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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

Modeling Internet Topology

Ellen W. Zegura

College of Computing

Georgia Tech

outline
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
  • Reading list

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networking background
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
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
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
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
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
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
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
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
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)
    • limiting traceroute load
  • Postprocessing challenges:
    • resolving aliases (which IP addresses belong to same router)

S1

D1

destination 0

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projects
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
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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outline14
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
  • Reading list

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foundations of degree based generation mihail 2002
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
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
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
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
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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example20
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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notes about models
Notes about models
  • Variants on evolutionary models
  • Variants on degree-driven models
  • Appeal of evolutionary
  • Relationship to work on “networks” in general

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outline22
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
  • Reading list

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evaluation
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
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
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
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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distinguishing between types of generators tangmunarunkit 2001
Distinguishing between types of generators (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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model signatures
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
Measure of hierarchy
  • 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

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outline30
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
  • Reading list

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semantics policy based routes
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
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
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
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
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
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
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
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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outline39
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
  • Reading list

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visualization netvisor eagan 2002
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
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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outline43
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
  • Reading list

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open problems
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
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
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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reading list 1 of 3
Reading List (1 of 3)
  • [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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reading list 2 of 3
Reading List (2 of 3)
  • [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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reading list 3 of 3
Reading List (3 of 3)
  • [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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the end
The End

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