Community Detection on Networks with Ricci Flow

Ni, Chien-Chun; Lin, Yu-Chih; Luo, Feng; Gao, Jie

doi:10.1038/s41598-019-46380-9

Cited by 74 publications

(106 citation statements)

References 68 publications

(139 reference statements)

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“…www.nature.com/scientificreports/ Correlation with other edge-based measures. We explore the correlation between DD and three other established edge-based measures, namely edge betweenness centrality 39,40 , Forman-Ricci curvature ( R F ) 13,14 and Ollivier-Ricci curvature ( R O ) 11,12,14,16 for characterizing the local network geometry. These results are summarized in Fig.…”

Section: Dd Distribution In Directed Network Dd Distribution Can Bementioning

confidence: 99%

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Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Amirhossein

Samal

Jost

2020

Sci Rep

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Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. Here, we analyze a simple, elegant yet underexplored measure, ‘degree difference’ (DD) between vertices of an edge, to understand the local network geometry. We describe the connection between DD and global assortativity of the network from both formal and conceptual perspective, and show that DD can reveal structural properties that are not obtained from other such measures in network science. Typically, edges with different DD play different structural roles and the DD distribution is an important network signature. Notably, DD is the basic unit of assortativity. We provide an explanation as to why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing synthetic and real networks, we show that DD distribution can be used to distinguish between different types of networks including those networks that cannot be easily distinguished using degree sequence and global assortativity. Moreover, we show DD to be an indicator for topological robustness of scale-free networks. Overall, DD is a local measure that is simple to define, easy to evaluate, and that reveals structural properties of networks not readily seen from other measures.

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Section: Dd Distribution In Directed Network Dd Distribution Can Bementioning

confidence: 99%

“…[10][11][12][13][14]. Local clustering coefficient 7 , generalized degree, local assortativity 15 , Ollivier-Ricci curvature 11,12,14,16 , and Forman-Ricci curvature 13,14 are some of the notable measures characterizing the local structural properties of complex networks.…”

mentioning

confidence: 99%

Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Amirhossein

Samal

Jost

2020

Sci Rep

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“…on large empirical and model networks. The second one is the exploration of the clustering and community detection capabilities of the Ricci curvature notions introduce herein, and their comparison, with the results in [26] and [5,27] respectively. Furthermore, it would be interesting to explore the correlation between the notions of curvature introduced herein and hyperbolic embeddings of networks.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

A Simple Differential Geometry for Networks and Its Generalizations

Saucan

Samal

Jost

2019

Complex Networks and Their Applications VIII

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Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, directed or undirected networks, are far more intuitive and easier to compute, than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible. We also propose even simpler and more intuitive substitutes of the Haantjes curvature, that allow for even faster and easier computations in large-scale networks.

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“…u) − f (i)) − u: Cu=2 u∼i (f (u) − f (i)) + u: Cu=1 u∼j (f (u) − f (j)) − u: Cu=2 u∼j (f (u) − f (j)) + n in (f (i) − f (j)) − n 2 pout(f (i) − f (j)) .Then, taking the supremum over all 1-Lipschitz functions f , we obtainij∈E W1(pi(τ ), pj(τ ))(1 − δ(Ci, Cj)) ≈ e −λcτ (p in + pout) 1 + |p in − pout| 2(p in + pout)(23)Substituting this into Eq (18). and noting that p in + pout is constant we obtain at a fixed τP(C|κ) ∝ exp |p in − pout| 2(p in + pout) ij∈E δ(Ci, Cj) ,which up to a constant of proportionality equals the expression in Eq.…”

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confidence: 99%

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

Gosztolai

Arnaudon

2021

Preprint

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Defining the geometry of networks is typically associated with embedding in low-dimensional spaces such as manifolds. This approach has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, the choice of embedding space is network-specific, and incompatible spaces can result in information loss. Here, we define a dynamic edge curvature for the study of arbitrary networks measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps robustly encode communities until the phase transition of detectability, where spectral clustering methods fail. We use this insight to derive geometric modularity optimisation and demonstrate it on the European power grid and the C. elegans homeobox gene regulatory network finding previously unidentified communities on multiple scales. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.

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Community Detection on Networks with Ricci Flow

Cited by 74 publications

References 68 publications

Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Degree difference: a simple measure to characterize structural heterogeneity in complex networks

A Simple Differential Geometry for Networks and Its Generalizations

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

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