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Scaled Gromov Four-Point Condition for Network Graph Curvature Computation

Edmond Jonckheere, Poonsuk Lohsoonthorn, Fariba Ariaei
2011 Internet Mathematics  
In this paper, we extend the concept of 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  ... 
doi:10.1080/15427951.2011.601233 fatcat:rtp5lxnru5earjmxsvnxcblury

Why did the shape of your network change? (On detecting network anomalies via non-local curvatures) [article]

Bhaskar DasGupta, Mano Vikash Janardhanan, Farzane Yahyanejad
2019 arXiv   pre-print
Based on these measures, we precisely formulate several 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.  ... 
arXiv:1808.05676v3 fatcat:nlc4poac2jhurfoetsglwl2rne

A Geometric Distance Oracle for Large Real-World Graphs [article]

Deepak Ajwani, W. Sean Kennedy, Alessandra Sala, Iraj Saniee
2014 arXiv   pre-print
The second set leverages 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)  ... 
arXiv:1404.5002v1 fatcat:55auyg3o2jfdvdy74craz3siea

On the Hyperbolicity of Large-Scale Networks [article]

W. Sean Kennedy, Onuttom Narayan, Iraj Saniee
2013 arXiv   pre-print
Finally, we present practical 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.  ... 
arXiv:1307.0031v1 fatcat:xk5ejb66mjcd3gngpvjhnt367i

Ricci curvature of the Internet topology

Chien-Chun Ni, Yu-Yao Lin, Jie Gao, Xianfeng David Gu, Emil Saucan
2015 2015 IEEE Conference on Computer Communications (INFOCOM)  
It is a more local measure which allows us to understand the distribution of 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.  ... 
doi:10.1109/infocom.2015.7218668 dblp:conf/infocom/NiLGGS15 fatcat:tsobsp2rg5ft3ge3cj74eeyujy

Emergence of the Circle in a Statistical Model of Random Cubic Graphs [article]

Christy Kelly, Carlo Trugenberger, Fabio Biancalana
2021 arXiv   pre-print
We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular 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  ... 
arXiv:2008.11779v3 fatcat:4sopddxir5f45kr7aar33cix3a

A Review of and Some Results for Ollivier–Ricci Network Curvature

Nazanin Azarhooshang, Prithviraj Sengupta, Bhaskar DasGupta
2020 Mathematics  
In the penultimate section we review a method based on the linear sketching technique 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  ... 
doi:10.3390/math8091416 fatcat:epqbnmlnqzh6hixpfgob2vwnk4

Negatively Curved Graphs [article]

Matthew Yancey
2021 arXiv   pre-print
Many applications in 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.  ... 
arXiv:1512.01281v3 fatcat:hb5hkygsprehhj3g6hzzn3u3ay

Efficient estimation of a Gromov–Hausdorff distance between unweighted graphs [article]

Vladyslav Oles, Nathan Lemons, Alexander Panchenko
2021 arXiv   pre-print
The algorithm finds the mGH distances exactly on most 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.  ... 
arXiv:1909.09772v8 fatcat:w5jmdinvbbanpijv7osqpr26ue

Hyperbolic Deep Neural Networks: A Survey [article]

Wei Peng, Tuomas Varanka, Abdelrahman Mostafa, Henglin Shi, Guoying Zhao
2021 arXiv   pre-print
Recently, there has been a rising surge of momentum 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.  ... 
arXiv:2101.04562v3 fatcat:yqj4zohrqjbplpsdy5f5uglnbu

On the treeness of internet latency and bandwidth

Venugopalan Ramasubramanian, Dahlia Malkhi, Fabian Kuhn, Mahesh Balakrishnan, Archit Gupta, Aditya Akella
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 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  ... 
doi:10.1145/1555349.1555357 dblp:conf/sigmetrics/RamasubramanianMKBGA09 fatcat:p76v7oe2lne6hngrby4uguawy4

On the Hyperbolicity of Small-World and Tree-Like Random Graphs [chapter]

Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney
2012 Lecture Notes in Computer Science  
On the other hand, 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).  ... 
doi:10.1007/978-3-642-35261-4_31 fatcat:v4o3zxazprd6dgc3tmmkcvyjum

Convergence of Combinatorial Gravity [article]

Christy Kelly, Carlo Trugenberger, Fabio Biancalana
2022 arXiv   pre-print
Our construction relies crucially on the Ollivier 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  ... 
arXiv:2102.02356v3 fatcat:seaeq4kvznhk7cokaz3iji3ig4

Topological implications of negative curvature for biological and social networks

Réka Albert, Bhaskar DasGupta, Nasim Mobasheri
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."  ... 
doi:10.1103/physreve.89.032811 pmid:24730903 fatcat:aerrdi32ura5rf7ff4d2etwfwi

Correlations in cellular patterns

Marcelo O. Mangasco
1994 Philosophical Magazine B  
INTRODUCTION Motivations The most important open issue in the theory of soap froth and cellular patterns has been, 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.  ... 
doi:10.1080/01418639408240118 fatcat:qwq62g3xrnadvn5notyzxf3dgi
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