Arrigo, Francesca and Grindrod, Peter and Higham, Desmond J. and Noferini, Vanni
(2017)
*Nonbacktracking walk centrality for directed networks.*
[MIMS Preprint]

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## Abstract

The theory of zeta functions provides an expression for the generating fu nction of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is determined by the spect rum of a three-by-three block matrix involving the original adjacency matrix. This giv es a means to choose appropriate values of the attenuation parameter. We find that three important a dditional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation para meter. Second, because the radius of convergence of the generating function is invariant under the remov al of certain types of nodes, we can gain computational efficiencies through reducing the dimension of t he resulting eigenvalue problem. Third, the dimension of the linear system defining the centrali ty measures may be reduced in the same manner. We show that the new centrality measure may be interp reted as standard Katz on a modified network, where self loops are added, and where nonreciproca l edges are augmented with negative weights. We also give a multilayer interpretation, wh ere negatively weighted walks between layers compensate for backtracking walks on the only non-emp ty layer. Studying the limit as the attenuation parameter approaches its upper bound allows us to propose an eigenvector-based nonbacktracking centrality measure in this directed network setting. We find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrat e the centrality measure on a synthetic network, where it is shown to eliminate a localization effect p resent in standard Katz centrality. Finally, we give results for real networks.

Item Type: | MIMS Preprint |
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Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory |

Depositing User: | Dr V Noferini |

Date Deposited: | 14 Mar 2017 |

Last Modified: | 20 Oct 2017 14:13 |

URI: | http://eprints.maths.manchester.ac.uk/id/eprint/2535 |

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