## Hyperbolic Embedding of Internet Graphs for Distance Estimation and Overlay Construction

### Cached

### Download Links

Venue: | IEEE/ACM Transactions on Networking, To Appear |

Citations: | 12 - 0 self |

### BibTeX

@ARTICLE{Shavitt_hyperbolicembedding,

author = {Yuval Shavitt and Senior Member and Tomer Tankel and Student Member},

title = {Hyperbolic Embedding of Internet Graphs for Distance Estimation and Overlay Construction},

journal = {IEEE/ACM Transactions on Networking, To Appear},

year = {}

}

### OpenURL

### Abstract

Abstract—Estimating distances in the Internet has been studied in the recent years due to its ability to improve the performance of many applications, e.g., in the peer-to-peer realm. One scalable approach to estimate distances between nodes is to embed the nodes in some dimensional geometric space and to use the pair distances in this space as the estimate for the real distances. Several algorithms were suggested in the past to do this in low dimensional Euclidean spaces. It was noted in recent years that the Internet structure has a highly connected core and long stretched tendrils, and that most of the routing paths between nodes in the tendrils pass through the core. Therefore, we suggest in this work, to embed the Internet distance metric in a hyperbolic space where routes are bent toward the center. We found that if the curvature, that defines the extend of the bending, is selected in the adequate range, the accuracy of Internet distance embedding can be improved. We demonstrate the strength of our hyperbolic embedding with two applications: selecting the closest server and building an application level multicast tree. For the latter, we present a distributed algorithm for building geometric multicast trees that achieve good trade-offs between delay (stretch) and load (stress). We also present a new efficient centralized embedding algorithm that enables the accurate embedding of short distances, something that have never been done before. I.

### Citations

2917 | A Scalable Content-Addressable Network
- Ratnasamy, Francis, et al.
- 2001
(Show Context)
Citation Context ...group nodes. Otherwise, one may build a tree where the delay to some nodes is a large multiple of the unicast delay. The first scalable approach for building application layer multicast trees was CAN =-=[18]-=-, [19] and its derivatives [20], [21]. Due to the high accuracy of our embedding we are presenting smaller stretch factors for distances, i.e., the delays on our trees are shorter, while maintaining g... |

2400 | Emergence of scaling in random networks
- Barabási, Albert
- 1999
(Show Context)
Citation Context ...retical models to describe its evolvement. Faloutsos et al. [7] experimentally discovered that the degree distribution of the Internet AS and router level graphs obey a power law. Barabási and Albert =-=[8]-=-, [9] developed an evolutionary model of preferential attachment, that can be used for generating topologies with power-law degree distributions. Fig. 1. An eight-node graph example in . Around the co... |

1320 | On Power-law Relationships of the Internet Topology
- Faloutsos, Faloutsos, et al.
- 1999
(Show Context)
Citation Context ...et structure has been the subject of many recent works. Researchers have looked at various features of the Internet graph, and proposed theoretical models to describe its evolvement. Faloutsos et al. =-=[7]-=- experimentally discovered that the degree distribution of the Internet AS and router level graphs obey a power law. Barabási and Albert [8], [9] developed an evolutionary model of preferential attach... |

631 | Scalable application layer multicast
- Banerjee, Bhattacharjee, et al.
- 2002
(Show Context)
Citation Context ...ibution, namely, most of our tree links are not congested. Fig. 9. RouteViews AS graph with random weights: 3-D multicast tree layout. An alternative tree-first approach is NICE application multicast =-=[22]-=-, which creates a hierarchy of clusters while selecting the same or adjacent cluster for all nodes that are “close by”. CAN and NICE both have low, and thus scalable, link stress and control overhead.... |

546 | Predicting internet network distance with coordinates-based approaches
- Ng, Zhang
- 2002
(Show Context)
Citation Context ...er nodes is calculated by minimizing the distortion of the distances from this node to several or all Tracers. Euclidean embedding for predicting network distances was first suggested by Ng and Zhang =-=[1]-=-, which named it Global Network Positioning (GNP). Lim et al. [2] suggest to use uncorrelated and orthogonal Cartesian coordinates to replace the minimization suggested at GNP. Tang and Crovella [3] s... |

354 | Characterizing the Internet hierarchy from multiple vantage points
- Subramanian, Agarwal, et al.
- 2002
(Show Context)
Citation Context ...reduces the tree distance significantly, then the node rewires to the new parent. The algorithm 2 Following are the parameter values for the IGT algorithm Name [Fig #,line] Value Stretch × ηTP”-” CTh =-=[10,5]-=- 2 DEGMax [10,8] 15 DTh [11,2 − 4] .8 ≤ 12 DTh 1.0 ≤ 18 DTh 1.3 ≤ 24 DTh 1.5 > 24 RewireTh [12,7] 1.3 UnRewTh [12,7] 1.01 AbortRewire [12,14] .07 RewireSweeps [12,3] 10 Algorithm IGT(V = {v1 . . . vm}... |

304 | ALMI: An Application level Multicast Infrastructure
- Pendarakis, Shi, et al.
- 2001
(Show Context)
Citation Context ...et, our performance even slightly supersedes IDMaps, achieving, at least, 99 correct answers in 99 of the mirror group experiments. C. Application Level Multicast In application level multicast [16], =-=[17]-=-, we wish to build a multicast tree without network support. To make the tree efficient, we need to know the distances among the multicast group nodes. Otherwise, one may build a tree where the delay ... |

265 | Idmaps: A global internet host distance estimation service
- Francis, Jamin, et al.
(Show Context)
Citation Context ...s of peering data sets were collected weekly starting March 2001. To increase the confidence each experiment was conducted using 3 sets of random weights per each of the peering topologies. Following =-=[15]-=- we randomly selected 10 mirror servers and estimated the closet mirror to each of the rest of the graph nodes Fig. 7. DIMES router map relative error: (a) delay and (b) minimum hop. acting as clients... |

236 |
Yoid: Extending the Internet Multicast Architecture. Available at http://www.aciri.org/yoid/docs/index.html
- FRANCIS
- 2000
(Show Context)
Citation Context ... dataset, our performance even slightly supersedes IDMaps, achieving, at least, 99 correct answers in 99 of the mirror group experiments. C. Application Level Multicast In application level multicast =-=[16]-=-, [17], we wish to build a multicast tree without network support. To make the tree efficient, we need to know the distances among the multicast group nodes. Otherwise, one may build a tree where the ... |

203 | Topology of evolving networks: Local events and universality
- Albert, Barabasi
- 2000
(Show Context)
Citation Context ...al models to describe its evolvement. Faloutsos et al. [7] experimentally discovered that the degree distribution of the Internet AS and router level graphs obey a power law. Barabási and Albert [8], =-=[9]-=- developed an evolutionary model of preferential attachment, that can be used for generating topologies with power-law degree distributions. Fig. 1. An eight-node graph example in . Around the core of... |

164 | Virtual Landmarks for the Internet
- Tang, Crovella
- 2003
(Show Context)
Citation Context ...g [1], which named it Global Network Positioning (GNP). Lim et al. [2] suggest to use uncorrelated and orthogonal Cartesian coordinates to replace the minimization suggested at GNP. Tang and Crovella =-=[3]-=- suggest to use Lipschitz embedding, which ignore the distance between Tracers, and thus is less accurate. Manuscript received January 21, 2005; revised December 14, 2005, and July 19, 2006; approved ... |

160 | DIMES: Let the internet measure itself
- Shavitt, Shir
- 2005
(Show Context)
Citation Context ..., and its gradient with respect to is given by III. HYPERBOLIC EMBEDDING IN REAL TOPOLOGIES (22) In Section III-B we use the Internet router topology extracted from Tel-Aviv University DIMES database =-=[13]-=- dated October 23–24, 2005. In Sections III-C and III-D we use the AS topology instance from the University of Oregon RouteViews database dated March 31, 2001. A. Experiment Details and Legend We use ... |

139 |
The Geometry and Topology of Three Manifolds
- Thurston
- 1979
(Show Context)
Citation Context ... for the case of . The hyperbolic distance between , denoted , is given [12], 4.1 by where With the contracted element of arc length 1 This picture and the discussion of Poincaré disk, are taken from =-=[11]-=-, 2.1. (1) (2) (3) (4)s4 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 1, FEBRUARY 2008 the hyperbolic distance is also contracted by , i.e., Let denote an input metric, being embedded into a Hype... |

134 | Big-bang simulation for embedding network distances in euclidean space
- Shavitt, Tankel
- 2003
(Show Context)
Citation Context ...neering, Tel-Aviv University, Israel (e-mail: shavitt@eng.tau.ac.il; tankel@eng.tau.ac.il). Digital Object Identifier 10.1109/TNET.2007.899021 1063-6692/$25.00 © 2008 IEEE Recently, we introduced BBS =-=[4]-=-, a new numerical method for Euclidean embedding. Euclidean BBS had achieved the lowest embedding distortion with relatively low complexity compared to other numerical methods. In particular for AP em... |

97 | The origin of power laws in internet topologies revisited
- Chen, Chang, et al.
- 2002
(Show Context)
Citation Context ...tance range. B. Server Selection This experiment used two Internet AS data sets, the U. of Oregon RouteViews dataset and the combined RouteView with looking glass and router registry, as described in =-=[14]-=-. The nine couples of peering data sets were collected weekly starting Mar. 2001. To increase the confidence each experiment was conducted using 3 sets of random weights per each of the peering topolo... |

79 | Building Topology-Aware Overlays Using Global Soft-State
- Xu, Tang, et al.
- 2003
(Show Context)
Citation Context ... a tree where the delay to some nodes is a large multiple of the unicast delay. The first scalable approach for building application layer multicast trees was CAN [18], [19] and its derivatives [20], =-=[21]-=-. Due to the high accuracy of our embedding we are presenting smaller stretch factors for distances, i.e., the delays on our trees are shorter, while maintaining good stress factor distribution, namel... |

70 | A simple conceptual model for the Internet topology
- Tauro, Palmer, et al.
(Show Context)
Citation Context ...While we achieved good embedding in [4] the results are far from perfect due to the Internet AS topology structure, which was shown to have a core in the middle and many tendrils connected to it [5], =-=[6]-=-. To understand the problem, consider embedding of the Internet in two dimensions. If the tendrils are placed with the correct distance from the core and are well spaced in all directions, the distanc... |

42 |
The origin of power laws
- Chen, Chang, et al.
- 2002
(Show Context)
Citation Context ...nge. B. Server Selection This experiment used two Internet AS data sets, the University of Oregon RouteViews dataset and the combined RouteView with looking glass and router registry, as described in =-=[14]-=-. The nine couples of peering data sets were collected weekly starting March 2001. To increase the confidence each experiment was conducted using 3 sets of random weights per each of the peering topol... |

42 | Building low-maintenance expressways for p2p systems
- Xu, Zhang
- 2002
(Show Context)
Citation Context ... build a tree where the delay to some nodes is a large multiple of the unicast delay. The first scalable approach for building application layer multicast trees was CAN [18], [19] and its derivatives =-=[20]-=-, [21]. Due to the high accuracy of our embedding we are presenting smaller stretch factors for distances, i.e., the delays on our trees are shorter, while maintaining good stress factor distribution,... |

32 |
Hyperbolic Geometry
- Anderson
- 1999
(Show Context)
Citation Context ...lidean image of a hyperbolic object, Fig. 2 right pan, as it moves away from the origin, shrinks in size roughly in proportion to the Euclidean distance from (when this distance is small). 1 Anderson =-=[12]-=- covers in details the upper half-plane model and has a chapter on the Poincaré disk model in case . B. Analysis of Hyperbolic Space In order to be able to embed an input metric in a geometric space, ... |

28 | A Study of the Performance Potential of DHT-based Overlays
- Jain, Mahajan, et al.
- 2003
(Show Context)
Citation Context ...rect unicast path. Link Stress is defined per link of the underlying topology and counts the number of identical packets sent between members of M over that link. This definition of stress, following =-=[23]-=-, is from the network’s perspective, rather than the application’s. Fig. 9(a) depicts the three-dimensional multicast tree constructed by our algorithm. Fig. 9(b) and (c) depict the hyperbolic distanc... |

8 |
A Scalable Content-Addressable Network. SIGCOMM’01
- Ratnasamy, Francis, et al.
- 2001
(Show Context)
Citation Context ...group nodes. Otherwise, one may build a tree where the delay to some nodes is a large multiple of the unicast delay. The first scalable approach for building application layer multicast trees was CAN =-=[18]-=-, [19] and its derivatives [20], [21]. Due to the high accuracy of our embedding we are presenting smaller stretch factors for distances, i.e., the delays on our trees are shorter, while (b) Complemnt... |

6 |
Topology-Aware Overlay Construction and Server Selection
- Ratnasamy, Handley, et al.
- 2002
(Show Context)
Citation Context ...nodes. Otherwise, one may build a tree where the delay to some nodes is a large multiple of the unicast delay. The first scalable approach for building application layer multicast trees was CAN [18], =-=[19]-=- and its derivatives [20], [21]. Due to the high accuracy of our embedding we are presenting smaller stretch factors for distances, i.e., the delays on our trees are shorter, while maintaining good st... |

2 |
On the curvature of the interent and its usage for overlay construction and distance estimation,” in
- Shavitt, Tankel
- 1986
(Show Context)
Citation Context ...dimensional Hyperbolic space, and run our IGT algorithm from 40 different source nodes. We also use TP embedding in five-dimensional Euclidean space, and run the Geometric-Multicast-Tree algorithm of =-=[24]-=-, Fig. 10 from these source nodes. Fig. 13(a) depicts the calculation results on the AS topology instance from the University of Oregon RouteViews database dated January 2, 2000, for a group of member... |

2 | Provisioning of network distance service on the internet - Lim, Hou, et al. - 2005 |

1 | Provisioning of network distance service on - Lim, Hou, et al. - 2005 |