166 research outputs found
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.Comment: 26 pages, 11 figures, 4 table
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
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
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
Predicting triadic closure in networks using communicability distance functions
We propose a communication-driven mechanism for predicting triadic closure in
complex networks. It is mathematically formulated on the basis of
communicability distance functions that account for the quality of
communication between nodes in the network. We study real-world networks
and show that the proposed method predicts correctly of triadic closures
in these networks, in contrast to the predicted by a random mechanism.
We also show that the communication-driven method outperforms the random
mechanism in explaining the clustering coefficient, average path length, and
average communicability. The new method also displays some interesting features
with regards to optimizing communication in networks
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
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
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