Characterizing the Analogy Between Hyperbolic Embedding and Community Structure of Complex Networks

Faqeeh, Ali; Osat, Saeed; Radicchi, Filippo

doi:10.1103/physrevlett.121.098301

Cited by 49 publications

(46 citation statements)

References 48 publications

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“…Taking a geometric view of complex networks is an emerging trend, as shown in a number of recent work. For example, the community structures were used as a coarse version of its embedding in a hidden space with hyperbolic geometry [55]. Topological data analysis, a typical geometric approach for data analysis, has been applied for analyzing complex systems [56].…”

Section: Related Workmentioning

confidence: 99%

Community Detection on Networks with Ricci Flow

Lin

Luo³

et al. 2019

Sci Rep

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Many complex networks in the real world have community structures – groups of well-connected nodes with important functional roles. It has been well recognized that the identification of communities bears numerous practical applications. While existing approaches mainly apply statistical or graph theoretical/combinatorial methods for community detection, in this paper, we present a novel geometric approach which enables us to borrow powerful classical geometric methods and properties. By considering networks as geometric objects and communities in a network as a geometric decomposition, we apply curvature and discrete Ricci flow, which have been used to decompose smooth manifolds with astonishing successes in mathematics, to break down communities in networks. We tested our method on networks with ground-truth community structures, and experimentally confirmed the effectiveness of this geometric approach.

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Section: Related Workmentioning

confidence: 99%

Community Detection on Networks with Ricci Flow

Lin

Luo³

et al. 2019

Sci Rep

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“…This leads to generated topologies without community structure [7,46,47]. On the other hand, the considered real layers (Table I) exhibit community structure and trans-layer community correlations, which are manifested in their embeddings as groups of nodes that are similar-close along the angular similarity direction-in both layers simultaneously [14,48]. It would be interesting to modify the assignment of angular coordinates in the GMM-LP along the lines of [46] so that the model can also generate synthetic layers with community correlations.…”

Section: Discussionmentioning

confidence: 99%

Link persistence and conditional distances in multiplex networks

Papadopoulos

Kleineberg

2019

Phys. Rev. E

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Recent progress towards unraveling the hidden geometric organization of real multiplexes revealed significant correlations across the hyperbolic node coordinates in different network layers, which facilitated applications like trans-layer link prediction and mutual navigation. But are geometric correlations alone sufficient to explain the topological relation between the layers of real systems? Here we provide the negative answer to this question. We show that connections in real systems tend to persist from one layer to another irrespectively of their hyperbolic distances. This suggests that in addition to purely geometric aspects the explicit link formation process in one layer impacts the topology of other layers. Based on this finding, we present a simple modification to the recently developed Geometric Multiplex Model to account for this effect, and show that the extended model can reproduce the behavior observed in real systems. We also find that link persistence is significant in all considered multiplexes and can explain their layers' high edge overlap, which cannot be explained by coordinate correlations alone. Furthermore, by taking both link persistence and hyperbolic distance correlations into account we can improve trans-layer link prediction. These findings guide the development of multiplex embedding methods, suggesting that such methods should be accounting for both coordinate correlations and link persistence across layers.A first step towards this direction is the finding that if the layers comprising real multiplexes are independently embedded into hyperbolic spaces, their coordinates exhibit significant correlations [14]. This finding motivated new applications, like multidimensional community de-1 2β r , gives the power law degree distribution, P (k) ≈ ρ(r(k))|r (k)| ∝ k −γ , γ = 1 + 1/β > 2. Without loss of generality, we assume here a hyperbolic plane of curvature K = −1. See [2] for further details.

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“…This picture is completely different for GR networks, reported in figure 6(b). GR networks show strong community organization at the topological level, resulting in large values of Q as measured by the Louvain method, which is induced by structural constraints imposed by the geometric models [46]. However, as expected, the critical gap does not detect soft communities, as demonstrated by the non-significant values of the modularity, compatible with zero, over different realizations of the randomization process.…”

Section: Community Structurementioning

confidence: 54%

Geometric randomization of real networks with prescribed degree sequence

2019

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We introduce a model for the randomization of complex networks with geometric structure. The geometric randomization (GR) model assumes a homogeneous distribution of the nodes in a hidden similarity space and uses rewirings of the links to find configurations that maximize a connection probability akin to that of the popularity-similarity geometric network models. The rewiring preserves exactly the original degree sequence, thus preventing fluctuations in the degree cutoff. The GR model is manifestly simple as it relies upon a single free parameter controlling the clustering of the rewired network, and it does not require the explicit estimation of hidden degree variables. We demonstrate the applicability of GR by implementing it as a null model for the analysis of community structure. As a result, we find that geometric and topological communities detected in real networks are consistent, while topological communities are also detected in randomized counterparts as an effect of structural constraints.

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Characterizing the Analogy Between Hyperbolic Embedding and Community Structure of Complex Networks

Cited by 49 publications

References 48 publications

Community Detection on Networks with Ricci Flow

Community Detection on Networks with Ricci Flow

Link persistence and conditional distances in multiplex networks

Geometric randomization of real networks with prescribed degree sequence

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