modeling internet topology

Modeling Internet Topology

Ellen W. Zegura

College of Computing

Georgia Tech

Zegura - Mar 2002 IPAM Workshop Tutorial 2

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


Networking background

access networks

hosts/endsystems

routers

domains/autonomous systems exchange point

stub domains

transit domains

border routerspeering

lowly worm


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


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


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

v

u d(u,v)


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


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


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


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


Router-level measurement

• 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)

source 0

destination 0

S1

D1


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


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)



– Background


– Example: mathematical foundations



– Visualization


• Reading list


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


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 v arbitrary


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


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


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

v

s

t

u

v

s

t


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


Notes about models

• Variants on evolutionary models

• Variants on degree-driven models

• Appeal of evolutionary

• Relationship to work on “networks” in general



– Background





– Visualization


• Reading list


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!”


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


• 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

Small world topologies (Bu 2002)

v k nodes


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


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)


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


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



– Background





– Visualization


• Reading list


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


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


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

AS2 AS6AS3

AS1 AS7

AS4 AS5


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)


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

AS2 AS6AS3

AS1 AS7

AS4 AS5


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


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


Example result

• Dense core - 20 ASes

• Transit core - 162 ASes

• Outer core - 675 ASes

• Small regional ISPs - 950 ASes

• Customers - 8852 ASes



– Background





– Visualization


• Reading list


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


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

core layers

legs



– Background





– Visualization


• Reading list


Open Problems

• Evaluation– what metrics are important?

• Useful modeling/scaling– what topologies should be used for

simulations?

• Semantics– let’s move beyond simple topology


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


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


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.


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.


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.


The End

modeling internet topology

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