We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the p-resistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p^* and p^** such that if p < p^*, then the p-resistance depends on meaningful global properties of the graph, whereas if p > p^**, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^* = 1 + 1/(d-1) and p^** = 1 + 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^** is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1/p^* + 1/q = 1.

Author(s): |
Alamgir, M. and von Luxburg, U. |

Book Title: |
Advances in Neural Information Processing Systems 24 |

Pages: |
379-387 |

Year: |
2011 |

Day: |
0 |

Editors: |
J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger |

Department(s): |
Empirical Inference |

Bibtex Type: |
Conference Paper (inproceedings) |

Event Name: |
Twenty-Fifth Annual Conference on Neural Information Processing Systems (NIPS 2011) |

Event Place: |
Granada, Spain |

Digital: |
0 |

Links: |
PDF
Web |

@inproceedings{Alamgirv2011, title = {Phase transition in the family of p-resistances}, author = {Alamgir, M. and von Luxburg, U.}, booktitle = {Advances in Neural Information Processing Systems 24}, pages = {379-387}, editors = {J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger}, year = {2011} } |