Centrality measures for node-weighted networks via line graphs and the matrix exponential

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

doi:10.1007/s11075-020-01050-0

Cited by 10 publications

(10 citation statements)

References 37 publications

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“…The quantity 1/ cos θ is referred to as the condition number of ρ and denoted by κ(ρ); see Wilkinson [28,Chapter 2]. Equality is attained in (9) for C = yx T . Consider increasing the entry w ij , i = j, of A slightly by ε > 0.…”

Section: Perron Network Communicabilitymentioning

confidence: 99%

“…Sometimes, additional information about the vertices and/or edges is indispensable for a fuller and more realistic understanding of a complex system. Examples include the use of weighted networks [4,9,20], in which edges between vertices are assigned different numerical values, so-called "weights." In our setting, a higher weight for a given edge corresponds to a higher communication capacity between the nodes it connects.…”

Section: Introductionmentioning

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: Perron Network Communicabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Communication in complex networks

Cabrera

Jin

Noschese

et al. 2022

Applied Numerical Mathematics

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“…where W is the Wilkinson perturbation defined by (10) and D is the cone of all nonnegative block-diagonal matrices in R N L×N L with L blocks in R N ×N .…”

Section: Propositionmentioning

confidence: 99%

“…where W is the Wilkinson perturbation defined by (10) and S is the cone of all nonnegative block-diagonal matrices in R N L×N L with L blocks in R N ×N having the same sparsity structure as the diagonal block matrices of the given matrix B (6).…”

Section: Propositionmentioning

confidence: 99%

“…The investigation of the importance of nodes and edges, as well as the sensitivity of the communicability within a network to changes in the edge weights of the network with only one kind of nodes and edges has received considerable attention in the literature; see, e.g., [4,5,8,9,10,11,12,17,19] and references therein. Several of the techniques discussed evaluate the exponential of the adjacency matrix of the network, or the exponential of adjacency matrix determined by the line graph associated with the given network.…”

Section: Introductionmentioning

confidence: 99%

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Perron communicability and sensitivity of multilayer networks

El-Halouy,

Noschese,

Reichel

2022

Preprint

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Modeling complex systems that consist of different types of objects leads to multilayer networks, where nodes in the different layers represent different kind of objects. Nodes are connected by edges, which have positive weights. A multilayer network is associated with a supra-adjacency matrix. This paper investigates the sensitivity of the communicability in a multilayer network to perturbations of the network by studying the sensitivity of the Perron root of the supra-adjacency matrix. Our analysis sheds light on which edge weights to make larger to increase the communicability of the network, and which edge weights can be made smaller or set to zero without affecting the communicability significantly.

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Perron communicability and sensitivity of multilayer networks

El-Halouy

Noschese²,

Reichel³

2022

Numer Algor

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Modeling complex systems that consist of different types of objects leads to multilayer networks, where nodes in the different layers represent different kinds of objects. Nodes are connected by edges, which have positive weights. A multilayer network is associated with a supra-adjacency matrix. This paper investigates the sensitivity of the communicability in a multilayer network to perturbations of the network by studying the sensitivity of the Perron root of the supra-adjacency matrix. Our analysis sheds light on which edge weights to make larger to increase the communicability of the network, and which edge weights can be made smaller or set to zero without affecting the communicability significantly.

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Centrality measures for node-weighted networks via line graphs and the matrix exponential

Cited by 10 publications

References 37 publications

Communication in complex networks

Communication in complex networks

Perron communicability and sensitivity of multilayer networks

Perron communicability and sensitivity of multilayer networks

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