### Abstract

'Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that, asymptotically, uniformly samples loopy graphs with any given sequence.

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

Pages (from-to) | 927-947 |

Number of pages | 21 |

Journal | Journal of Complex Networks |

Volume | 6 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2018 |

### Fingerprint

### Keywords

- Configuration model
- Double-edge swaps
- Graph sampling
- MCMC
- Self-loops

### ASJC Scopus subject areas

- Computer Networks and Communications
- Management Science and Operations Research
- Control and Optimization
- Computational Mathematics
- Applied Mathematics

### Cite this

**The connectivity of graphs of graphs with self-loops and a given degree sequence.** / Nishimura, Joel.

Research output: Contribution to journal › Article

*Journal of Complex Networks*, vol. 6, no. 6, pp. 927-947. https://doi.org/10.1093/comnet/cny008

}

TY - JOUR

T1 - The connectivity of graphs of graphs with self-loops and a given degree sequence

AU - Nishimura, Joel

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 'Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that, asymptotically, uniformly samples loopy graphs with any given sequence.

AB - 'Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that, asymptotically, uniformly samples loopy graphs with any given sequence.

KW - Configuration model

KW - Double-edge swaps

KW - Graph sampling

KW - MCMC

KW - Self-loops

UR - http://www.scopus.com/inward/record.url?scp=85059481122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059481122&partnerID=8YFLogxK

U2 - 10.1093/comnet/cny008

DO - 10.1093/comnet/cny008

M3 - Article

VL - 6

SP - 927

EP - 947

JO - Journal of Complex Networks

JF - Journal of Complex Networks

SN - 2051-1310

IS - 6

ER -