Matching Exponential-Based and Resolvent-Based Centrality Measures

Aprahamian, Mary and Higham, Desmond J. and Higham, Nicholas J. (2016) Matching Exponential-Based and Resolvent-Based Centrality Measures. Journal of Complex Networks, 4 (2). pp. 157-176.

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Abstract

The relative importance of nodes in a network can be quantified via functions of the adjacency matrix. Two popular choices of function are the exponential, which is parameter-free, and the resolvent function, which yields the Katz centrality measure. Katz centrality can be the more computationally efficient, especially for large directed networks, and has the benefit of generalizing naturally to time-dependent network sequences, but it depends on a parameter. We give a prescription for selecting the Katz parameter based on the objective of matching the centralities of the exponential counterpart. For our new choice of parameter the resolvent can be very ill conditioned, but we argue that the centralities computed in floating point arithmetic can nevertheless reliably be used for ranking. Experiments on \revised{six} real networks show that the new choice of Katz parameter leads to rankings of nodes that \revised{generally} match those from the exponential centralities well in practice.

Item Type: Article
Uncontrolled Keywords: Katz centrality; Katz parameter; adjacency matrix; matrix exponential; matrix resolvent; network analysis; inverse iteration; condition number
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 05 Combinatorics
MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 07 Oct 2016
Last Modified: 20 Oct 2017 14:13
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2507

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