Analysis of directed networks via the matrix exponential

Cabrera, Omar De la Cruz; Matar, Mona; Reichel, Lothar

doi:10.1016/j.cam.2019.01.015

Cited by 30 publications

(11 citation statements)

References 16 publications

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“…The quantity [exp 0 (A)] ii is commonly referred to as the subgraph centrality of the vertex v i ; it measures the ease of leaving node v i and returning to this node by following the edges of the graph; see [10,15], though we remark that these references apply the matrix exponential instead of (2). The subgraph centrality is an appropriate measure for undirected graphs; a discussion about directed graphs is provided in [8].…”

Section: The Modified Matrix Exponential and Network Communicabilitymentioning

confidence: 99%

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Communication in complex networks

Cabrera

Jin

Noschese

et al. 2022

Applied Numerical Mathematics

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One of the properties of interest in the analysis of networks is global communicability, i.e., how easy or difficult it is, generally, to reach nodes from other nodes by following edges. Different global communicability measures provide quantitative assessments of this property, emphasizing different aspects of the problem. This paper investigates the sensitivity of global measures of communicability to local changes. In particular, for directed, weighted networks, we study how different global measures of communicability change when the weight of a single edge is changed; or, in the unweighted case, when an edge is added or removed. The measures we study include the total network communicability, based on the matrix exponential of the adjacency matrix, and the Perron network communicability, defined in terms of the Perron root of the adjacency matrix and the associated left and right eigenvectors. Finding what local changes lead to the largest changes in global communicability has many potential applications, including assessing the resilience of a system to failure or attack, guidance for incremental system improvements, and studying the sensitivity of global communicability measures to errors in the network connection data.

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Section: The Modified Matrix Exponential and Network Communicabilitymentioning

confidence: 99%

“…Then A = X Y T . The vector y is a rescaling of the left Perron vector y in (8). This normalization of the columns of Y yields y T x = 1 and y T j x j = 1 for j = 2, 3, .…”

Section: Definitionmentioning

confidence: 99%

Communication in complex networks

Cabrera

Jin

Noschese

et al. 2022

Applied Numerical Mathematics

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“…over the years [3,20] and that everything discussed here for nodes easily translates to address the case of edges by working on the line graph [10]. The simplest measure of centrality for nodes is degree centrality.…”

Section: Centrality Measuresmentioning

confidence: 99%

Mittag--Leffler Functions and their Applications in Network Science

Arrigo¹,

Durastante²

2021

SIAM J. Matrix Anal. Appl.

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We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag-Leffler functions. This overarching theory includes as special cases wellknown centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag-Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modelling and computational issues, and provide guidelines on parameter selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided.

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“…In order to quantify this idea of importance, entities are assigned a nonnegative score, or centrality [15]: the higher its value, the more important the entity is within the graph. We will focus here on centrality measures for nodes, although we note that several centrality measures for edges have been defined over the years [3,20] and that everything discussed here for nodes easily translates to address the case of edges by working on the line graph [10]. The simplest measure of centrality for nodes is degree centrality.…”

Section: Centrality Measuresmentioning

confidence: 99%

Mittag-Leffler functions and their applications in network science

Arrigo,

Durastante

2021

Preprint

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Analysis of directed networks via the matrix exponential

Cited by 30 publications

References 16 publications

Communication in complex networks

Communication in complex networks

Mittag--Leffler Functions and their Applications in Network Science

Mittag-Leffler functions and their applications in network science

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