## Abstract

Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we define high order random walks: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension d has vertices, edges, triangles, pyramids, up to d-dimensional cells. For any 0 ≤ i < d, a high order random walk on dimension i moves between neighboring i-faces (e.g., edges) of the complex, where two i-faces are considered neighbors if they share a common (i + 1)-face (e.g., a triangle). The case of i = 0 recovers the well studied random walk on graphs. We provide a local-To-global criterion on a complex which implies rapid convergence of all high order random walks on it. Specifically, we prove that if the 1-dimensional skeletons of all the links of a complex are spectral expanders, then for all 0 ≤ i < d the high order random walk on dimension i converges rapidly to its stationary distribution. We derive our result through a new notion of high dimensional combinatorial expansion of complexes which we term colorful expansion. This notion is a natural generalization of combinatorial expansion of graphs and is strongly related to the convergence rate of the high order random walks. We further show an explicit family of bounded degree complexes which satisfy this criterion. Specifically, we show that Ramanujan complexes meet this criterion, and thus form an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.

Original language | English |
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Title of host publication | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 |

Editors | Christos H. Papadimitriou |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770293 |

DOIs | |

State | Published - 1 Nov 2017 |

Event | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 - Berkeley, United States Duration: 9 Jan 2017 → 11 Jan 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 67 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 |
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Country/Territory | United States |

City | Berkeley |

Period | 9/01/17 → 11/01/17 |

### Bibliographical note

Funding Information:∗ This work was partially supported by ERC and BSF.

### Funding

∗ This work was partially supported by ERC and BSF.

Funders | Funder number |
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European Research Council | |

United States-Israel Binational Science Foundation |

## Keywords

- Expander graphs
- High dimensional expanders
- Random walks