subgraphs within an anatomical connectivity network from neuroscience

Edge manipulation techniques for complex networks with applications to communicability and triadic closure.

Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks. Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges. Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process. Extensive numerical tests are presented to validate our approaches

Edge manipulation techniques for complex networks with applications to communicability and triadic closure.

Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks. Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges. Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process. Extensive numerical tests are presented to validate our approaches

Edge modification criteria for enhancing the communicability of digraphs

We introduce new broadcast and receive communicability indices that can be used as global measures of how effectively information is spread in a directed network. Furthermore, we describe fast and effective criteria for the selection of edges to be added to (or deleted from) a given directed network so as to enhance these network communicability measures. Numerical experiments illustrate the effectiveness of the proposed techniques

Tapping Environmental History to Recreate America’s Colonial Hydrology

To properly remediate, improve, or predict how hydrological systems behave, it is vital to establish their histories. However, modern-style records, assembled from instrumental data and remote sensing platforms, hardly exist back more than a few decades. As centuries of data is preferable given multidecadal fluxes of both meteorology/climatology and demographics, building such a history requires resources traditionally considered only useful in the social sciences and humanities. In this Feature, Pastore et al. discuss how they have undertaken the synthesis of historical records and modern techniques to understand the hydrology of the Northeastern U.S. from Colonial times to modern day. Such approaches could aid studies in other regions that may require heavier reliance on qualitative narratives. Further, a better insight as to how historical changes unfolded could provide a “past is prologue” methodology to increase the accuracy of predictive environmental models