Normalized discrete Ricci flow used in community detection

Lai, Xin Ji; Bai, Shuliang; Lin, Yong

doi:10.1016/j.physa.2022.127251

Cited by 4 publications

(8 citation statements)

References 45 publications

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“…Classic approaches include studying the degree distribution, clustering coefficient, and shortest path between nodes, all of which provide insights into the network’s geometry 46 . However, to study the geometric and topological properties of networks more deeply, discrete adaptations of differential geometry have become widely applied 19 , 22 , 30 – 33 , 36 , 41 . In differential geometry curvature is a key actor, describing the local behaviour of a manifold, and geometric flows can be employed to perturb this important property and examine the consequences.…”

Section: Discussionmentioning

confidence: 99%

“…In differential geometry curvature is a key actor, describing the local behaviour of a manifold, and geometric flows can be employed to perturb this important property and examine the consequences. By treating networks as discrete counterparts of manifolds, we can view them as geometric objects and discrete curvatures and flows on networks have proven effective tools for addressing common network theory questions 31 – 33 , 36 .…”

Section: Discussionmentioning

confidence: 99%

“…In a seminal contribution to the field, Hamilton introduced Ricci flow as a tool to study the topological implications of deforming a metric on a manifold according to its Ricci curvature 26 , which led subsequently to the striking solution of the Poincaré conjecture by Perelman 27 , 28 . Like curvature Ricci flow can also be defined in a discrete setting 29 , and recently discrete Ricci flows and curvatures have been applied to problems in network theory 30 – 32 such as network alignment 33 , community detection 34 – 36 , functional community inference for biological networks 37 and phase transitions in time-varying complex networks 38 .…”

Section: Introductionmentioning

confidence: 99%

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Charting cellular differentiation trajectories with Ricci flow

Baptista,

MacArthur,

Banerji

2024

Nat Commun

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Complex biological processes, such as cellular differentiation, require intricate rewiring of intra-cellular signalling networks. Previous characterisations revealed a raised network entropy underlies less differentiated and malignant cell states. A connection between entropy and Ricci curvature led to applications of discrete curvatures to biological networks. However, predicting dynamic biological network rewiring remains an open problem. Here we apply Ricci curvature and Ricci flow to biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as previously suggested, and provide complementary rather than interchangeable information. We next employ Ricci flow to derive network rewiring trajectories from stem cells to differentiated cells, accurately predicting true intermediate time points in gene expression time courses. In summary, we present a differential geometry toolkit for understanding dynamic network rewiring during cellular differentiation and cancer.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Charting cellular differentiation trajectories with Ricci flow

Baptista,

MacArthur,

Banerji

2024

Nat Commun

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“…Sia et al [ 23 ] constructed a community detection algorithm by removing negative curved edges step by step. Lai et al [ 24 ] leveraged a DRC-based Ricci flow to deform a graph, then intracommunity nodes became closer and intercommunity nodes dispersed. The DRC is capable of finding the underlying relationship between nodes, characterizing them to clusters with identical or distinct properties.…”

Section: Related Workmentioning

confidence: 99%

“…Based on the homophily assumption of most graphs, the mainstream graph-based tasks, such as node classification, link prediction and graph classification/regression, tend in essence to strengthen the connection between nodes with the same property and discriminate against nodes with different properties. To describe the geometric relationships of nodes from intra-/intercommunities, we draw inspiration from recent research focusing on developing community detection algorithms [ 22 , 23 , 24 ] with the help of a geometric notion, i.e., the discrete Ricci curvature (DRC) [ 25 ].…”

Section: Introductionmentioning

confidence: 99%

Deeper Exploiting Graph Structure Information by Discrete Ricci Curvature in a Graph Transformer

Lai

Liu

Qian

et al. 2023

Entropy

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Graph-structured data, operating as an abstraction of data containing nodes and interactions between nodes, is pervasive in the real world. There are numerous ways dedicated to extract graph structure information explicitly or implicitly, but whether it has been adequately exploited remains an unanswered question. This work goes deeper by heuristically incorporating a geometric descriptor, the discrete Ricci curvature (DRC), in order to uncover more graph structure information. We present a curvature-based topology-aware graph transformer, termed Curvphormer. This work expands the expressiveness by using a more illuminating geometric descriptor to quantify the connections within graphs in modern models and to extract the desired structure information, such as the inherent community structure in graphs with homogeneous information. We conduct extensive experiments on a variety of scaled datasets, including PCQM4M-LSC, ZINC, and MolHIV, and obtain a remarkable performance gain on various graph-level tasks and fine-tuned tasks.

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Sobolev spaces on locally finite graphs

Shao,

Yang,

Zhao

2024

Proc. Amer. Math. Soc.

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In this paper, we focus on the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability and Sobolev inequalities. We introduce a linear space composed of vector-valued functions with variable dimensions such that the gradients of functions on graphs happen to fit into such a space and we can get the desired properties of various Sobolev spaces along this line. Moreover, we also derive several Sobolev inequalities under certain assumptions on measures or weights of graphs. Although these results are within the framework of functional analysis, the key is that we provide an appropriate perspective for applying variational methods on graphs. As fundamental analytical tools, all these results are highly applicable and useful for partial differential equations on locally finite graphs.

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Normalized discrete Ricci flow used in community detection

Cited by 4 publications

References 45 publications

Charting cellular differentiation trajectories with Ricci flow

Charting cellular differentiation trajectories with Ricci flow

Deeper Exploiting Graph Structure Information by Discrete Ricci Curvature in a Graph Transformer

Sobolev spaces on locally finite graphs

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