Comparative analysis of two discretizations of Ricci curvature for complex networks

Samal, Areejit; Sreejith, R.P.; Gu, Jiao; Liu, Shiping; Saucan, Emil; Jost, Jürgen

doi:10.1038/s41598-018-27001-3

Cited by 108 publications

(219 citation statements)

References 62 publications

(105 reference statements)

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“…Ollivier's definition for metric graphs assigns a probability measure to each node and the Ricci curvature of an edge is related to the optimal transportation cost between two probability measures defined on the vertices of the edge. Various definitions of Ricci curvatures on networks have been used in graph analysis for applications such as anomaly detection, detection of backbone edges or cancer related proteins [22,23,24,25,26,27,28,29,30].…”

Section: Our Contributionmentioning

confidence: 99%

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Community Detection on Networks with Ricci Flow

Lin

Luo³

et al. 2019

Sci Rep

103

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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: Our Contributionmentioning

confidence: 99%

“…It is easier and faster to compute than Ollivier-Ricci curvature, but is less geometrical. It is more suitable for large scale network analysis [23,24,44,45] and image processing [46]. We have also experimented with Forman curvature for community detection.…”

Section: Related Workmentioning

confidence: 99%

Community Detection on Networks with Ricci Flow

Lin

Luo³

et al. 2019

Sci Rep

103

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“…From the point of view of networks, it is found that graph connectivity is closely correlated to the existence of points of large negative curvature [45]. For any graph ω with adjacency matrix A [59],…”

Section: Mean Field Dynamicsmentioning

confidence: 99%

“…Locality comes from the fact that the Ollivier curvature of an edge is totally specified by the network structure of a known core neighbourhood of the edge. Much of the utility of the Ollivier curvature in discrete settings arises from the fact that it admits rather explicit formulations in terms of purely combinatorial variables for certain classes of graphs [40][41][42][43], though its definition in terms of optimal transport theory [44] makes it a burdensome drain on computational power when exact combinatorial expressions are unknown [45]. We present a new exact result (20) for the Ollivier curvature of an edge in terms of the number of short cycles (to be explained fully below) supported on that edge.…”

Section: Introductionmentioning

confidence: 99%

Self-assembly of geometric space from random graphs

Kelly

Trugenberger

Biancalana

2019

Class. Quantum Grav.

117

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We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.

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“…On the other hand, Forman-Ricci curvature has been introduced as a tool for undirected [29,40] as well as directed [41] network analyses. Irrespective of their applicability and handling the large networks, the two curvature notions demonstrated high correlation [42]. However, researchers suggested that Forman-Ricci curvature is a faster computation method and can be utilized in larger real-networks to gain preliminary insight into Ollivier-Ricci curvature that is much more computationally extensive.…”

Section: Curvature-based Methods For Complex Networkmentioning

confidence: 99%

Integrative Computational Framework for Understanding Metabolic Modulation in Leishmania

Chauhan

Singh

2019

Preprint

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The integration of computational and mathematical approaches is used to provide a key insight into the biological systems. Here, we seek to find detailed and more robust information on Leishmanial metabolic network by performing mathematical characterization in terms of Forman/Forman-Ricci curvature measures combined with flux balance analysis (FBA). The model prototype developed largely depends on its structure and topological components. The correlation of curvature measures with various network statistical properties revealed the structural-functional framework. The analyses helped us to identify the importance of several nodes and detect sub-networks. Our results revealed several key high curvature nodes (metabolites) belonging to common yet crucial metabolic, thus, maintaining the integrity of the network which signifies its robustness. Further analysis revealed the presence of some of these metabolites in redox metabolism of the parasite. MGO, an important node, has highly cytotoxic and mutagenic nature that can irreversibly modify DNA, proteins and enzymes, making them nonfunctional, leading to the formation of AGEs and MGO •-. Being a component in the glyoxalase pathway, we further attempted to study the outcome of the deletion of the key enzyme (GLOI) mainly involved in the neutralization of MGO by utilizing FBA. The model and the objective function both kept as simple as possible, demonstrated an interesting emergent behavior. The nonfunctional GLOI in the model contributed to 'zero' flux which signifies the key role of GLOI as a rate limiting enzyme. This has led to several fold increase production of MGO, thereby, causing an increased level of MGO •generation. Hence, the integrated computational approaches has deciphered GLOI as a potential target both from curvature measures as well as FBA which could further be explored for kinetic modeling by implying various redox-dependent constraints on the model. Designing various in vitro experimental perspectives could churn the therapeutic importance of GLOI.Leishmaniasis, one of the most neglected tropical diseases in the world, is of primary concern due to the increased risk of emerging drug resistance. To design novel drugs and search effective molecular drug targets with therapeutic importance, it is important to decipher the relation among the components responsible for leishmanial parasite survival inside the host cell at the metabolic level. Here, we have attempted to get an insight in the leishmanial metabolic network and predict the importance of key metabolites by applying mathematical characterization in terms of curvature measures and flux balance analysis (FBA). Our results identified several metabolites playing significant role in parasite's redox homeostasis. Among these MGO (methylglyoxal) caught our interest due to its highly toxic and reactive nature of irreversibly modifying DNA and proteins. FBA results helped us to look into the important role of GLOI (Glyoxalase I), the enzyme that catalyses the detoxification of MGO, in the pathway tha...

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Comparative analysis of two discretizations of Ricci curvature for complex networks

Cited by 108 publications

References 62 publications

Community Detection on Networks with Ricci Flow

Community Detection on Networks with Ricci Flow

Self-assembly of geometric space from random graphs

Integrative Computational Framework for Understanding Metabolic Modulation in Leishmania

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