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OutlineOutline

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

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)
- limiting traceroute load
- 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
- Reading list

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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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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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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

- 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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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”

- 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

- 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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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

- 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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- 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)

- 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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- 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

- 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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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)

- [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)

- [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

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