As on arXiv
Abstract
In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Evolution of weights on a connected finite graph
Topics Surgery, Discrete & graph Ricci, Long-time existence & convergence · Same authors Jicheng Ma, Yunyan Yang · Shared terms “community detection”, “Ollivier Ricci”, “algorithm”
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Core detection via Ricci curvature flows on weighted graphs
Topics Discrete & graph Ricci · Same authors Jicheng Ma, Yunyan Yang · Shared terms “weighted graphs”, “detection”, “algorithm”
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Evolving edge weights via local entropy flow and cohesion flow on graphs
Topics Discrete & graph Ricci, Long-time existence & convergence, Stability & uniqueness · Same authors Jicheng Ma, Yunyan Yang · Shared terms “weighted graphs”, “community detection”, “Ollivier Ricci”
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Equivalence of Lin–Lu–Yau curvature and 1/2-Ollivier curvature on weighted graphs
Topics Discrete & graph Ricci, Long-time existence & convergence, Stability & uniqueness · Same author Yunyan Yang · Shared terms “weighted graphs”, “Ollivier”, “Lin”
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The Ollivier Ricci flow with prescribed curvature on infinite graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “Ollivier Ricci”, “Yau Ricci”, “Lin”
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The weighted Forman and Lin-Lu-Yau Ricci flow on graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “Yau Ricci”, “Lin”
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Ricci flow with surgery on three-manifolds
Topics Surgery, Discrete & graph Ricci
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Ricci Flow on Weighted Digraphs with Balancing Factor
Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “weighted graphs”, “community detection”, “algorithm”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.7Hamilton’s Program in 2002
What was proved, what was missing, and where Perelman came in.
- 12B.5The Canonical Neighbourhood Theorem
Every high-curvature point looks like a neck or a cap.
- 12B.7The Singular Time: Horns and Strong Necks
The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.