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Topological combinatorics and simplicial networks : 위상수학적 조합론과 고차원 네트워크
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 국웅 | - |
| dc.contributor.author | 이강주 | - |
| dc.date.accessioned | 2019-10-21T03:38:58Z | - |
| dc.date.available | 2019-10-21T03:38:58Z | - |
| dc.date.issued | 2019-08 | - |
| dc.identifier.other | 000000158275 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/162423 | - |
| dc.identifier.uri | http://dcollection.snu.ac.kr/common/orgView/000000158275 | ko_KR |
| dc.description | 학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅. | - |
| dc.description.abstract | 그래프의 일반화인 단체의 복합체는 고차원 조합론 분야에서 연구되는 대상이다. 여러 점들 사이의 상호작용을 고려하기 위해 고차원 네트워크가 등장하였다. 본 논문에서는 단체의 복합체의 조합론적인 측면과 네트워크 이론적 측면을 다룬다. 생성나무와 관련된 단체의 복합체를 연구하여 조합론적 공식을 얻는다. 전기 네트워크 이론을 이용하여 고차원 네트워크를 분석하기 위한 도구를 개발한다. | - |
| dc.description.abstract | Simplicial complexes as a generalization of graphs have been studied in high-dimensional combinatorics. Simplicial networks emerged due to demands for considering interactions among more than two vertices. This thesis covers combinatorial aspects and network-theoretic aspects of simplicial complexes. We present combinatorial formulas for simplicial complexes concerning spanning trees. We provide tools for analyzing simplicial networks based on electrical networks. | - |
| dc.description.tableofcontents | 1 Introduction 1
1.1 Combinatorics of simplicial complexes . . . 1 1.2 Simplicial networks and electrical networks . . . 3 1.3 Organization of the thesis . . . 5 2 Mobius coinvariants and bipartite edge-rooted forests . . . 6 2.1 Introduction . . . 6 2.2 Background: Mobius coinvariant . . . 9 2.3 Combinatorial interpretations for \mu^{\bot}(Km+1) and \mu^{\bot}(Km+1;n+1) . . . 11 2.4 Formulas for \mu^{\bot}(Km+1) and \mu^{\bot}(Km+1;n+1) . . . 15 2.5 Homology of I(Km+1;n+1) . . . 22 2.6 The Mobius coinvariants of bi-coned graphs . . . 29 3 A formula for simplicial tree-numbers of matroid complexes . . . 31 3.1 Introduction . . . 31 3.2 Preliminaries . . . 33 3.3 The main result . . . 36 3.4 Examples . . . 39 4 A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes . . . 42 4.1 Introduction . . . 42 4.2 Preliminaries . . . 47 4.3 Proof of the main formula . . . 51 4.4 Another tree count for hypercubes . . . 53 5 Weighted tree-numbers of matroid complexes . . . 61 5.1 Introduction . . . 61 5.2 Weighted tree-numbers of simplicial complexes . . . 63 5.3 Weighted tree-numbers of matroid complexes . . . 72 5.4 Application: complete colorful complexes . . . . 79 6 Simplicial networks and effective resistance . . . 83 6.1 Introduction . . . 83 6.2 Preliminaries . . . 85 6.3 Simplicial networks and effective resistance . . . 89 6.4 Effective resistance via simplicial potential . . . 93 6.5 Effective resistance via high-dimensional trees . . . 97 7 High-dimensional networks and spanning forests . . . 104 7.1 Introduction . . . 104 7.2 Preliminaries . . . 107 7.3 An acyclization in codimension 1, and spanning forests in higher dimensions . . . 109 7.4 High-dimensional electrical networks and combinatorics . . . 111 8 Kirchhoff index of simplicial networks . . . 120 8.1 Introduction . . . 121 8.2 Preliminaries . . . 123 8.3 Electrical networks and Kirchhoff index . . . 126 8.4 Robustness of simplicial complexes . . . 132 8.5 An integral expression and a dynamical system . . . 135 8.6 An open problem: another proof of Theorem 8.3.4 . . . 139 | - |
| dc.language.iso | eng | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject | simplicial complex | - |
| dc.subject | Laplacian | - |
| dc.subject | spanning tree | - |
| dc.subject | matroid | - |
| dc.subject | network | - |
| dc.subject | centrality | - |
| dc.subject.ddc | 510 | - |
| dc.title | Topological combinatorics and simplicial networks | - |
| dc.title.alternative | 위상수학적 조합론과 고차원 네트워크 | - |
| dc.type | Thesis | - |
| dc.type | Dissertation | - |
| dc.contributor.AlternativeAuthor | Kang-Ju Lee | - |
| dc.contributor.department | 자연과학대학 수리과학부 | - |
| dc.description.degree | Doctor | - |
| dc.date.awarded | 2019-08 | - |
| dc.identifier.uci | I804:11032-000000158275 | - |
| dc.identifier.holdings | 000000000040▲000000000041▲000000158275▲ | - |
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