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Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

2011
*
Internet Mathematics
*

In this paper, we extend the concept of

doi:10.1080/15427951.2011.601233
fatcat:rtp5lxnru5earjmxsvnxcblury
*scaled**Gromov*hyperbolic*graph*, originally developed*for*the Thin Triangle*Condition*(TTC), to the*computationally*simplified, but less intuitive,*Four*-*Point**Condition*... Here we develop various ways of*scaling*the 4-*point*delta, and develop upper bounds*for*the*scaled*4-*point*delta in various spaces. ... It transpires from Fig. 4 that the theoretically established Conclusions We have shown that*scaling*the*Gromov**Four*-*Point**Condition*in various ways and requiring the various*scaled*quantities to ...##
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Why did the shape of your network change? (On detecting network anomalies via non-local curvatures)
[article]

2019
*
arXiv
*
pre-print

Based on these measures, we precisely formulate several

arXiv:1808.05676v3
fatcat:nlc4poac2jhurfoetsglwl2rne
*computational*problems related to anomaly detection in static or dynamic*networks*, and provide non-trivial*computational*complexity results*for*these ...*For*this purpose, we use two measures of*network**curvatures*which depend on non-trivial global properties, such as distributions of geodesics and higher-order correlations among nodes, of the given*network*... Acknowledgements We thank Anastasios Sidiropoulos and Nasim Mobasheri*for*very useful discussions. This research work was partially supported by NSF grants IIS-1160995 and IIS-1814931. ...##
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A Geometric Distance Oracle for Large Real-World Graphs
[article]

2014
*
arXiv
*
pre-print

The second set leverages

arXiv:1404.5002v1
fatcat:55auyg3o2jfdvdy74craz3siea
*Gromov*-type tree contraction of the*graph*with the additive error guaranteed not to exceed 2δn, where δ is the hyperbolic constant of the*graph*. ... In this paper, we present a novel and scalable distance oracle that leverages the hyperbolic core of real-world large*graphs**for*fast and scalable distance approximation. ... (b) Simpler to*compute*is the the equivalent 4-*point**condition*, where*for*any*four**points*w, x, y, z, the largest sum L of opposite pairs (xz+yw in this picture) minus the middle sum M (=xy + wz here) ...##
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On the Hyperbolicity of Large-Scale Networks
[article]

2013
*
arXiv
*
pre-print

Finally, we present practical

arXiv:1307.0031v1
fatcat:xk5ejb66mjcd3gngpvjhnt367i
*computational*means*for*detection of hyperbolicity and show how the test itself may be*scaled*to much larger*graphs*than those we examined via renormalization group methodology ... These observations indicate that d-hyperbolicity is a common feature of large-*scale**networks*. ... Acknowledgements The authors wish to thank Vladimir Marbukh*for*helpful discussions. This work was supported by grants FA9550-11-1-0278 and 60NANB10D128 from AFOSR and NIST, respectively. ...##
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Ricci curvature of the Internet topology

2015
*
2015 IEEE Conference on Computer Communications (INFOCOM)
*

It is a more local measure which allows us to understand the distribution of

doi:10.1109/infocom.2015.7218668
dblp:conf/infocom/NiLGGS15
fatcat:tsobsp2rg5ft3ge3cj74eeyujy
*curvatures*in the*network*. ... Analysis of Internet topologies has shown that the Internet topology has negative*curvature*, measured by Gromov's "thin triangle*condition*", which is tightly related to core congestion and route reliability ... all Ricci*curvatures*in such*graphs**scales*in the linear order of the number of edges of the*network*, thus suitable*for*large-*scale**networks*. ...##
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Emergence of the Circle in a Statistical Model of Random Cubic Graphs
[article]

2021
*
arXiv
*
pre-print

We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular

arXiv:2008.11779v3
fatcat:4sopddxir5f45kr7aar33cix3a
*graphs*and making using of the Ollivier*curvature*, a coarse analogue of the Ricci*curvature*... configurations converges in the sense of*Gromov*-Hausdorff to S^1_r. ... Indeed, the Forman*curvature*[40, 87, 88, 89] , an alternative notion of*curvature*defined*for**networks*(strictly speaking regular CW -complexes), is the basis*for*a recent proposal of*network*gravity ...##
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A Review of and Some Results for Ollivier–Ricci Network Curvature

2020
*
Mathematics
*

In the penultimate section we review a method based on the linear sketching technique

doi:10.3390/math8091416
fatcat:epqbnmlnqzh6hixpfgob2vwnk4
*for*efficient approximate*computation*of the Ollivier–Ricci*network**curvature*. ... In the next section we provide some "not-previously-published" bounds on the exact and approximate*computation*of the*curvature*measure. ... One notion of*network**curvature*that has been well-studied in the*network*theory literature, first suggested by*Gromov*in a non-*network*group theoretic context [4] , is the*Gromov*-hyperbolic*curvature*...##
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Negatively Curved Graphs
[article]

2021
*
arXiv
*
pre-print

Many applications in

arXiv:1512.01281v3
fatcat:hb5hkygsprehhj3g6hzzn3u3ay
*network*science have recently been discovered*for*the "*curvature*" of a*network*, but there is no consensus on the definition*for*this term. ... A connection between one type of*curvature*– Gromov's hyperbolicity – and a tree representation has been known*for*decades, and recently it has also been connected*for*unweighted*graphs*to a logical center ... We model traffic on a*computer**network*using uniformly distributed demand. We assume all*graphs*are connected. ...##
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Efficient estimation of a Gromov–Hausdorff distance between unweighted graphs
[article]

2021
*
arXiv
*
pre-print

The algorithm finds the mGH distances exactly on most

arXiv:1909.09772v8
fatcat:w5jmdinvbbanpijv7osqpr26ue
*graphs*with the*scale*-free property. We use the*computed*mGH distances to successfully detect outliers in real-world social and*computer**networks*. ...*Gromov*-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g.*point*clouds, manifolds, or*graphs*. ...*Computing*the shape of brain*networks*using*graph*filtration and*gromov*-hausdorff metric. ...##
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Hyperbolic Deep Neural Networks: A Survey
[article]

2021
*
arXiv
*
pre-print

Recently, there has been a rising surge of momentum

arXiv:2101.04562v3
fatcat:yqj4zohrqjbplpsdy5f5uglnbu
*for*deep representation learning in hyperbolic spaces due to theirhigh capacity of modeling data like knowledge*graphs*or synonym hierarchies, possessing ... We refer to the model as hyperbolic deep neural*network*in this paper. ... ACKNOWLEDGMENTS The authors would like to thank Octavian-Eugen Ganea, from MIT*for*the useful discussion. ...##
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On the treeness of internet latency and bandwidth

2009
*
Proceedings of the eleventh international joint conference on Measurement and modeling of computer systems - SIGMETRICS '09
*

Acknowledgments We thank our shepherd, Jia Wang, and reviewers

doi:10.1145/1555349.1555357
dblp:conf/sigmetrics/RamasubramanianMKBGA09
fatcat:p76v7oe2lne6hngrby4uguawy4
*for*their insightful comments and feedback. ... There is a convenient*condition*called the*Four**Points**Condition*(4PC) to verify whether a set of measures is a tree metric. ... Figure 1 : 1*Four*-*Points**Condition*:*For*any*four*nodes and the three sums of distinct pairs of distances between them, the highest sum is equal to the second highest iff the nodes and distances are embeddable ...##
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On the Hyperbolicity of Small-World and Tree-Like Random Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

On the other hand,

doi:10.1007/978-3-642-35261-4_31
fatcat:v4o3zxazprd6dgc3tmmkcvyjum
*for*a class of tree-like*graphs*called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the small-world*graph*constructions ... Here, we consider Gromov's notion of δ-hyperbolicity, and we establish several positive and negative results*for*small-world and tree-like random*graph*models. ... Definition 1 (Gromov's*four*-*point**condition*). ...##
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Convergence of Combinatorial Gravity
[article]

2022
*
arXiv
*
pre-print

Our construction relies crucially on the Ollivier

arXiv:2102.02356v3
fatcat:seaeq4kvznhk7cokaz3iji3ig4
*curvature*of optimal transport theory. Our methods also allow us to define an analogous discrete action*for*Klein-Gordon fields. ... to the manifold in the sense of*Gromov*-Hausdorff). ... As an immediate corollary we thus have: There are*four*intrinsic*scales*in this problem: the*Gromov*-Hausdorff error ε, the radius δ of the balls defining the the uniform measures in G and M, the distance ...##
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Topological implications of negative curvature for biological and social networks

2014
*
Physical Review E
*

*Network*measures that reflect the most salient properties of complex large-

*scale*

*networks*are in high demand in the

*network*research community. ... In this paper we adapt a combinatorial measure of negative

*curvature*(also called hyperbolicity) to parameterized finite

*networks*, and show that a variety of biological and social

*networks*are hyperbolic ... so-called "

*four*-

*point*

*condition*." ...

##
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Correlations in cellular patterns

1994
*
Philosophical Magazine B
*

INTRODUCTION
Motivations The most important open issue in the theory of soap froth and cellular patterns has been,

doi:10.1080/01418639408240118
fatcat:qwq62g3xrnadvn5notyzxf3dgi
*for**four*decades now, that of the*scaling*state. ... The reason*for*this is that its sum over the*network*is a constant, independent of the*network*itself, by virtue of the Euler theorem. ...
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