Modeling Internet Topology

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Modeling Internet Topology. Ellen W. Zegura College of Computing Georgia Tech. Outline. 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

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Modeling Internet Topology

Ellen W. Zegura

College of Computing

Georgia Tech

Outline
• 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

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Networking background

transit domains

domains/autonomous systems

exchange point

border routers

peering

hosts/endsystems

routers

stub domains

lowly worm

access networks

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Topology modeling
• 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

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Survey of models
• 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

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Waxman model (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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Transit-stub model (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

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Real data: AS topology
• 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

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Inet (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

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BRITE (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

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Router-level measurement

source 0

• General technique: traceroute, returns list of IP addresses on a path from source to destination
• Collection challenges:
• obtaining sufficient traceroute origin points
• deciding set of destination IP addresses (for coverage)
• Postprocessing challenges:
• resolving aliases (which IP addresses belong to same router)

S1

D1

destination 0

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Projects
• 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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What is known? (hard to say)
• 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)

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Outline
• 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

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Foundations of degree-based generation (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

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Construction algorithm
• 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 varbitrary

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Sparse/dense core
• 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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Example
• 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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Random instance
• 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

u

s

v

t

u

s

v

t

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Example
• 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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Outline
• 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

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Evaluation
• 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!”

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Metrics
• 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

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Small world topologies (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
• C = average clustering coefficient
• C >> C_random, L  L_random

k nodes

v

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Findings
• 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

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• 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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Model “signatures”
• 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

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Measure of hierarchy
• see paper for details…
• bottom line: degree-based generators contain loose notion of hierarchy that is somewhat similar to loose notion in Internet

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Outline
• 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

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Semantics: policy-based routes
• 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

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Transit-stub
• 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

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BGP peering relationships (Gao 2000)
• Problem: Routes determined by routing policy, including AS-level contractual agreements
• Idea: label edges in AS-level graph as
• 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)
• Use BGP routing table entries

AS1

AS7

AS2

AS3

AS6

AS4

AS5

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Principles
• e.g., routing table entry = AS path 1849 702 701 1
• downhill path: all edges provider-to-customer or sibling-to-sibling
• uphill path: all edges customer-to-provider or sibling-to-sibling
• 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)

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Examples
• 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

AS1

AS7

AS2

AS3

AS6

AS4

AS5

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Basic algorithm sketch
• 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-to-peer edges

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Hierarchical classification (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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Example result
• Dense core - 20 ASes
• Transit core - 162 ASes
• Outer core - 675 ASes
• Small regional ISPs - 950 ASes
• Customers - 8852 ASes

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Outline
• 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

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Visualization: netvisor (Eagan 2002)
• Tool for router-level layout
• Combines automatic placement with user-assisted placement
• Understands domain semantics
• Collaboration between Information Visualization experts and Networking experts

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Visualization: conceptual model (Faloutsos 2002)
• Idea: simple representation of AS-level 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

layers

core

legs

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Outline
• 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

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Open Problems
• Evaluation
• what metrics are important?
• Useful modeling/scaling
• what topologies should be used for simulations?
• Semantics
• let’s move beyond simple topology

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Are AS-level topologies useful?
• 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

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Observations on level of detail
• 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

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• [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.

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• [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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