## Network Topology Generators: Degree-Based vs. Structural (2002)

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Citations: | 164 - 14 self |

### BibTeX

@MISC{Tangmunarunkit02networktopology,

author = {Hongsuda Tangmunarunkit and Ramesh Govindan and Sugih Jamin and Scott Shenker},

title = {Network Topology Generators: Degree-Based vs. Structural},

year = {2002}

}

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### Abstract

Following the long-held belief that the Internet is hierarchical, the network topology generators most widely used by the Internet research community, Transit-Stub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal paper by Faloutsos et al. revealed that the Internet's degree distribution is a power-law. Because the degree distributions produced by the Transit-Stub and Tiers generators are not power-laws, the research community has largely dismissed them as inadequate and proposed new network generators that attempt to generate graphs with power-law degree distributions.

### Citations

2090 | Emergence of scaling in random networks
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Citation Context ...er generators. This conclusion holds for all other degree-based generators we tested. Figure 2(j-l) shows our three metrics for four other proposed degree-based generators: Brite version 1.0 [28], BA =-=[4]-=-, BT [8] and Inet [23]. All of these can be classified, along with the PLRG, as having high expansion and resilience, and low distortion. These generators all produce graphs with a power-law degree di... |

1906 |
Collective dynamics of ’small-world’ networks
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Citation Context ...(Section 1). Two other recent pieces of work examine local properties of network topologies. Bu and Towsley [8] find that degree-based generators differ significantly in their clustering coefficients =-=[47]-=-. Their work proposes an alternative degree-based generator that more closely matches the clustering behavior of the measured AS graph. For completeness, we have incorporated both the clustering metri... |

1789 | Random Graphs
- BOLLOBAS
- 2001
(Show Context)
Citation Context ...f real networks. The first network topology generator to become widely used in protocol simulations was developed by Waxman [48]. This generator is a variant of the classical Erdos-Renyi random graph =-=[6]-=-; its link creation probabilities are biased by Euclidean distance between the link endpoints. A later line of research, noting that real network topologies have a non-random structure, emphasized the... |

1247 | On power-law relationships of the Internet topology
- Faloutsos, Faloutsos, et al.
(Show Context)
Citation Context ...hese as structural generators because of their focus on the hierarchical structure of networks. These structural generators reigned supreme until the appearance of a seminal paper by Faloutsos et al. =-=[17]-=- in 1999. In that paper, the authors used measurements of the router-level and AS-level Internet graphs—the former having routers as nodes and the latter having ASs as nodes—to investigate (among othe... |

796 | A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
- Karpis, Kumar
(Show Context)
Citation Context ...and some graph-theoretic metrics that have plausible networking interpretations, are listed below: Neighborhood size (or expansion) [35]. Resilience, the size of a cut-set for a balanced bi-partition =-=[24]-=-. Distortion, or the minimum communication cost spanning tree [21]. Node diameter (or eccentricity) distribution [51]. Eigenvalue distribution [17]. Size of a vertex cover [33]. Biconnectivity (number... |

760 | Collective Dynamics of ‘Small-World’ networks. Nature 393:440–442 - Watts, Strogatz - 1998 |

639 |
Routing of multipoint connections
- Waxman
- 1988
(Show Context)
Citation Context ...t to create network topologies that embody the fundamental characteristics of real networks. The first network topology generator to become widely used in protocol simulations was developed by Waxman =-=[48]-=-. This generator is a variant of the classical Erdos-Renyi random graph [6]; its link creation probabilities are biased by Euclidean distance between the link endpoints. A later line of research, noti... |

417 | Modeling Internet topology
- Calvert, Doar, et al.
- 1997
(Show Context)
Citation Context .... In other words, there is a hierarchy imposed on nodes... This reasoning quickly became accepted wisdom and, for many years, the network generators resulting from this line of research, Transit-Stub =-=[10]-=- and Tiers [14], were considered state-of-the-art. In what follows, we will refer to these as structural generators because of their focus on the hierarchical structure of networks. These structural g... |

417 | On inferring autonomous system relationships in the Internet - Gao - 2001 |

346 | Characterizing the internet hierarchy from multiple vantage points
- Subramanian, Agarwal, et al.
(Show Context)
Citation Context ...e literature on routing hierarchies, we are not aware of much work that has attempted to measure (as opposed to create, or utilize) hierarchy in network topologies. Two notable, and related, examples =-=[18, 40]-=-, describe techniques for inferring hierarchical relationships (e.g., provider-customer) in the AS topology. The latter work also classifies ASs into a five-level hierarchy. Somewhat orthogonal to the... |

333 | BRITE: An approach to universal topology generation
- Medina, Lakhina, et al.
- 2001
(Show Context)
Citation Context ...number of proposals for topology generators that are designed primarily to match the Internet’s degree distribution and do not attempt to model the Internet’s hierarchical structure; for example, see =-=[23, 28, 2, 31, 1, 8]-=-. These degree-based topology generators embody the implicit assumption that it is more important to match a certain local property—the degree distribution—than to capture the large-scale hierarchical... |

332 | A random graph model for massive graphs
- Aiello, Lu
- 2000
(Show Context)
Citation Context ...number of proposals for topology generators that are designed primarily to match the Internet’s degree distribution and do not attempt to model the Internet’s hierarchical structure; for example, see =-=[23, 28, 2, 31, 1, 8]-=-. These degree-based topology generators embody the implicit assumption that it is more important to match a certain local property—the degree distribution—than to capture the large-scale hierarchical... |

329 | Heuristics for internet map discovery
- Govindan, Tangmunarunkit
- 2000
(Show Context)
Citation Context ...ond representation is at the router level, where routers are nodes and an edge indicates that the corresponding routers are separated by one IP-level hop. The router graph comes from the SCAN project =-=[20]-=- which uses a series of traceroute measurements to map the Internet. The router graph represents the Internet at a much finer level of granularity, and has roughly 17 times more nodes and links than t... |

315 | Probabilistic approximation of metric spaces and its algorithmic applications
- Bartal
- 1996
(Show Context)
Citation Context ...keep computation times reasonable) other nodes, then average the sizes and distortion values of all subgraphs of the same radius. We also use a simple divide and conquer algorithm suggested by Bartal =-=[5]-=-. This approach is known to compute distortions to ; /2143 within of the optimal solution. We should note that for all the topologies except mesh our own heuristics resulted in smaller distortion valu... |

305 | The web as a graph: Measurements, models, and methods
- Kleinberg, Kumar, et al.
- 1999
(Show Context)
Citation Context .... Watts and Strogatz [47] found that many real-world networks, such as the actor collaboration network and a section of the power grid, are well-modeled by the smallworld phenomenon. Kleinberg et al. =-=[25]-=- analyzed properties of the World-Wide Web graph and proposed a new family of random graph models. Aiello et al. [1] proposed a random graph model for massive graphs and showed that this model capture... |

265 | The end-to-end effects of internet path selection
- Savage, Collins, et al.
- 1999
(Show Context)
Citation Context ...hey also only reflect the topology and do not contain any information about the speed of the links. We do, however, approximately model an aspect of reality that has been shown to impact path lengths =-=[43, 38]-=- in Internet topologies—policy routing. To measure the properties of the Internet graphs and the generated graphs, we use a set of three topology metrics described in Section 3.2. These metrics are in... |

237 | Emergence of Scaling - Barabási, Albert - 1999 |

222 | A quantitative comparison of graph-based models for Internet topology
- Zegura, Calvert, et al.
- 1997
(Show Context)
Citation Context ...lidean distance between the link endpoints. A later line of research, noting that real network topologies have a non-random structure, emphasized the fundamental role of hierarchy. The following from =-=[51]-=- reflects this observation: It should be noted that sometimes topology generators are used to tickle subtle bugs in protocols. However, for this purpose the emphasis is not on finding realistic topolo... |

207 | On distinguishing between internet power law topology generators
- Bu, Towsley
- 2002
(Show Context)
Citation Context ...number of proposals for topology generators that are designed primarily to match the Internet’s degree distribution and do not attempt to model the Internet’s hierarchical structure; for example, see =-=[23, 28, 2, 31, 1, 8]-=-. These degree-based topology generators embody the implicit assumption that it is more important to match a certain local property—the degree distribution—than to capture the large-scale hierarchical... |

192 | Measuring Link Bandwidths Using a Deterministic Model of Packet Delay
- Lai, Baker
- 2000
(Show Context)
Citation Context ...ther the RL nor the AS graph contains any indication of the capacity of the underlying transmission link (or shared medium). Although techniques for estimating link capacities along a path are known (=-=[15, 26]-=-), they are reported to be fairly time consuming and, to our knowledge, no one has attempted to annotate the router-level graph of the entire Internet with link capacity information. We don’t know how... |

189 | On the Origin of Power Laws in Internet Topologies
- Medina, Matta, et al.
(Show Context)
Citation Context ...iformly or exponentially assigned link weights. Some of the topology metrics used in our paper are based on the metrics introduced in these papers. Also directly relevant is the work of Medina et al. =-=[29]-=-. They too compare random graph generators (such as Waxman), and hierarchical generators (such as Transit-Stub) to degree-based generators (such as the BRITE generator [28]). Their metrics for compari... |

185 | Topology of evolving networks: local events and universality
- Albert, Barabasi
(Show Context)
Citation Context |

157 | A Better Model for Generating Test Networks
- Doar
(Show Context)
Citation Context ...s, there is a hierarchy imposed on nodes... This reasoning quickly became accepted wisdom and, for many years, the network generators resulting from this line of research, Transit-Stub [10] and Tiers =-=[14]-=-, were considered state-of-the-art. In what follows, we will refer to these as structural generators because of their focus on the hierarchical structure of networks. These structural generators reign... |

143 | Inet: Internet Topology Generator
- Jin, Chen, et al.
- 2000
(Show Context)
Citation Context |

121 |
On Routes and Multicast Trees in the Internet
- Pansiot, Grad
- 1998
(Show Context)
Citation Context ...possible without developments in Internet router-level topology discovery. Early work in this area used traceroutes from a small set of sources to several thousand hosts to compute a router-level map =-=[32]-=-. Subsequent work improved the coverage of the Internet address space by randomly selecting IP addresses [39], randomly selecting addresses from route entries in BGP tables [9], using a precomputed se... |

106 | Scaling of multicast trees: Comments on the Chuang-Sirbu scaling law
- Phillips, Shenker, et al.
- 1999
(Show Context)
Citation Context ...ork also documents several techniques for improving completeness of the inferred topologies. Several papers have addressed the impact of topology on protocol performance. For example, Phillips et al. =-=[35]-=- showed that graphs with exponentially increasing neighborhood sizes (i.e., number of nodes within a certain radius increases exponentially with radius) approximately obey the Chuang-Sirbu multicast s... |

80 |
Mapping the internet
- Burch, Cheswick
- 1999
(Show Context)
Citation Context ...ute a router-level map [32]. Subsequent work improved the coverage of the Internet address space by randomly selecting IP addresses [39], randomly selecting addresses from route entries in BGP tables =-=[9]-=-, using a precomputed set of Web sites [13], or using heuristics to infer addressable parts of the IP space [20]. This last work also documents several techniques for improving completeness of the inf... |

71 | Optimum communication spanning trees
- Hu
- 1974
(Show Context)
Citation Context ...terpretations, are listed below: Neighborhood size (or expansion) [35]. Resilience, the size of a cut-set for a balanced bi-partition [24]. Distortion, or the minimum communication cost spanning tree =-=[21]-=-. Node diameter (or eccentricity) distribution [51]. Eigenvalue distribution [17]. Size of a vertex cover [33]. Biconnectivity (number of biconnected components) [51]. The average pairwise shortest pa... |

62 | Analysis of the autonomous system network topology
- Magoni, Pansiot
- 2001
(Show Context)
Citation Context ... in Section 1, our work merely assumes that the degree distribution is well approximated by a heavy tail and does not depend on the exact mathematical form of the distribution. Finally, Magoni et al. =-=[27]-=- study various graph theoretic and time evolution properties of the AS topology. Our work would not have been possible without developments in Internet router-level topology discovery. Early work in t... |

56 |
Attack and error tolerance of complex networks
- Albert, Jeong, et al.
- 2000
(Show Context)
Citation Context ...ead but extend their study using a larger collection of metrics, adding measured networks and degree-based generators, and explicitly analyzing the degree of hierarchy. More recently, Barabasi et al. =-=[3]-=- have attempted to quantify the attack and error tolerance of random graphs and real-world “scale-free” networks. Finally, van Mieghem et al. [45] have shown that the Internet’s hop count distribution... |

52 | Modeling the branching characteristics and efficiency gains in global multicast trees
- Chalmers, Almeroth
- 2001
(Show Context)
Citation Context ...ood sizes (i.e., number of nodes within a certain radius increases exponentially with radius) approximately obey the Chuang-Sirbu multicast scaling law. In closely related work, Almeroth and Chambers =-=[11]-=- considered a variety of metrics for the efficiency of multicast trees. Wong and Katz [49] found that the amount of multicast state from randomly placed receivers differs qualitatively with different ... |

47 | Internet path inflation due to policy routing - Tangmunarunkit, Govindan, et al. - 2001 |

40 | On characterizing network topologies and analyzing their impact on protocol design - Radoslavov, Tangmunarunkit, et al. - 2000 |

32 | An analysis of multicast forwarding state scalability
- Wong, Katz
- 2000
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Citation Context ...us) approximately obey the Chuang-Sirbu multicast scaling law. In closely related work, Almeroth and Chambers [11] considered a variety of metrics for the efficiency of multicast trees. Wong and Katz =-=[49]-=- found that the amount of multicast state from randomly placed receivers differs qualitatively with different topologies. Radoslavov et al. [36] found similar results for other kinds of protocol perfo... |

28 | dispersion graphs - As - 2012 |

24 |
Constructing disjoint paths on expander graphs
- Peleg, Upfal
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(Show Context)
Citation Context ..., not the radius of the ball itself—to factor out the fact that An node bipartite graph from a vertex set to a vertex set is said to be an expander if, every set of nodes in has at least neighbors in =-=[34]-=-. For each node in the network, we grow balls with increasing radius. For the subgraph formed by nodes within a ball, we compute the number of nodes as well as the resilience of the subgraph. We repea... |

22 | On Inferring AS-level Connectivity from BGP Route Tables
- CHANG, GOVINDAN, et al.
- 2001
(Show Context)
Citation Context ... Claffy [7] find that various properties of real-world graphs, including the degree distribution, are wellmodeled by a Weibull distribution. Using extensive measurements of the AS graph, Chang et al. =-=[12]-=- show that the degree distribution of the AS graph deviates significantly from a strict power-law fit. As we have discussed in Section 1, our work merely assumes that the degree distribution is well a... |

16 | A spectral analysis of the Internet topology
- Vukadinovic, Huang, et al.
- 2001
(Show Context)
Citation Context ... matches the clustering behavior of the measured AS graph. For completeness, we have incorporated both the clustering metric and the proposed generator in our analyses (Section 4). Vukadinovic et al. =-=[46]-=- evaluate the Laplacian eigenvalue spectrum of a variety of graphs, and conclude that the multiplicity of eigenvalues of value 1 differentiates AS graphs from grids and random trees. However, as claim... |

15 |
On the Enumeration of Trees
- RÉNYI
(Show Context)
Citation Context ...d with resilience, we define the + distortion for a topology to be the average distortion of a subgraph of nodes within a “ball” around a node in the topology. Computing the distortion can be NP-hard =-=[37]-=-. For the results described in this paper, we use the smallest distortion obtained by applying our own heuristics. 9 The tree ,%-' has . The random graph and the mesh each have .0/2143 [19]. Summary T... |

14 | Lecture notes on approximation algorithms", Volume I
- Motwani
- 1992
(Show Context)
Citation Context ... intuitive notion of the hierarchy in those graphs. To use this metric in the presence of multiple shortest paths, we had to use a weighted vertex cover. 18 We use well-known approximation algorithms =-=[30]-=- for computing weighted vertex covers. For all topologies, we compute link values using shortest path routing. In addition, for the AS and RL topologies, we use the simple policy model described in Se... |

13 |
Using pathchar to estimate link characteristics
- Downey
- 1999
(Show Context)
Citation Context ...ther the RL nor the AS graph contains any indication of the capacity of the underlying transmission link (or shared medium). Although techniques for estimating link capacities along a path are known (=-=[15, 26]-=-), they are reported to be fairly time consuming and, to our knowledge, no one has attempted to annotate the router-level graph of the entire Internet with link capacity information. We don’t know how... |

13 |
The Impact of Policy on Internet Paths
- TANGMUNARUNKIT, GOVINDAN, et al.
- 2001
(Show Context)
Citation Context ... ball is less than or equal to , and only include links that lie on policy-compliant paths to those nodes. To do so, we use a policy model that is slightly more sophisticated than the one reported in =-=[43]-=-. At the AS level, this policy model computes the shortest AS path between two nodes that does not violate provider-customer relationships (an example of a path that would violate these relationship i... |

11 | Extending Greedy Multicast Routing to Delay Sensitive Applications
- GOEL, MUNAGALA
- 1999
(Show Context)
Citation Context ...be NP-hard [37]. For the results described in this paper, we use the smallest distortion obtained by applying our own heuristics. 9 The tree ,%-' has . The random graph and the mesh each have .0/2143 =-=[19]-=-. Summary To more fully understand the distinctions made by our three metrics, we consider two other standard networks: a fullyconnected network and a linear chain. A fully-connected network has extre... |

9 |
Discovering Internet Topology. Unpublished manuscript
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(Show Context)
Citation Context ...routes from a small set of sources to several thousand hosts to compute a router-level map [32]. Subsequent work improved the coverage of the Internet address space by randomly selecting IP addresses =-=[39]-=-, randomly selecting addresses from route entries in BGP tables [9], using a precomputed set of Web sites [13], or using heuristics to infer addressable parts of the IP space [20]. This last work also... |

8 |
Thoughts on Router-level Topology Modeling. The end2end-interest Mailing List Archives
- Zegura
- 2001
(Show Context)
Citation Context ...ther large graphs (the smallest generated graph had 1000 nodes). Choosing a small (less than, say, 100 node) topology on which to run network simulations is an entirely separate question. As noted in =-=[50]-=-, a power-law distribution is almost meaningless if the number of nodes is small. With only a few nodes, it is unlikely that the degree distribution will be able to create the implicit hierarchy neces... |

7 |
A Scaling Law for the Hopcount
- Mieghem, Hooghiemstra, et al.
- 2000
(Show Context)
Citation Context ...he degree of hierarchy. More recently, Barabasi et al. [3] have attempted to quantify the attack and error tolerance of random graphs and real-world “scale-free” networks. Finally, van Mieghem et al. =-=[45]-=- have shown that the Internet’s hop count distribution (the distribution of path lengths in hops) is well modeled by that of a random graph with uniformly or exponentially assigned link weights. Some ... |

4 |
Internet Topology: Local Properties
- BROIDO, CLAFFY
- 2001
(Show Context)
Citation Context ...findings (and with the commonly held intuition that the AS graph is neither mesh-like nor tree-like). Also relevant to our work is recent work on the analysis of graph measurements. Broido and Claffy =-=[7]-=- find that various properties of real-world graphs, including the degree distribution, are wellmodeled by a Weibull distribution. Using extensive measurements of the AS graph, Chang et al. [12] show t... |

4 |
On Power-Laws In Network Topologies
- PALMER, STEFFEN
- 2000
(Show Context)
Citation Context |

3 |
Impact of topology on traceback techniques. Private communication
- PARK
(Show Context)
Citation Context ...a balanced bi-partition [24]. Distortion, or the minimum communication cost spanning tree [21]. Node diameter (or eccentricity) distribution [51]. Eigenvalue distribution [17]. Size of a vertex cover =-=[33]-=-. Biconnectivity (number of biconnected components) [51]. The average pairwise shortest path between nodes in the largest component under random failure (when nodes are removed from the graph randomly... |

3 | On the Enumeration of Trees - ENYI, A |