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Michele Benzi

Publications and source records attributed to Michele Benzi.

At least 37 records · Page 2Linked to original sources
arXiv2021

Refined decay bounds on the entries of spectral projectors associated with sparse Hermitian matrices

Spectral projectors of Hermitian matrices play a key role in many applications, and especially in electronic structure computations. Linear scaling methods for gapped systems are based on the fact that these special matrix functions are localized, which means that the entries decay exponentially away from the main diagonal or with respect to more general sparsity patterns. The relation with the sign function together with an integral representation is used to obtain new decay bounds, which turn out to be optimal in an asymptotic sense. The influence of isolated eigenvalues in the spectrum on the decay properties is also investigated and a superexponential behaviour is predicted.

math.NA↗
arXiv2021

Fully Atomistic Modeling of Realistic Plasmonic Materials: Assessing the Performance of Iterative Solvers

The fully atomistic modeling of real-size plasmonic nanostructures is computationally demanding, therefore most calculations are limited to small-to-medium sized systems. However, plasmonic properties strongly depend on the actual shape and size of the samples. In this paper we substantially extend the applicability of classical, fully atomistic approaches by exploiting state-of-the-art numerical iterative Krylov-based techniques. In particular, we focus on the recently developed $ω$FQ model, when specified to carbon nanotubes, graphene-based nanostructures and metal nanoparticles. The performance of Generalized Minimal Residual (GMRES) and Quasi-Minimum Residual (QMR) algorithms is studied, with special emphasis on the dependence of the convergence rate on the dimension of the structures (up to 1 million atoms) and the physical parameters entering the definition of the atomistic approach.

physics.comp-ph↗
arXiv2020

Rational Krylov methods for fractional diffusion problems on graphs

In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a non-analytic function involving fractional powers and $\boldsymbol{b}$ is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

math.NA↗
arXiv2020

Fast Iterative Solution of the Optimal Transport Problem on Graphs

In this paper, we address the numerical solution of the Optimal Transport Problem on undirected weighted graphs, taking the shortest path distance as transport cost. The optimal solution is obtained from the long-time limit of the gradient descent dynamics. Among different time stepping procedures for the discretization of this dynamics, a backward Euler time stepping scheme combined with the inexact Newton-Raphson method results in a robust and accurate approach for the solution of the Optimal Transport Problem on graphs. It is found experimentally that the algorithm requires solving between $\mathcal{O}(1)$ and $\mathcal{O}(M^{0.36})$ linear systems involving weighted Laplacian matrices, where $M$ is the number of edges. These linear systems are solved via algebraic multigrid methods, resulting in an efficient solver for the Optimal Transport Problem on graphs.

math.NA↗
arXiv2020

Nonlocal network dynamics via fractional graph Laplacians

We introduce nonlocal dynamics on directed networks through the construction of a fractional version of a nonsymmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.

cs.SI↗
arXiv2020

Risk-dependent centrality in economic and financial networks

Node centrality is one of the most important and widely used concepts in the study of complex networks. Here, we extend the paradigm of node centrality in financial and economic networks to consider the changes of node "importance" produced not only by the variation of the topology of the system but also as a consequence of the external levels of risk to which the network as a whole is submitted. Starting from the "Susceptible-Infected" (SI) model of epidemics and its relation to the communicability functions of networks we develop a series of risk-dependent centralities for nodes in (financial and economic) networks. We analyze here some of the most important mathematical properties of these risk-dependent centrality measures. In particular, we study the newly observed phenomenon of ranking interlacement, by means of which two entities may interlace their ranking positions in terms of risk in the network as a consequence of the change in the external conditions only, i.e., without any change in the topology. We test the risk-dependent centralities by studying two real-world systems: the network generated by collecting assets of the S\&P 100 and the corporate board network of the US top companies, according to Forbes in 1999. We found that a high position in the ranking of the analyzed financial companies according to their risk-dependent centrality corresponds to companies more sensitive to the external market variations during the periods of crisis.

q-fin.MF↗
arXiv2017

What is the meaning of the graph energy after all?

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $λ_{1}\geqλ_{2}\geq\cdots\geqλ_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|λ_{j}|$. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such interpretation and prove that the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the traces of even powers of the adjacency matrix. We then use such result to find bounds for the energy in terms of subgraphs contributing to it. The new bounds are studied for some specific simple graphs, such as cycles and fullerenes. We observe that including contributions from subgraphs of sizes not bigger than 6 improves some of the best known bounds for the energy, and more importantly gives insights about the contributions of specific subgraphs to the energy of these graphs.

math.CO↗
arXiv2016

Dynamic communicability and epidemic spread: a case study on an empirical dynamic contact network

We analyze a recently proposed temporal centrality measure applied to an empirical network based on person-to-person contacts in an emergency department of a busy urban hospital. We show that temporal centrality identifies a distinct set of top-spreaders than centrality based on the time-aggregated binarized contact matrix, so that taken together, the accuracy of capturing top-spreaders improves significantly. However, with respect to predicting epidemic outcome, the temporal measure does not necessarily outperform less complex measures. Our results also show that other temporal markers such as duration observed and the time of first appearance in the the network can be used in a simple predictive model to generate predictions that capture the trend of the observed data remarkably well.

physics.soc-ph↗
arXiv2015

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.

math.NA↗
arXiv2015

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.

cs.SI↗
arXiv2015

Core-satellite Graphs. Clustering, Assortativity and Spectral Properties

Core-satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts-Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In addition, we completely describe the spectrum of the adjacency and Laplacian matrices associated with core-satellite graphs. Finally, we introduce the class of generalized core-satellite graphs, and we analyze the spectral properties of such graphs.

cs.SI↗
arXiv2015

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.

cs.SI↗
arXiv2015

On the limiting behavior of parameter-dependent network centrality measures

We consider a broad class of walk-based, parameterized node centrality measures for network analysis. These measures are expressed in terms of functions of the adjacency matrix and generalize various well-known centrality indices, including Katz and subgraph centrality. We show that the parameter can be "tuned" to interpolate between degree and eigenvector centrality, which appear as limiting cases. Our analysis helps explain certain correlations often observed between the rankings obtained using different centrality measures, and provides some guidance for the tuning of parameters. We also highlight the roles played by the spectral gap of the adjacency matrix and by the number of triangles in the network. Our analysis covers both undirected and directed networks, including weighted ones. A brief discussion of PageRank is also given.

math.NA↗
arXiv2015

Approximation of functions of large matrices with Kronecker structure

We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2 \otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular function such that $f({\cal A})$ is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential function, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications.

math.NA↗
arXiv2015

Decay bounds for functions of matrices with banded or Kronecker structure

We present decay bounds for a broad class of Hermitian matrix functions where the matrix argument is banded or a Kronecker sum of banded matrices. Besides being significantly tighter than previous estimates, the new bounds closely capture the actual (non-monotonic) decay behavior of the entries of functions of matrices with Kronecker sum structure. We also discuss extensions to more general sparse matrices.

math.NA↗
arXiv2014

Are Social Networks Really Balanced?

There is a long-standing belief that in social networks with simultaneous friendly/hostile interactions (signed networks) there is a general tendency to a global balance. Balance represents a state of the network with lack of contentious situations. Here we introduce a method to quantify the degree of balance of any signed (social) network. It accounts for the contribution of all signed cycles in the network and gives, in agreement with empirical evidences, more weight to the shorter than to the longer cycles. We found that, contrary to what is believed, many signed social networks -- in particular very large directed online social networks -- are in general very poorly balanced. We also show that unbalanced states can be changed by tuning the weights of the social interactions among the agents in the network.

physics.soc-ph↗
arXiv2013

Total communicability as a centrality measure

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. Extensive numerical studies are conducted in order to compare this centrality measure with the closely related ones of subgraph centrality [E. Estrada and J. A. Rodriguez-Velazquez, Phys. Rev. E, 71 (2005), 056103] and Katz centrality [L. Katz, Psychometrica, 18 (1953), pp. 39-43]. Both synthetic and real-world networks are used in the computations.

cs.SI↗