### Abstract

Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

Original language | English (US) |
---|---|

Pages (from-to) | 253-265 |

Number of pages | 13 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 1 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2009 |

### Fingerprint

### Keywords

- Network reliability
- network resilience
- series-parallel networks

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

**MULTI-TERMINAL NETWORK CONNECTEDNESS on SERIES-PARALLEL NETWORKS.** / Farley, Toni R.; Colbourn, Charles J.

Research output: Contribution to journal › Article

*Discrete Mathematics, Algorithms and Applications*, vol. 1, no. 2, pp. 253-265. https://doi.org/10.1142/S1793830909000208

}

TY - JOUR

T1 - MULTI-TERMINAL NETWORK CONNECTEDNESS on SERIES-PARALLEL NETWORKS

AU - Farley, Toni R.

AU - Colbourn, Charles J.

PY - 2009/6/1

Y1 - 2009/6/1

N2 - Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

AB - Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

KW - Network reliability

KW - network resilience

KW - series-parallel networks

UR - http://www.scopus.com/inward/record.url?scp=71749119130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71749119130&partnerID=8YFLogxK

U2 - 10.1142/S1793830909000208

DO - 10.1142/S1793830909000208

M3 - Article

VL - 1

SP - 253

EP - 265

JO - Discrete Mathematics, Algorithms and Applications

JF - Discrete Mathematics, Algorithms and Applications

SN - 1793-8309

IS - 2

ER -