Ellen W. Zegura
College of Computing
Georgia Tech

• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
– Example: mathematical foundations of degree-based generation
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
• Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 2

domains/autonomous systems transit domains exchange point border routers peering hosts/endsystems routers stub domains access networks
Zegura - Mar 2002 IPAM Workshop Tutorial lowly worm
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• Graph representation
• Router-level modeling
– vertices are routers
– edges are one-hop IP connectivity
• Domain- (AS-) level modeling
– vertices are domains (ASes)
– edges are peering relationships
Zegura - Mar 2002 IPAM Workshop Tutorial 4

• Waxman (Waxman 1988)
– router level model capturing locality
• Transit-stub (Zegura 1997), Tiers (Doar 1997)
– router level model capturing hierarchy
• Inet (Jin 2000)
– AS level model based on degree sequence
• BRITE (Medina 2000)
– AS level model based on evolution
Zegura - Mar 2002 IPAM Workshop Tutorial 5

(Waxman 1988)
• Router level model
• Nodes placed at random in
2-d space with dimension L
• Probability of edge (u,v):
– ae^{-d/(bL)}, where d is
Euclidean distance (u,v), a and b are constants
• Models locality u d(u,v) v
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(Zegura 1997)
• Router level model
• Transit domains
– placed in 2-d space
– populated with routers
– connected to each other
• Stub domains
– placed in 2-d space
– populated with routers
– connected to transit domains
• Models hierarchy
Zegura - Mar 2002 IPAM Workshop Tutorial 7

• Oregon route view server; peers with routers to collect
BGP routing tables
• Data publicly available from Nov 97 to present (nlanr.org, routeviews.org)
• Faloutsos 1999
– degree sequence approximated by power law
– i.e., let f(d) be fraction of nodes with degree d , then f(d)
 d^

• Chen 2002
– Oregon data incomplete (but so is theirs!)
– degree sequence highly variable but not strict power law
Zegura - Mar 2002 IPAM Workshop Tutorial 8

(Jin 2000)
• Generate degree sequence
• Build spanning tree over nodes with degree larger than 1, using preferential connectivity
– randomly select node u not in tree
– join u to existing node v with probability d(v)/
 d(w)
• Connect degree 1 nodes using preferential connectivity
• Add remaining edges using preferential connectivity
Zegura - Mar 2002 IPAM Workshop Tutorial 9

(Medina 2000)
• Generate small backbone, with nodes placed:
– randomly or
– concentrated (skewed)
• Add nodes one at a time
(incremental growth)
• New node has constant # of edges connected using:
– preferential connectivity and/or
– locality
Zegura - Mar 2002 IPAM Workshop Tutorial 10

• General technique: traceroute, returns list of IP addresses on a path from source to destination
• Collection challenges:
S1
– obtaining sufficient traceroute origin points
– deciding set of destination IP addresses (for coverage)
– limiting traceroute load
• Postprocessing challenges:
– resolving aliases (which IP addresses belong to same router) source 0
D1
Zegura - Mar 2002 IPAM Workshop Tutorial destination 0
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• Lucent (Burch 1999)
– single source (Lucent), ~100k destinations
– emphasis: longitudinal study, visualization
• Skitter (Broido 2001)
– 20 sources (“monitors”), ~400k destinations
– emphasis: measurement repository, analysis
• Mercator (Govindan 2000)
– single source (but uses source routing), 150k interfaces
– emphasis: heuristics for map construction
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• Caveat: router-level mapping clearly incomplete, so conclusions are weak
• Observations:
– qualitatively similar to AS graph on a number of measures
– Weibull distributions good fit for number of quantities (including degree distribution)
Zegura - Mar 2002 IPAM Workshop Tutorial 13

• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
–
Example: mathematical foundations
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
• Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 14

(Mihail 2002)
• Given degree sequence d(1) >= d(2) >= … >= d(n)
• A degree sequence is realizable if there is a simple graph
(no self-loops or multiple links) with this sequence
• Necessary and sufficient condition for degree sequence to be realizable:
– for each subset of k highest degree nodes, degrees can be
“absorbed” within the nodes and the outside degrees
15 Zegura - Mar 2002 IPAM Workshop Tutorial

• Maintain residual degrees of vertices, d(v)
• Repeat until all vertices have been chosen:
– pick arbitrary vertex v
– add edges from v to d(v) vertices of highest residual degree
– update residual degrees
•
Note: order to pick v arbitrary
Zegura - Mar 2002 IPAM Workshop Tutorial 16

• Dense core
– pick v’s starting with high degree vertices
– will tend to connect high degree vertices
• Sparse core
– pick v’s starting with low degree vertices
– less likely to connect high degree vertices
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• Large topology (11000+ nodes, 32000+ edges)
• Dense core
– diameter 5
– average path length 3.6
• Sparse core
– diameter 29
– average path length 17.9
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• Start from any realization of degree sequence
• Pick two edges at random, (u,v) and
(s,t), with distinct endpoints
• If doesn’t disconnect graph, remove edges and insert (u,s) and (v,t)
• Result satisfies degree sequence
•
In the limit, reaches every possible connected realization with equal probability
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• Different starting points
• Snapshots, 25k, 50k, 100k, 300k, 600k iters
• Large topology, sparse initial core
– diameter: 29, 13, 11, 11, 10, 10
– avgspl: 5.6, 3.6, 3.4, 3.4, 3.4, 3.4
• Large topology, dense initial core
– diameter: 5, 10, 10, 10, 10, 10
– avgspl: 3.6, 3.2, 3.2, 3.4, 3.4, 3.4
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• Variants on evolutionary models
• Variants on degree-driven models
• Appeal of evolutionary
• Relationship to work on “networks” in general
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• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
– Example: mathematical foundations
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
• Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 22

• Question: what determines whether a topology generator is “good”?
• Essentially an unsolved (and hard) problem
– depends on what topologies are used for
• NOT “degree sequence follows a power law!”
Zegura - Mar 2002 IPAM Workshop Tutorial 23

• Path-related metrics
– diameter, shortest path length
• Clustering metrics
– neighborhood size (“expansion”), eigenvalue decomposition, clustering coefficient
• Robustness metrics
– resilience
• Hierarchy metrics
– link usage, size of layers
Zegura - Mar 2002 IPAM Workshop Tutorial 24

(Bu 2002)
• Defined by two measures:
– characteristic path length L = number of edges in shortest path between two vertices, averaged over all vertex pairs
– clustering coefficient C:
• take vertex v with k 
1 neighbors
• at most k(k-1)/2 edges among neighbors
• C(v) = fraction of k(k-1)/2 edges present v
• C = average clustering coefficient
• C >> C_random, L 
L_random k nodes
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• AS-level topologies satisfy small-world test
• Example Mar 00:
– L=3.7, L_random=3.8
– C=.39, C_random=.0023
• Example, Sept 01:
– L= 3.6, L_random=3.6
– C=.47, C_random=.0015
Zegura - Mar 2002 IPAM Workshop Tutorial 26

(Tangmunarunkit 2001)
• Goal: large-scale metrics that distinguish between classes of graphs
• Proposal: Expansion, resilience and distortion
– differentiate between canonical graphs (mesh, tree, random graph)
– differentiate between three types of generators
• random graph (e.g., Waxman)
• structural (e.g., Transit-Stub, Tiers)
• degree-based (e.g., PLRG, BRITE)
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• Signature: expansion, resilience, distortion
• Waxman: H H H (like random)
• Tiers: L H L
• Transit-stub: H L L (like tree)
• PLRG: H H L (like complete graph)
• Also: real topologies and other degree-based generators have H H L signature
Zegura - Mar 2002 IPAM Workshop Tutorial 28

• link-value measure
• see paper for details…
• bottom line: degree-based generators contain loose notion of hierarchy that is somewhat similar to loose notion in Internet
29 Zegura - Mar 2002 IPAM Workshop Tutorial

• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
– Example: mathematical foundations
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
• Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 30

• Internet routes are not hop-based shortest paths
• General policies:
– path between two nodes in a domain remains in that domain
– path between two nodes in two different domains traverses zero or more transit domains
Zegura - Mar 2002 IPAM Workshop Tutorial 31

• Use edge weights so that shortest-paths obey general policies
• Four weights (in order)
– intra-domain edges
– T-T edges
– S-T edges
– S-S edges
Zegura - Mar 2002 IPAM Workshop Tutorial 32

(Gao 2000)
• Problem: Routes determined by routing policy, including AS-level contractual
AS1 agreements
• Idea: label edges in AS-level graph as AS2
– provider-to-customer (customer pays provider for connectivity to rest of Internet)
– peer-to-peer (exchange traffic between customers free of charge)
– sibling-to-sibling (provide connectivity to rest of Internet for each other)
AS4
AS3
• Use BGP routing table entries
AS7
AS5
AS6
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• e.g., routing table entry = AS path 1849 702 701 1
• downhill path: all edges provider-to-customer or siblingto-sibling
• uphill path: all edges customer-to-provider or sibling-tosibling
• An AS path of a BGP routing table is:
– an uphill path followed by a downhill path (either path segment may be empty)…or...
– an uphill path followed by a peer-to-peer edge followed by a downhill path (either path segment may be empty)
Zegura - Mar 2002 IPAM Workshop Tutorial 34

• an uphill path followed by a downhill path
– AS4-AS2-AS1-AS3-AS5
– AS7-AS1-AS2
• an uphill path followed by a peer-to-peer edge followed by a downhill path
– AS5-AS6-AS3-AS5
– AS6-AS3-AS2-AS4
Zegura - Mar 2002 IPAM Workshop Tutorial
AS2
AS1
AS4
AS3
AS7
AS5
35
AS6

• Compute degrees for each AS
• For each routing table path:
– find highest degree AS (“top provider” T)
– AS edge (u,v) to left of T assigned value 1
– AS edge (u,v) to right of T assigned value 1
• For each edge (u,v):
– if (u,v) =1 and (v,u) = 1 then sibling-to-sibling
– else if (v,u) = 1 then provider-to-customer
– else if (u,v) = 1 then customer-to-provider
• Note: complete algorithm also identifies peer-topeer edges
Zegura - Mar 2002 IPAM Workshop Tutorial 36

(Subramanian 2002)
• Idea: partition ASes into hierarchical levels using directed graph of peering relationships
• Process:
– identify and remove nodes with out-degree 0 (customers)
– recursively identify and remove nodes with out-degree 0
(small
ISPs)
– identify dense core as largest subset of nodes that is “almost a clique” (in and out-degree at least half nodes)
– identify transit core as smallest subset of nodes that peer primarily with each other and ASes in dense core
– remaining nodes are outer core
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• Dense core - 20 ASes
• Transit core - 162 ASes
• Outer core - 675 ASes
• Small regional ISPs - 950 ASes
• Customers - 8852 ASes
Zegura - Mar 2002 IPAM Workshop Tutorial 38

• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
– Example: mathematical foundations
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
• Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 39

(Eagan 2002)
• Tool for router-level layout
• Combines automatic placement with userassisted placement
• Understands domain semantics
• Collaboration between Information
Visualization experts and Networking experts
Zegura - Mar 2002 IPAM Workshop Tutorial 40

Zegura - Mar 2002 IPAM Workshop Tutorial 41

(Faloutsos 2002)
• Idea: simple representation of ASlevel topology, useful for intuitive understanding (and NY Times publication!)
• e.g., bowtie model for web
• jellyfish model
– highly connected core
– layers (“shells”)
– degree one nodes form legs
– length of legs denotes density core layers legs
Zegura - Mar 2002 IPAM Workshop Tutorial 42

• Part I - Modeling topology
– Background
– Survey of models + what is known about topology
– Example: mathematical foundations
– Evaluation of topologies
• Part II - Reality check
– Beyond simple topology
– Visualization
•
Open questions/Bold statements/Random thoughts
• Reading list
Zegura - Mar 2002 IPAM Workshop Tutorial 43

• Evaluation
– what metrics are important?
• Useful modeling/scaling
– what topologies should be used for simulations?
• Semantics
– let’s move beyond simple topology
Zegura - Mar 2002 IPAM Workshop Tutorial 44

• Many interesting problems arise due to large scale of Internet, hence need simulations that are “big enough”
• AS-level topology (about 10,000 nodes) manageable for some simulations
• But…representation of every AS as a comparable node (especially in 2-d space!) is a gross simplification
Zegura - Mar 2002 IPAM Workshop Tutorial 45

• AS level models are limited (useless?)
– not enough distinction (all ASes look alike)
– not suitable for packet level simulations
• router level models are limited (useless?)
– too small to be realistic…or...
– too large for simulations
• need alternative models
– intermediate (border routers, exchange points,…)
– fluid flow network model??
• need better understanding of scaling
Zegura - Mar 2002 IPAM Workshop Tutorial 46

• [Broido 2001] Broido and Claffy, “Internet topology: local properties”, SPIE
ITCom 2001.
• [Bu 2002] Bu and Towsley, “Distinguishing between Internet power-law generators”, IEEE Infocom 2002.
• [Burch 1999] Burch and Cheswick, “Mapping the Internet”, IEEE Computer,
April 1999.
• [Chen 2002] Chen, Chang, Govindan, Jamin, Shenker and Willinger, “The origin of power laws in Internet topologies revisited”,
• [Calvert 1997] Calvert, Doar and Zegura, “Modeling Internet topology”, IEEE
Communications Magazine, June 1997.
• [Doar 1997] Doar and Leslie, “How bad is naïve multicast routing”, IEEE
Infocom 1993.
•
[Eagan 2002] Netvisor. http://www.cc.gatech.edu/gvu/ii/netviz/
• [Faloutsos 1999] Faloutsos, Faloutsos and Faloutsos, “On power-laws relationships of the Internet topology”, ACM Sigcomm 1999.
Zegura - Mar 2002 IPAM Workshop Tutorial 47

• [Gao 2000] Gao, “On inferring autonomous system relationships in the
Internet”, IEEE Infocom 2000.
• [Govindan 2000] Govindan and Tangmunarunkit, “Heuristics for Internet map discovery”, IEEE Infocom 2000.
• [Jin 2000] Jin, Chen and Jamin, “Inet: Internet topology generator”, U.
Michigan technical report CSE-TR-433-00, September 2000.
• [Medina 2000] Medina, Matta and Byers, “On the origin of power-laws in
Internet topologies”, ACM CCR, April 2000.
• [Mihail 2002] Mihail, Gkantsidis, Saberi, Zegura, “On semantics of Internet topologies”, GT technical report, January 2002.
•
[Subramanian 2002] Subramanian, Agarwal, Rexford and Katz,
“Characterizing the Internet from multiple vantage points”, IEEE Infocom
2002.
• [Tauro 2002] Tauro, Palmer, Siganos and Faloutsos, “A simple conceptual model for the Internet topology”, Global Internet 2001.
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• [Tangmunarunkit 2001] Tangmunarunkit, Govindan, Jamin, Shenker and
Willinger, “Network topologies, power laws, and hierarchy”, USC technical report 01-746, 2001.
• [Waxman 1988] Waxman, “Routing of multipoint connections”, IEEE JSAC,
1988.
• [Zegura 1997] Zegura, Calvert and Donahoo, “A quantitative comparison of graph-based models for Internet topology”, IEEE/ACM Transactions on
Networking, December 1997.
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