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© James P.G. Sterbenz ITTC
03 February 2017 © 2002-2017 James P.G. Sterbenzrev. 17.1
Science of Communication NetworksThe University of Kansas EECS 784
Internet Topology and Graph Spectra
James P.G. Sterbenz
Department of Electrical Engineering & Computer Science
Information Technology & Telecommunications Research Center
The University of Kansas
http://www.ittc.ku.edu/~jpgs/courses/scinets
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-2
Internet Topology and Graph SpectraOutline
ST.1 Multilevel graphs
ST.2 Internet topology
ST.3 Graph spectra
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-3
Internet Topology Complexity
• Modelling the Internet
– Internet is really complex system (more later)
• scale: number of nodes (graph theoretic order)
• hierarchy: backbones, regional access, subscribers
• differing structural (graph theoretic properties) of each
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-4
Internet Topology Multilevel
• Modelling the Internet
– Internet is really complex system (more later)
• scale: number of nodes (graph theoretic order)
• hierarchy: backbones, regional access, subscribers
• differing structural (graph theoretic properties) of each
– Internet is multilevel graph
• physical infrastructure (fibre, wireless links, L2 muxes, switches)
• engineering underlay (MPLS)
• IP (layer 3) router topology
• AS (service provider) graph
• end-to-end (layer 4) transport flows
• social interaction graph
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-5
Internet TopologyTomography
• No Internet map
– service provider secret
• Impedes research on
– physical structure
– traffic patterns
– application use
• Internet tomography– probe to infer structure
– logical maps only, e.g.
• IP address structure
• AS relationships [Britt wikipedia Image:Internet_map_1024.jpg]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-6
Internet TopologyApproximating Router Topology
• IP and routing tomography
– infer router interconnectivity of a particular service provider
– provides an approximate map of L3 connectivity
– does not provide L2 and physical topology
– does not provide peering information
• Rocketfuel ISP topology mapping engine
[http://www.cs.washington.edu/research/networking/rocketfuel]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-7
Multilevel Network TopologyExample: Sprint L3 IP PoP Topology
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-8
Multilevel Network TopologyExample: Sprint L3 overlay on L2.5
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-9
Multilevel Network TopologyExample: Sprint L2.5 MPLS PoP Topology
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-10
Multilevel Network TopologyExample: Sprint L2.5 overlay on L2/1
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-11
Multilevel Network TopologyExample: Sprint L1 Physical Fiber Topology
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-12
Multilevel Network TopologyExample: Sprint L1–3 Topology
L3
L2.5
L1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-13
Multilevel Network Analysis Abstraction of Internet Topology
physical level topology
ISP 1 ISP 2
ISP 3 ISP 4
IXP 2
IXP 1
IXP 3
router level topology
AS level topologyAS 1
AS 3
AS 2
AS 4
E2E topology
users and apps
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-14
Multilevel Network AnalysisMultilevel Graph Model
• Multilevel model for unweighted & undirected graphs
• Two requirements for multilevel graph model:
– nodes at the above level are subset of lower level
– nodes that are disconnected below are disconnected above
2 51
3
6
4
1
3
6
4
3 4
2 51
3
6
4
3
6
4
3 4
1
2 51
3
6
4
3
6
4
3 4
1
Connected network Disconnected network Partitioned network
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-15
Topology GenerationFlexible and Realistic Topology Generation
• KU-LoCGen
– location and cost constrained[TR2009, TSJ2011]
• Level 1: backbone realms
– nodes distributed based on location constraints
– links generated using various models under cost constraints
• Level 2: access network realms
– distributed around backbone nodes
– access network connectivity: ring, star, mesh
• Level 3: subscribers
– distributed around access node with preferential attachment
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-16
KU-LoCGen2-level example using Sprint PoP Locations
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-17
KU-LoCGen REAL POPSPopulation Based Node Generation Example
actual US PoPs:
Sprint + AT&T + Level3
population-based [CCNet-GLOBECOM 2010]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-18
KU-LoCGenPopulation Based Node Generation Accuracy
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-19
KU-LoCGenNode Generation with Technology Penetration
Delhi
Chennai
Hyderabad
Mumbai
Bangalore
Kolkata
Patna
actual city location
generated PoP no
generated PoP with
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-20
Internet Topology and Graph SpectraInternet Topology
ST.1 Multilevel graphs
ST.2 Internet topology
ST.3 Graph spectra
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-21
Internet PrehistoryEarly Packet Switching Research
• 1961-1973: Early packet-switching principles
– 1961: Kleinrock – queueing theoryshows effectiveness of packet-switching
– 1964: Baran – packetised voice switching in military nets
– 1965: Davies proposes UK packet switched network
• coins term packet
– 1967: NPL begins construction of UK experimental network
– 1967: ARPANET conceived by US DOD ARPA
– 1972: CYCLADES project planning in France
– 1973: first CYCLADES packet switch demonstration
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-22
Internet PrehistoryEarly Packet Switching: NPL Topology
Plymouth
BrightonSouthampton
HullLeeds
Liverpool
Manchester
Norwich
London
BristolCardiff
Swansea
Birmingham
Dudley
PeterboroughLeicester
GreenwichTeddington
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-23
Internet PrehistoryEarly Packet Switching: CYCLADES 1978
Grenoble
NiceToulouse
Lyon
ParisNancy
Rennes
St. Etienne
CIGALE
packet switch
[adapted from Pouzin-1982]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-24
Internet HistoryEarly Packet Switching: ARPANET
• 1967-1972: ARPANET emergence
– 1967: ARPANET conceived by US DoD ARPA
– 1969:
• BBN awarded contract to build IMPs (interface msg processors)
• first ARPANET nodes operational at UCLA, SRI, UCSB, Utah
– 1972:
• ARPANET demonstrated publicly
• NCP (Network Control Protocol) first host-host protocol
• first e-mail program
• ARPANET has 15 nodes
– 1979: ARPANET has 200 nodes
• but access limited to research institutions with ARPA contracts
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-25
Internet HistoryARPANET Design Principles
• Minimalism, autonomy
– no internal changes required to interconnect networks
• Best effort service model
• Robust to failures (or attack)
– stateless gateways (routers)
– decentralized control
– most functionality in end systems
• note: very different from end-to-end arguments!
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-26
Internet HistoryARPANET 1969
SRI
UCLA
Utah
UCSB
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-27
Internet HistoryARPANET 1970
SRI
UCLA
Utah
UCSB
BBN
MIT
Harvard
SDC
Rand
LA
Boston
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-28
Internet HistoryARPANET 1971
SRI
UCLA
Utah
UCSB
BBN
MIT
Harvard
SDC
Rand
CMUUIUC
Case
Lincoln
MitreBurroughs
Ames
Stanford
Boston
LA
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-29
Internet HistoryARPANET 1975
CMUUIUC
Burroughs
UCLASDC
Rand
Utah
USC-ISI
USC
SCRLHawaii
(SATNET)
Ames
LBLMoffett
SRI
LLLSRIXeroxTymshare
FNWCSumex
ISI
UCSD
DOCB
AFWL
AFGWC
ArgonnePurdue
RML Patrick
Elgin
Gunter
MitreARPA
SDAC
NSA
London
NORSAIR(SATNET)
NBS
ETAC
AberdeenBelvoirSDAC
WPAFB
BBN
MIT-MAC
Lincoln
RADC
Rutgers HarvardNYU
RCCNCC
BBNIOX
CCA
MIT-IPC
Boston
LA
Si Valley
Washington
UCSB
Stanford
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-30
Internet HistoryCSNET
• 1981–1996: CSNET (Comp. Sci. Research Network)– 1979: first link meeting at Wisc. [Landweber]
– 1981: initial funding from NSF
– network for researchers without ARPANET access• ARPANET restricted to certain DOD research contracts
– TCP/IP and other services from ARPANET over:• X.25 public networks (initially Telenet)
• Phonenet (MMDF over leased lines)
• ARPANET for institutions with ARPA contracts
– 1989: merged with CSNET into CREN• Corporation for Research and Educational Networking
[Denning, Hearn, Kern, ACM SIGCOMM 1983]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-31
Internet HistoryCSNET
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-32
EASYnet
Internet HistoryGatewayed Networks to ARPANET/CSNET
• Simplified logical structure– some nets use links of others (e.g. PSTN dialup/leased lines)
• Gateways interconnect– no seamless addressing– mixed formats through gateways (e.g. %-hack)
ARPANET
CSNET
BITNET
UUCPNET
GW
GW
Telenet
VNET
corporate
enterprise
networks
public
data
networks
GW
GWInternet
GW
GW
PSTN
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-33
Internet HistoryARPANETNSFNET
• 1986: NSFNET begun– NREN (national research and engineering network)
• HPCC (high perf. computing & communication) act “Gore bill”
– funded by National Science Foundation
– limited to academic institutions and a few govt. contractors
• Progression of – 1986: 56kb/s switched by LSI-11 Fuzzball routers
– 1989: T1 links switched by TR interconnected IBM PC-RTs
– 1992: T3 links switched by FDDI interconnected IBM RS/6Ks
• Late 1980’s: Gigabit testbeds– research testbeds to increase network performance
• Early 1990’s: ARPANET decommissioned
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-34
Internet HistoryRole of Al Gore
• Al Gore was critical in funding the Internet (NSFNET)
– he never claimed to have invented the Internet, rather:
“During my service in the United States Congress, I took the initiative in creating the Internet. I took the initiative in moving forward a whole range of initiatives that have proven to be important to our country's economic growth and environmental protection, improvements in our educational system.”– Al Gore, 9 March 1999 on CNN
– his role & statements have been defended by those who did
“Bob [Kahn] and I believe that the vice president deserves significant credit for his early recognition of the importance of what has become the Internet. … Gore was talking about and promoting the Internet long before most people were listening.” – Bob Kahn and Vint Cerf, 28 Sep. 2000
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-35
EASYnet
Internet HistoryNSFNET Emerges
• Simplified logical structure– some nets use links of others (e.g. PSTN dialup/leased lines)
• Gateways interconnect– no seamless addressing– mixed formats through gateways (e.g. %-hack)
– UUNET becomes main UUCPNET–Internet gateway
ARPANET
CSNET
BITNET
UUCPNET
GW
GW
VNET
corporate
enterprise
networks
GWInternet
GW
GW
PSTN
MILNET
NSFNET
UUNET
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-36
Internet History56 kb/s NSFNET Backbone
Princeton
Ithaca
Pittsburgh
Urbana-Champaign
Boulder
San Diego
FuzzballRouter
DEC LSI-11
56 kb/s leased PSTN lines
1986 – 1988
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-37
Internet History448 kb/s over T1 NSFNET Backbone
Princeton
Ithaca
PittsburghUrbana-Champaign
Boulder
San Diego
RouterIBM PC-RTtoken ring
448 kb/s link multiplexed on T1 lines
Seattle
Palo Alto
Houston
Salt Lake City
Lincoln
Ann Arbor
Washington
1988 – 1989
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-38
Internet History1.5 Mb/s T1 NSFNET Backbone
Princeton
Ithaca
Pittsburgh
Urbana-Champaign
Boulder
San Diego
RouterIBM PC-RTtoken ring
Seattle
Palo Alto
Houston
Salt Lake CityLincoln
Ann Arbor
Washington
1989 – 1990
15 Mb/s T1 lines
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-39
Internet History1.5 Mb/s T1 NSFNET Backbone
Princeton
Ithaca
Pittsburgh
Urbana-Champaign
Boulder
San Diego
RouterIBM PC-RTtoken ring
15 Mb/s T1 lines
Seattle
Palo Alto
Houston
Salt Lake CityLincoln
Ann Arbor
Washington
1990 – 1992
Atlanta
Boston
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-40
Internet History45 Mb/s T3 NSFNET Backbone
Ithaca
Pittsburgh
Urbana-Champaign
Boulder
San Diego
core edgeRouter
IBM RS/6000FDDI
45 Mb/s T3 core network links
Seattle
Palo Alto
Houston
Salt Lake City
Lincoln
Ann Arbor
45 Mb/s T3 edge network links
1992 – 1994
Atlanta
Boston
SF
LA
DENAmes
STL
CHI
Argonne
CLE NYCHFD
GRO
WAS College ParkFIX-east
Princeton
SEA
HOS
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-41
Internet HistoryNSFNET Hierarchical Structure
• NSFNET Backbone– evolved from 56kb/s leased lines to 45Mb/s DS3
• Regional networks– MIDnet, NEARnet, NYSERnet, SURAnet, etc.
• Campus networks– KU, WashUStL, UMass, etc.
NSFNET backbone
NEARnet
KU
MIDnet
WUStL
UMass
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-42
Internet HistoryNSFNET Hierarchical Structure
• Strict hierarchal structure
– advantages?
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-43
Internet HistoryNSFNET Hierarchical Structure
• Strict hierarchal structure
– simple to understand, measure, and analyse
– traffic locality exploited
– each level engineered for
• local traffic characteristics
• transit traffic between lower levels
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-44
Internet HistoryNSFNET Regional Network Examples
MIDnet:Midwest Network
NEARnet: New England
Academic and Research Network
MIDNET nodes
UIUC
WUStL
KUKSU
UNL
ISUUI
UArk
OU
OSUTulsa
NSFNETaccess
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-45
EASYnet
Internet HistoryNSFNET and MILNET become the US Internet
• Simplified logical structure– some nets use links of others (e.g. PSTN dialup/leased lines)
• Global Internet with NSFNET as backbone– seamless addressing using user@DNS and IP addresses– corporate networks convert to TCP/IP– MILNET is separate but interconnected US military backbone
MILNET
NSFNET
VNETcorporate
enterprise
networks
Internet
PSTN
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-46
Internet HistoryPrivatisation
• 1991: NSF lifts access restrictions on NSFNET
– use of email and netnews explodes
• 1993: NSF solicitation for privatisation plans
• 1995: NSF awards for
– NAPs: network access point for ISP traffic exchange
– RA: routing arbiter
– vBNS: very high performance backbone network service
• 1995 April: NSFNET backbone decomissioned
• 1995 Sept.: NSF ends DNS registration subsidies
• 1998: NAPs and RA transferred to private sector
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-47
Internet HistoryResearch Networks
• Internet precursors were research networks
– privatisation constrained use by researchers
– drove need for research infrastructure
• separate from but attached to public Internet
• Research networks
– US: Internet2 and NLR
– Canada: CANARIE (Canadian Advanced Network and Research for Industry and Education)
– Europe: GÉANT
– UK: JANET
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-48
Research NetworksInternet2 Overview
• 1996: Internet2 forms among 34 universities
– not an alternate or replacement for the Internet
• this is a common misconception by the public
– consortium of Universities and other research institutions
– operates network infrastructure with fee to access
• 1999: Abilene OC-48 links upgraded in 2004 to OC-192 links
• 2006: Internet2 network with 10×10Gb/s WDM links
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-49
Research NetworksAbilene Network
Seattle
Palo Alto
Houston
Atlanta
Washington
New York
2.4 Gb/s OC-48 links
Los Angeles
1999 – 2003Internet2GigaPoP
Kansas City
DenverIndianapolis
Chicago
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-50
Research NetworksAbilene Network
Seattle
Palo Alto
Houston
Atlanta
Washington
New York
10 Gb/s OC-192 links
Los Angeles
2004 – 2006
Kansas City
DenverIndianapolis
Chicago
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-51
Research NetworksInternet2 Network
Sunnyvale
Houston
Atlanta
Indianapolis Washington
New York
1010 Gb/s links
Los Angeles
2006 –
Portland
El Paso
Baton Rouge
Jacksonville
Charlotte
Philadelphia
Boston
Tulsa
Salt Lake
Seattle
Louisville
Nashville
Kansas City
Denver
Albuquerque
ChicagoCleveland
Pittsburgh
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-52
Research NetworksInternet2 Network with L2
Sunnyvale
Houston
Atlanta
Indianapolis Washington
New York
Los Angeles
2006 –
Portland
El Paso
Baton Rouge
Jacksonville
Charlotte
Philadelphia
Boston
Tulsa
Salt Lake
Seattle
Louisville
Nashville
1010 Gb/s links
Kansas City
Denver
Albuquerque
ChicagoCleveland
Pittsburgh
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-53
Research NetworksRegional Network Example: KanREN
• 1991: KanREN (KS Research and Education Network)
– formed to expand connectivity in KS beyond MIDnet
– initial connectivity in 1993 funded by NSF
– GPN (1997) formed as successor to MIDnet
– KanREN attaches to Inernet2 through GPN
– 2003 partnership with Kan-Ed for K-12 network
KUKansas State U
Ft. Hays SU
Wichita SUPittsburg SU
Emporia SU
KUMC
JCCC
Internet
GPNI2
GpENIInternet
dual 1 Gb/s Ethernet links
single 1 Gb/s Ethernet links
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-54
Internet ArchitectureCurrent Structure: Tier 1
• Loose hierarchy but with many interconnections
– no longer a map of the Internet; ISP internals proprietary
• Tier-1 backbone providers
– largest with national or international high-speed backbones
– interconnect with one-another at IXPs
• Internet exchange points formerly NAP (network access points)
• peering without charging one-another for traffic
– by bilateral agreement, e.g. Sprint, AT&T
• or sometimes not: Level3–Cogent peering war in 2005
– do not purchase transit service from anyone else
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-55
Internet ArchitectureCurrent Structure: Tier-1 Providers
• Tier 1 backbone providers [Wikipedia Feb 2017] degree
– at&t AS7018 2137– Centurylink (Qwest + SAVVIS + Exodus) AS0209 1689– Cogent AS0174 4681– Deutsche Telekom (Germany) AS3320 504– Global Telecom (Tinet/Tiscali Italy + nLayer) AS3257/4436 1274– KPN (Netherlands) AS286 250– Level 3 with Global Crossing (GX) AS1/3356/3549 4190– Liberty AS6830 607– NTT Communications (formerly Verio – Japan) AS2914 1353– Orange (France) AS5511 159– Sprint AS1239 591– Tata (+Teleglobe – India) AS6543 688– Telecom Italia (Italy) AS6762 536– Telefonica (Spain) AS12956 268– Telia (formerly TeliaSonera – Sweden / Finland) AS1299 1315– Verizon Business (formerly UUNET and MCI/WorldComm) AS0701/702/703 1251– XO AS2828 1070– Zayo (formerly AboveNet) AS6461 1504
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-56
Internet ArchitectureCurrent Structure Tiers 2 and 3
• Loose hierarchy but with many interconnections
– no longer a map of the Internet; ISP internals proprietary
• Tier-1 backbone providers
• Tier-2 ISPs
– purchases some transit traffic from tier-1 providers
– peer directly with some networks (tier-1 and tier-2)
– e.g. BT (AS5400) purchases from GX and Sprint
• Tier-3 local ISPs buys all transit service
– purchases all transit traffic (from tier-2 and perhaps tier-1)
– provide local Internet access
– e.g. Sunflower from Sprint & Level3 [cidr-report.org]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-57
Internet ArchitectureCurrent Structure
tier-1 providers
NAP NAP
tier-2
providers
local ISPs
access lines
peering points
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-58
Social NetworkGraph Structure
• Social graph
– vertices are people or organisatons
– edges are relationships
• Edge relationships
– bidirectional: symmetric, e.g. Facebook friends
– directed: e.g. Twitter follow, Facebook page like
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-59
WebGraph Structure
• Web graph
– vertices
– edges are hyperlinks (URLs)
• Edge relationships
– directed
– bidirectional only when pages link to one-another
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-60
Web GraphBow-Tie Structure
Tubes
Disconn. comp.
Tendrils (out)
[BKM+2000]Deep Web
undiscoverable
Tendrils (in)GSCC
(all pages reachable)
OUTIN
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-61
Web GraphBow-Tie Structure
IN44M nodes 21%
1.2G nodes 32%
OUT44M nodes 21%
1.2G nodes 32%
GSCC56M nodes 28%
1.8G nodes 51%
Tubes
Disconn. comp.
17M 8%
208M 6%
Tendrils
44M nodes 22%
164M nodes 5%
1999 Altavista [BKM+2000]
2012 Common Crawl [MVL+2014]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-62
Internet Topology and Graph SpectraGraph Spectra
ST.1 Multilevel graphs
ST.2 Internet topology
ST.3 Graph spectra
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-63
Representation of GraphsMatrix Types
• Adjacency matrix A (G )
– if nodes i and j are connected, aij=1 ; else 0
• Laplacian matrix– L (G ) = D (G ) − A (G )
– D (G ) is diagonal matrix of node degrees
– L+ (G ) = Q (G ) is signless Laplacian
• Normalised Laplacian matrix
1 , if i = j and di≠ 0
L (G ) = –1 / √didj , if vi and vj are adjacent
0 , otherwise
– allow us to compare structure of graphs of different sizes
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-64
Matrix TypesExamples
v0
v1
v2
v3
e1
e2
e4
e3
1 0 -0.5 -0.40 1 0 -0.6
-0.5 0 1 -0.4-0.4 -0.6 -0.4 1
v0v1v2v3
v0 v1 v2 v3
0 0 1 10 0 0 11 0 0 11 1 1 0
v0v1v2v3
v0 v1 v2 v3
2 0 0 00 1 0 00 0 2 00 0 0 3
v0v1v2v3
v0 v1 v2 v3
2 0 -1 -10 1 0 -1
-1 0 2 -1-1 -1 -1 3
v0v1v2v3
v0 v1 v2 v3
A (G ) D (G )
L (G ) L (G )
2 0 1 10 1 0 11 0 2 11 1 1 3
v0v1v2v3
v0 v1 v2 v3
Q (G )
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-65
Graph SpectraEigenvalues and Eigenvectors
• Given a matrix M , eigenvalues , and eigenvectors x
• Eigenvalues and eigenvectors satisfy, – M x = x
• Eigenvalues are roots of characteristic polynomial
– det (M− I ) for x ≠ 0
• Spectrum of M is its eigenvalues and multiplicities
– multiplicity is the number of occurrences of an eigenvalue
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-66
Graph SpectraImportant Characteristics of NLS
• Eigenvalues of normalised Laplacian spectra (NLS)
– {0 = 1 ≤ 2 ≤ … ≤ n ≤ 2}
– number of 0s represent number of connected components
• Quasi-symmetric about 1
• Spectral radius (L ) : largest eigenvalue
– if (L ) = 2, then the graph is bipartite
– closer to 2 means nearly bipartite
• 2(L ) ≤ 1 for non-complete (non-full-mesh) graphs
– 2(L ) ≥ 1/(2e D) ≥ 0
• where e is the graph size (# of links) and D is graph diameter
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-67
Topological DatasetPoP-Level Topologies – AT&T and Sprint
• Included only 48 US contiguous states [Rocketfuel]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-68
Topological DatasetUS Interstate Highways
• Added 5 highways, 6 interchange nodes, 2 pendants [AASHTO]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-69
Topological DatasetFibre-Optic Routes – AT&T and Sprint
• Included only 48 US contiguous states [KMI]
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-70
Topology Analysis of Graphs Metrics
• Baseline networks Lecture SCN-RN– star, linear, tree, ring, grid, toroid, mesh
– order (number of nodes) of the baseline graphs: 10 and 100
• Average degree is same for star, linear, tree
– do these graphs share same structural properties?
• Clustering coefficient is too coarse for baseline graphs
– except for mesh which the value is 1, rest is 0
• Algebraic connectivity a (G ):
– second smallest eigenvalue of the Laplacian matrix
– a (G ) = 0.1 for n=10 (linear) and n=100 (grid)
– not very useful for comparing different order graphs
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-71
Topological Analysis via MetricsBaseline Networks with n=10
Topology Star Linear Tree Ring Grid Toroid Mesh
Number of nodes 10 10 10 10 10 10 10
Number of links 9 9 9 10 13 15 45
Maximum degree 9 2 3 2 3 3 9
Average degree 1.8 1.8 1.8 2 2.6 3 9
Degree assortativity -1 -0.13 -0.53 1 0.28 1 1
Node closeness 0.58 0.29 0.37 0.36 0.44 0.53 1
Clustering coefficient 0 0 0 0 0 0 1
Algebraic connectivity 1 0.1 0.18 0.38 0.38 1.38 10
Network diameter 2 9 5 5 5 3 1
Network radius 1 5 3 5 3 3 1
Average hop count 1.8 3.67 2.82 2.78 2.3 1.89 1
Node betweenness 36 20 26 8 11 4 0
Link betweenness 9 25 24 13 12 6 1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-72
Topological Analysis via MetricsBaseline Networks with n=100
Topology Star Linear Tree Ring Grid Toroid Mesh
Number of nodes 100 100 100 100 100 100 100
Number of links 99 99 99 100 180 200 4950
Maximum degree 99 2 3 2 4 4 99
Average degree 1.98 1.98 1.98 2 3.6 4 99
Degree assortativity -1 -0.01 -0.34 1 0.57 1 1
Node closeness 0.51 0.03 0.13 0.04 0.15 0.20 1
Clustering coefficient 0 0 0 0 0 0 1
Algebraic connectivity 1 0.001 0.01 0.004 0.1 0.38 100
Network diameter 2 99 12 50 18 10 1
Network radius 1 50 6 50 10 10 1
Average hop count 1.98 33.7 7.8 25.3 6.67 5.05 1
Node betweenness 4851 2450 3068 1201 616 200 0
Link betweenness 99 2500 2496 1250 341 200 1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-73
Topology Analysis of Graphs Spectra
• Normalised Laplacian spectrum– since normalised, eigenvalues range [0,2]
• Calculate:– RF: relative frequency [BJ2009]
– RCF: relative cumulative frequency [VHE2002]
• Eigenvalue of 2 indicates how bipartite a graph is
• Eigenvalue 1 multiplicity indicates node duplications– nodes having similar neighbours
• Spectrum is symmetric around 1
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-74
Graph SpectrumRF for Baseline Networks with n=100
• RF of eigenvalue multiplicities is noisy– mesh and star graphs look similar (except = 2 eigenvalue)
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-75
Graph SpectrumRCF for Baseline Networks with n=100
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-76
Topology Analysis of Real Networks Metrics and Spectra
• Communication and transportation networks
• Metrics indicate:
– physical topologies are closer to transportation network
– difference between physical and logical topologies
• higher number of nodes in physical topologies
• rich connectivity in logical topologies
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-77
Topological Analysis via MetricsReal Networks
Topology Sprint Physical
Sprint Logical
AT&T Physical
AT&T Logical
US Highways
Number of nodes 263 28 361 107 400
Number of links 311 76 466 140 540
Maximum degree 6 14 7 23 7
Average degree 2.37 5.43 2.58 2.62 2.7
Degree assortativity -0.17 -0.23 -0.16 -0.4 0.11
Node closeness 0.07 0.48 0.08 0.3 0.08
Clustering coefficient 0.03 0.41 0.05 0.09 0.05
Algebraic connectivity 0.0053 0.6844 0.0061 0.1324 0.0059
Network diameter 37 4 37 6 40
Network radius 19 2 19 3 21
Average hop count 14.78 2.19 13.57 3.38 13.34
Node betweenness 11159 100 15970 2168 22798
Link betweenness 9501 27 14270 661 18585
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-78
Graph SpectrumRCF for Real Networks
• Spectrum of physical topologies resemble motorways
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-79
Network TopologyFiber Relation to Other Infrastructure
L1
Class 1 rail mainlines
Interstate freeways
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-80
Spectral Properties of Real NetworkscTGD vs. Algebraic Conn. & Spectral RadiusTopology cTGD cTGD
Ranka(G) a(G)
Rank(L) (L)
Rank
Level 3 0.4494 1 0.9518 3 1.5033 2
AboveNet 0.4386 2 0.9645 2 1.4978 1
Exodus 0.3617 3 1.0083 1 1.7408 4
EBONE 0.3113 4 0.6477 4 1.7335 3
Tiscali 0.2641 5 0.5255 5 1.7470 5
Sprint 0.2407 6 0.3817 6 1.7853 6
Verio 0.2009 7 0.2448 8 1.8463 7
VSNL 0.1783 8 0.3402 7 1.9053 9
GEANT2 phys. 0.1668 9 0.1515 9 1.8518 8
AT&T 0.1446 10 0.1324 10 1.9127 10
Telstra 0.0941 11 0.0454 11 1.9797 11
AT&T phys. 0.0348 12 0.0061 12 1.9892 13
Sprint phys. 0.0307 13 0.0053 13 1.9839 12
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-81
Graph SpectrumRCF for Communication Networks
• Spectrum of physical topologies resemble motorways– Level 3 is richly connected and have small spectral radius
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-82
Analysis of Internet Infrastructure Summary
• Physical topologies resemble motorways
– known: but not rigorously studied
– grid-like structures
• The normalised Laplacian spectrum is powerful
– spectral radius indicates bipartiteness
– = 1 multiplicity indicates duplicates (i.e. star-like structures)
• Future work:
– study other physical critical infrastructures
• railways, power grid, pipelines
– investigate metrics and relationship to resiliency/connectivity
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-83
Analysis of Internet Infrastructure Review of Graph Spectra
• All but structural graphs have same nodes
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-84
Topology and SpectraReferences and Further Reading
• [BKM+2000] Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, Janet Wiener,“Graph structure in the Web”Computer Networks, Elsevier, vol.33, iss.1–6, June 2000, pp. 309–320
• [MVLB2014] Robert Meusel, Sebastiano Vigna, Oliver Lehmberg, and Christian Bizer,“Graph Strucutre in the Web – Revisited”,ACM WWW’14, Seoul, pp.427–432
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-85
Topology and SpectraReferences and Further Reading
• [CARS2012] Egemen K. Çetinkaya, Mohammed J.F. Alenazi, Justin P. Rohrer, and James P.G. Sterbenz, “Topology Connectivity Analysis of Internet Infrastructure Using Graph Spectra”, in Proc. of the IEEE/IFIP RNDM , St. Petersburg, October 2012
• [AASHTO] American Association of State Highway and Transportation Officials, “Guidelines for the Selection of Supplemental Guide Signs for Traffic Generators Adjacent to Freeways”, Washington, D.C., 2001
• [KMI] KMI Corporation, “North American Fiberoptic Long-haul Routes Planned and in Place”, 1999
• [KU-TopView] https://www.ittc.ku.edu/resilinets/maps
• [Rocketfuel] http://www.cs.washington.edu/research/networking/rocketfuel
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-86
Graph SpectraReferences and Further Reading
• [C1994] Fan R. K. Chung, Spectral Graph Theory , American Mathematical Society, 1994
• [M2011] Piet van Mieghem, Graph Spectra for Complex Networks , Cambridge University Press, 2011
• [BH2012] Andries E. Brouwer and Willem H. Haemers, Spectra of Graphs , Springer, 2012
• [CRS2010] Dragoš Cvetković, Peter Rowlinson, and Slobodan Simić, An Introduction to the Theory of Graph Spectra , London Mathematical Society, 2010
• [B1993] Norman Biggs, Algebraic Graph Theory , Cambridge University Press, 1993
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-87
Graph SpectraReferences and Further Reading
• [GN2006] M. T. Gastner and M. E. Newman, “The spatial structure of networks”,The European Physical Journal B , vol. 49, no. 2, January 2006, pp. 247 – 252
• [JWM2006] Almerima Jamaković, Huijuan Wang, and Piet van Mieghem, “Topological Characteristics of the Dutch Road Infrastructure”, in Infrastructure Reliability Seminar , Delft, June 2006
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-88
Graph SpectraReferences and Further Reading
• [CS2011] Dragoš Cvetković and Slobodan Simić, “Graph spectra in Computer Science”, Linear Algebra and its Applications , vol. 434, no. 6, March 2011, pp. 1545 – 1562
• [VHE2002] Danica Vukadinović, Polly Huang, and Thomas Erlebach, “On the Spectrum and Structure of Internet Topology Graphs”, in Proc. of the IICS , June 2002, pp. 83 – 95
• [BJ2009] Anirban Banerjee and Jürgen Jost, “Spectral Characterization of Network Structures and Dynamics”, Dynamics On and Of Complex Networks , 2009, pp. 117 – 132
© James P.G. Sterbenz ITTC
03 February 2017 KU EECS 784 – Science of Nets – Topology & Spectra SCN-TS-89
End of Foils