# Path Laplacian matrices : introduction and application to the analysis of consensus in networks

Estrada, Ernesto
(2012)
*Path Laplacian matrices : introduction and application to the analysis of consensus in networks.*
Linear Algebra and Its Applications, 436 (9).
pp. 3373-3391.
ISSN 0024-3795

## Abstract

The concept of k-pathLaplacian matrix of a graph is motivated and introduced. The pathLaplacian matrices are a natural generalization of the combinatorial Laplacian of a graph. They are defined by using path matrices accounting for the existence of shortest paths of length k between two nodes. This new concept is motivated by the problem of determining whether every node of a graph can be visited by means of a process consisting of hopping from one node to another separated at distance k from it. The problem is solved by using the multiplicity of the trivial eigenvalue of the corresponding k-pathLaplacian matrix. We apply these matrices to the analysis of the consensus among agents in a networked system. We show how the consensus in different types of network topologies is accelerated by considering not only nearest neighbors but also the influence of long-range interacting ones. Further applications of pathLaplacian matrices in a variety of other fields, e.g., synchronization, flocking, Markov chains, etc., will open a new avenue in algebraic graph theory.

Item type: | Article |
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ID code: | 40662 |

Keywords: | graph theory, path matrices, laplacian matrix, consensusanalysis, synchronization, Mathematics, Discrete Mathematics and Combinatorics, Algebra and Number Theory, Geometry and Topology, Numerical Analysis |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Pure Administrator |

Date Deposited: | 02 Aug 2012 10:42 |

Last modified: | 04 May 2016 22:17 |

URI: | http://strathprints.strath.ac.uk/id/eprint/40662 |