Download Internet_Topology_Modelling_and_Analysis

ECODE FP7 Project
Training Seminar: Session 2a
Internet architecture (incl. Topology
structure, and models)
Dimitri Papadimitriou and Olivier Bonaventure
Alcatel-Lucent BELL - Universite catholique de Louvain (UCL)
September 1, 2008
Alcatel-Lucent BELL
Antwerpen, Belgium
Outline
1. Organization of the Internet
•
•
Topology
Types of domains
–
–
–
•
Transit domain
Stub domain
Example of domains
Internet Routing
2. Evolution of the Internet
•
•
Number of Hosts
IP address allocation
–
–
•
•
Number of AS
Routing tables
–
–
•
•
IPv4 address allocation
IPv6 address allocation
Size of the IPv4 BGP routing tables
Size of the IPv6 BGP routing tables
IP traffic flows
Bandwidth
Outline
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Outline
1. Organization of the Internet
•
•
Topology
Types of domains
–
–
–
•
Transit domain
Stub domain
Example of domains
Internet Routing
2. Evolution of the Internet
3. Internet Topology modeling
Organization of the Internet

Internet: infrastructure composed by an
interconnected set of (heterogeneous) networks
architected around a distributed routing system that
is partitioned into independently administrated
domains (autonomous systems)

A domain is a set of routers, links, hosts and local
area networks under the same administrative control
•
A domain can be very large...
–
•
A domain can be very small...
–

AS568: SUMNET-AS DISO-UNRRA contains 73154560 IP
addresses
AS2111: IST-ATRIUM TE Experiment a single PC running Linux...
Internet is composed of ~ 30.000 autonomous
systems (AS)
Organization of the Internet
 Domains
•
•
are interconnected in various ways
The interconnection of all domains should in theory allow
packets to be sent anywhere
Usually IP datagram will need to cross a few ASes (3 to 4,
average 3.4) to reach its destination
Evolution of the Internet Topology (1)






1986: NSF builds NSFNet as backbone, links 6 supercomputer centers,
56 kbps; huge increase of connections, especially from universities
1987: 10,000 hosts - 1989: 100,000 hosts - 1992: 1 million hosts
1988: NSFNet backbone upgrades to 1.5Mbps
1991: NSF lifts restrictions on the commercial use of the Net;
1994: NSF reverts back to research network (vBNS); the backbone of the
Internet consists of multiple private backbones
Before ‘95: Strict hierarchical network with single central
backbone
NSFNet Backbone
Regional
Campus
Campus
Regional
Campus
Regional
Campus
Evolution of the Internet Topology (2)




Between 1995-1999: increased meshedness between ISP
backbones and customers
Decentralization: from a single backbone network to a
conglomeration of 100s of backbone and 1000s ISP
Loss of hierarchy and abstraction: from hierarchical network
to increasingly meshed interconnection
Significant bandwidth increase: from T3 (45MB) and T1 (1MB)
to OC48 (2.5GB) and OC12 (622MB) link capacity
AS1
AS2
R2
R1
AS4
AS3
R3
R4
Evolution of the Internet Topology (3)
Can be viewed as structured into tiers
•
Tier-1 ISPs a.k.a backbone providers
–
–
–
•
Tier-2 ISPs
–
–
–
–
•
Dozen (12 to 20 AS) of large international or large national ISPs
interconnected by multiple private peering points (shared cost)
Provide transit service (no “upstream” provider)
Examples: AT&T, Verizon, Sprint, Level 3, etc.
Regional or National ISPs (order 1k AS)
Customer of T1 ISP(s) - at least 1 and often 2 - and Provider of T3 ISP(s)
Shared-cost with other T2 ISPs
Examples: France Telecom, BT, Belgacom
Tier-3 ISPs a.k.a stub AS
–
–
–
Smaller ISPs, Corporate Networks, Content providers (order 10k AS)
Customers of T2 or T1 ISPs (no transit service to other ISPs)
Shared-cost with other T3 ISPs
Interconnections
•
•
An ISP runs (private) Points of Presence (PoP) where its
customers and other ISPs connect to it
ISPs also connect at (public) Network Access Point (NAP)
called public peering
9
Tier-1 ISP
“Tier-1” ISPs (a.k.a. backbone providers e.g., AT&T, Verizon,
Sprint, Level 3, Qwest): national/ international coverage treating
each other as equals (peers)
Tier-1 providers
interconnect privately =
multiple private peering
Tier-1 providers also
interconnect at public
network access points
(NAPs) = public peering
Tier 1 ISP
NAP
Tier 1 ISP
Tier 1 ISP
Tier-2 ISP
“Tier-2” ISPs (often regional-national): ISPs that connect to one
or more Tier-1 ISPs, possibly other Tier-2 ISPs
Tier-2 ISP
Tier-2 ISP pays Tier-1
ISP for connectivity to
rest of Internet
Tier-2 ISP is customer
of Tier-1 ISP
Tier-2 ISP
Tier-2 ISPs also peer
privately with each
other, and publicly
interconnect at NAP
Tier 1 ISP
NAP
Tier 1 ISP
Tier 1 ISP
Tier-2 ISP
Tier-2 ISP
PoP
Tier-2 ISP
Tier-3 ISP
“Tier-3” ISPs: last hop (“access”) network (closest to end
systems)
Tier-3 ISP
Tier-3 ISP
Tier-2 ISP
Tier- 3 ISPs are
customers of
higher tier ISPs
connecting them
to rest of Internet
Tier-3 ISP
Tier-3 ISP
Tier-3 ISP
Tier-2 ISP
Tier 1 ISP
NAP
Tier 1 ISP
Tier 1 ISP
Tier-2 ISP
Tier-2 ISP
Tier-2 ISP
PoP
Tier-3 ISP
Tier-3 ISP
Tier-3 ISP
Tier-3 ISP
Organization of the Internet
Tier-1 ISPs
–
–
Dozen of large ISPs
interconnected by shared-cost
Provide transit service
–
Uunet, Level3, Sprint, ...
Tier-2 ISPs
–
–
–
–
Regional or National ISPs
Customer of T1 ISP(s)
Provider of T2 ISP(s)
Shared-cost with other T2 ISPs
–
France Telecom, BT, Belgacom
Tier-3 ISPs
–
–
–
Smaller ISPs, Corporate
Networks, Content providers
Customers of T2 or T1 ISPs
Shared-cost with other T3 ISPs
AS Ranking

Proposing two ranking methods:
•
•
Degree-based: ASes are ranked by their degrees
in the AS topology graph: http://as-rank.caida.org/
AS-relationship-based: ASes are ranked by their
customer cone sizes
See http://as-rank.caida.org/data
RouteViews BGP AS links annotated with inferred relationships
Dataset date: 20080818
Alpha parameter of inference algorithm: 0.01000
Format: <AS1> <AS2> <relationship>
where
<AS1> and <AS2> are AS numbers, and
<relationship> is -1 if AS1 is a customer of AS2,
0 if AS1 and AS2 are peers,
1 if AS1 is a provider of AS2, and
2 if AS1 and AS2 are siblings (the
same organization)
Summary

Based on AS connectivity and relationships, the
Internet routing infrastructure can be viewed as a
three tier hierarchy
•
•
•
Core: consisting of a dozen or so Tier-1 providers forming
the top level of the hierarchy
Middle: consisting of few thousands of ASes (Tier-2
providers) that provide transit service but are not part of
the core
Edge: 10 thousands of stub ASes that do not provide
transit service. Usually, local ISP, ASP and CSP
Outline
1. Organization of the Internet
•
•
Topology
Types of domains
–
–
–
•
Transit domain
Stub domain
Example of domains
Internet Routing
2. Evolution of the Internet
3. Internet Topology modeling
Types of domains

The Internet consists of routing domains:
Autonomous Systems (AS) interconnected with
each other:
•
•
Transit domain: provider, hooking many AS together
Stub domain: smaller corporation/domain:
–
–

At least one and usually two connections to other domain
No transit service to other domains
Two-level routing:
•
•
Intra-domain: administrator responsible for choice of
routing protocol within network (usually link-state routing
protocol)
Inter-domain: standard for interdomain routing: BGP
Types of domains (1)

Transit domain
•
A transit domain allows external domains to use its
own infrastructure to send packets to other domains
S1
S2

T2
T1
T3
S4
S3
Examples
•
UUNet, OpenTransit, GEANT, Internet2, RENATER,
EQUANT, BT, Telia, Level3,...
Types of domains (2)
Stub domains
•
•
A stub domain does not allow external domains to
use its infrastructure to send packets to other domains
A stub is connected to at least one transit domain
–
–
Single-homed stub : connected to one transit domain
Dual-homed stub : connected to two transit domains
S1
S2
•
T3
S4
S3
Content stub domain (Content Service Provider)
–
•
T2
T1
Large web servers : Yahoo, Google, MSN, TF1, BBC,...
Access-rich stub domain (Access Service Provider)
–
ISPs providing Internet access via CATV, ADSL, ...
Multihomed domains

Definition: use of redundant network links/connections
to the same or different domain for the purposes of
external connectivity

Objective:
•
•
•

Robustness in case of failure (link, upstream domain)
Performance (load balancing)
Cost
Multi-homed stub AS: connectivity to multiple
immediate upstream transit domains
T2
T1
T3

Multi-homed transit AS
S3
A transit domain : Easynet
A transit domain : GEANT
A transit domain : BT/IGnite
A large transit domain : UUNet
Composition of Internet paths

Most Internet paths contain a sequence of
•
•
•
0 or more Customer->Provider relationships
0 or 1 Peer-to-Peer relationships
0 or more Provider->Customer relationships
AS1
AS2
$
$
$
Shared-cost (peering)
$
$
AS9
$
AS4
AS3
AS8
$
$
AS7
Customer-provider
Outline
1. Organization of the Internet
•
•
Topology
Types of domains
–
–
–
•
Transit domain
Stub domain
Example of domains
Internet Routing
2. Evolution of the Internet
3. Internet Topology modeling
Internet Routing
Internet domains comprises devices called routers comprising a
routing and a forwarding engine (and a management agent)
Routing engine:
•
•
•
Process routing information (exchanged between routers using a routing
protocols such as BGP) so as to compute routes (using a shortest path
algorithms)
Routes entries (composed by a destination, a next-hop interface, and a
metric) are stored in routing information bases (RIB)
Routing entries are subsequently used by the forwarding engine
Forwarding engine:
•
Transfer incoming IP datagram to an outgoing interface directed towards
a router closer (next-hop) to the traffic destination by performing a
longest match prefix lookup on forwarding entries stored in forwarding
information base (FIB) using the incoming IP datagram destination
address
Architecture of a normal IP router
Routing
protocol
Routing table
Control
The "best" paths selected from the routing table
built by the routing protocols are installed in the
forwarding table
Shap.
IP packets
Forwarding
Table
IP packets
Class.
Pol
Forwarding
Shap.
Class.
Pol
IP packets
Forwarding decision based on longest prefix match
Update of TTL and checksum fields in IP datagrams (packets)
Internet Routing Protocols

Interior Gateway Protocol (IGP)
•
•
Routing of IP datagrams inside each domain
Only knows topology of its own domain (all routers within
given AS managed by a single admin unit)
Domain4
Domain2
Domain1

Domain3
Exterior Gateway Protocol (EGP)
•
•
Routing of IP packets between domains
Each domain is considered as a blackbox
Inter vs Intra-domain Routing Protocols
IGP: Intra-domain routing (within AS)
•
•
•
Allow routers to transmit IP packets towards their destination along the best
path = shortest-path (metrics: #hops, link cost)
IGP routing protocols: distance vector or link state
All routers exchange routing information: each domain router can obtain
routing information for the whole domain
eBGP
eBGP
IGP
eBGP
eBGP
iBGP
eBGP
eBGP
EGP: Inter-domain routing (between AS)
•
•
•
Routing policies based on business relationships
No common metrics, and limited cooperation
Policy-based, path-vector routing protocol: external/internal Border
Gateway Protocol (eBGP/iBGP)
Session 2a - Outline
1. Organization of the Internet
2. Evolution of the Internet
•
•
Number of Hosts
IP address allocation
–
–
•
•
Number of AS
Routing tables
–
–
•
•
IPv4 address allocation
IPv6 address allocation
Size of the IPv4 BGP routing tables
Size of the IPv6 BGP routing tables
IP traffic flows
Bandwidth
3. Internet Topology modelling
Growth in number of Internet hosts
Number of Hosts on the Internet:
Aug. 1981
213
Oct. 1984
1,024
Dec. 1987
28,174
Oct. 1990
313,000
Jul. 1993
1,776,000
Jul. 1996
19,540,000
Jul. 1999
56,218,000
Jul. 2004
285,139,000
Jul. 2005
353,284,000
Jul. 2006
439,286,000
Jul. 2007
489,774,000
Growth in number of Internet users
Number of Users over Time (from Dec’95 to Mar’08)
Internet Users - Growth [1995,2008]
1407
1400
1129
1023
1200
888
1000
745
800
458
600
400
200
16 36
70
147
248
558 608
304
0
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De 5
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De 6
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De 7
c.
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De 8
c.
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M 0
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M 1
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M 2
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M 3
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M 4
ar
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M 5
ar
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M 6
ar
.0
M 7
ar
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8
Number of Users (in Milliion)
1600
Month/Year
Issues with the current
Internet architecture
Limited size of IPv4 addressing space
•
NAT, CIDR and IPv6 have been proposed to
overcome this limitation
Projected Address Consumption (/8s)
Source http://www.potaroo.net/tools/ipv4/index.html
Issues with the current
Internet architecture


Exhaustion date of first RIR available pool of addresses (and no
further numbers available in IANA unallocated pool to replenish
RIR's pool) - Best fit predictive model predicts occurrence on
Dec 2011
Exhaustion of IANA unallocated number pool - Model predicts
occurrence on Feb 2011
Source http://www.potaroo.net/tools/ipv4/index.html
IPv6 usage: advertised prefixes
source http://bgp.potaroo.net/v6/v6rpt.html
Current IPv6 usage: ASes using IPv6
Ratio: prefix/AS ~ 1
source http://bgp.potaroo.net/v6/v6rpt.html
Issues with the current
Internet architecture (2)
Interdomain routing scalability
•
Growth of BGP IPv4 routing tables
Growth is back again !
Internet bubble:
growth is back
Classless Inter-domain routing (CIDR) as
reaction to running out of class B: RFC 1338
(Jun.92) - RFC 1519 (Sep.93)
CIDR works well
Bubble explosion
pre-CIDR fast growth
Source : http://bgp.potaroo.net
Growth of Active BGP Entries in FIB (from
Jan’89 to Mar’08)
Jan.1 2006
–
–
–
–
–
FIB Size: 176,000 prefixes
Update Rate: 0.7M prefix updates / day
Withdrawal Rate: 0.4M prefix withdrawals / day
250Mbytes memory
30% of a 1.5Ghz processor
~25%
~15-20%
RIB/FIB ratio (779057/266725): 2.9208 (*)
Jan.1 2009
-
FIB size: [275,000;300,000] prefixes
Update Rate: 1.7M prefix updates / day
Withdrawal Rate: 0.9M withdrawals / day
400Mbytes Memory
75% of a 1.5Ghz processor
Jan.1 2011 (low-end predictions)
- FIB Size: [370,000;400,000] prefixes
- Update Rate: 2.8M prefix updates / day
- Withdrawal Rate: 1.6M withdrawals per day
- 550Mbytes Memory
- 120% of a 1.5Ghz processor
09
(*) - RIB/FIB ratio can vary from ~3 to 30 (function of the number of BGP peering sessions at sample point)
Source: BGP Routing Table Analysis Reports on AS65000 - http://bgp.potaroo.net
Issues with the current
Internet architecture (3)
Reasons for the BGP growth
•
Number of distinct ASes ?
Number of unique ASN advertised in BGP routing table over Time
29.227
post-boom period
sharp growth during the
Internet boom period from
1999 until early 2001
pre-Internet boom prior to 1999
Source: http://www.potaroo.net/tools/asn32/
Ratio: prefix/AS ~ 10
Issues with the current
Internet architecture (3)
Unadvertised ASN count
= Assigned ASN count - Advertised ASN count
Number of advertised and
advertised ASs over Time
Ratio of unadvertised to
advertised ASN over Time
Expansion of Internet
between 2005 and 2006
Prefixes: 173,800 – 203,800 (+17%)
AS Numbers: 21,200 – 24,000 (+13%)
IPv4 in 2006
Total BGP FIB entries over Time
Addresses: 87.6 – 98.4 (/8) (+12%)
Average advertisement size: smaller (8,450 –
8,100)
Average prefixes per update: smaller (2.1 1.95)
Average address origination per AS: smaller
(69,600 – 69,150)
Average AS Path length: steady (3.4)
AS transit interconnection degree: growing
(2.56 – 2.60)
 IPv4 network becomes
 denser (more interconnections)
 finer levels of advertisement granularity
(more specific advertisements)
 Higher levels of path exploration before
stabilization on best available paths
Source: IEPG, <http://www.potaroo.net>
Issues with the current
Internet architecture (4)
Reasons for the BGP growth
•
Multihoming
Client : AS4567
I can reach
194.100.10.0/23
194.100.0.0/16
R2
R1
194.100.10.0/23
I can reach
194.100.0.0/16
and 194.100.10.0/23
Provider
AS123
I can reach
194.100.10.0/23
R3
200.0.0.0/16
Provider
AS789
I can reach
200.0.0.0/16 and 194.100.10.0/23
Internet
Issues with the current
Internet architecture (5)
Reasons for the BGP growth
•
Traffic engineering
I can reach
194.100.0.0/16
194.100.11.0/24
Client : AS4567
I can reach
and 194.100.10.0/23
R2
194.100.0.0/16
and 194.100.11.0/24
R1
and 194.100.10.0/23
194.100.10.0/23
Provider
AS123
I can reach
194.100.10.0/24
and 194.100.10.0/23
R3
200.0.0.0/16
Provider
AS789
I can reach
200.0.0.0/16 and 194.100.10.0/24
and 194.100.10.0/23
Internet
Issues with the current
Internet architecture (6)
BGP messages processed by routers
•
Hourly average prefix update rate (per second)
http://bgpupdates.potaroo.net/instability/bgpupd.html
Issues with the current
Internet architecture (7)
BGP messages processed by routers
•
Hourly peak of per second prefix update rate
http://bgpupdates.potaroo.net/instability/bgpupd.html
Issues with the current
Internet architecture (8)
Interdomain routing security
•
Only Best Current Practices from network operators
prevent a customer network from using BGP to
announce the prefix of someone else
–
Misconfigurations (fat fingers) are frequent
Evolution-Internet-Architecture/2008
http://www.ripe.net/news/study-youtube-hijacking.html
Internet architectural evolution:
Sequence of reactive updates

1978-1983

1982

1980s

1988

1989

1992
TCP split into TCP and IP - Cutover
from NCP to TCP/IP as a reaction to
the limitations of NCP
DNS as a reaction to the net getting too
large for hosts.txt files
EGP, and OSPF as reactions to scaling
problems with earlier routing protocols
TCP congestion control in response to
congestion collapse
BGP as a reaction to the need for
policy routing in NSFnet
CIDR as a reaction to running out of
class B
Internet architectural evolution:
Sequence of reactive updates
… as the Internet become bigger, it becomes a lot harder to change
while the Internet is accumulating problems faster than they are being fixed
ARPANET in 1974
NA Internet in March 2006
62 host computers (37 nodes)
Red: Verizon, Blue: AT&T, Yellow: Qwest, Green is other
backbone players like Level 3 & Sprint Nextel, Black: entire
cable industry together, Gray: everyone else
Source: http://som.csudh.edu/cis/lpress/history/arpamaps/
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Internet Modeling

Dimensions: Topology - Traffic - Protocols
Topology
Traffic
Routing protocols
Protocols
Congestion control
Forwarding protocols
Goals of Internet topology modeling
Better understand Internet topology and its evolution
•
•
•
Find simple fundamental properties underlying Internet
“complex system”
Understand why these properties appear and their effects
Internet structure fundamentally affects functionality: derive
Topology metrics
Objectives
•
Design more efficient routing protocols
–
•
Create more accurate models
–
•
Because routing protocol efficiency depends on topology
Simulation tools for testing old and new routing protocols
Speculate future internet topology
–
–
Will the current protocols perform well in the future?
How many high-class, mid-class and simple routers to produce?
Why is it important to model the network
topology
1. Performance evaluation of protocols
2. Topology constrains applications and services that
run on top of it
•
Traffic engineering, capacity/resource planning,
provisioning and management
3. Understanding large-scale properties
•
•
Network reliability and robustness to accidents, failures
and attacks on network components (security)
New routing protocol design, development, and testing,
e.g. of scalability, stability and convergence properties
Internet Topology modeling

Graph Representation
•
•

Most analysis considers them as un-weighted graphs
Not annotated with capacity or latency
Internet topology graphs: two levels of granularity
•
•
Router level modeling  router graphs
Inter-domain level modeling  AS graphs
Internet Topology modeling
1. Router-level modeling
•
Router-level adjacencies graph
Vertices/nodes represent routers
– Edges represent one-hop IP connectivity
–
•
Examples
Waxman (Waxman 1988): router level model capturing locality
– Transit-stub/GT-ITM (Clavert/Zegura, 1995), Tiers (Doar, 1996): router
level model capturing hierarchy
–
2. Domain- (AS-) level modeling
•
AS level BGP peering Graph
Vertices/nodes represent domains (AS)
– Edges represent peering relationships between domains (AS)
–
•
Examples
Inet (Jin 2000): AS level model based on degree sequence
– BRITE (Medina 2000): AS level model based on evolution
–
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Models for Network Topology
1988
1996
Spatial/Graph Models
No clue era
Structural Models
Common
sense era
Pre power law era
Power law era
1999
Degree-based Models
Random Graphs models
Model
Pure random model
(ER model)
Waxman model
Exponential model
Locality model



Year
p
1960
P(u,v) =  e-d/( L)
P(u,v) =  e-d/(L-d)
 if d < r
 if d ≥ r
1988
d is the distance from u to v d(u,v)
L is the maximum distance between any two nodes
0<, 0<1
•
•

Probability
Increasing  increases the number of edges in the graph
increasing  increases the ratio of long edges to short edges
r is the boundary
Random Graph Model
Erdös-Renyi (ER) Model
Basic random graph model: given n vertices, an edge
between any two vertices exists with a probability p,
independent of any other edge in the network
•
•
•
Graph G(V,E)
number of vertices (or nodes) V : n
number of edges (or links) E : m
Probability : 0 ≤ p ≤ 1
For each pair (i,j), generate the edge (i,j) independently with
probability p  ensemble Gn,p with average number of edges
n(n - 1)
m p
2
n(n -1)
where
is the number of candidate edges
2
Random Graph Model
Erdös-Renyi (ER) Model
Probability p(k) that a node has a degree k is Binomial:
 k  k
 p (1 - p) n-1-k
p(k )  
 n - 1
In practice, this is the Poisson distribution
p(k ) 
lk e - l
k!
For large n (n >> k z) where l is the mean
average degree l = 2m/n = p(n-1) ≈ pn
The expected value and variance of a Poisson-distributed
random variable P(k) is equal to λ and so is its variance: P(k,l)
Random Graph Model
Erdös-Renyi (ER) Model
The measurements on real networks are usually compared
against those on “random networks”
Problem: find the probability distribution that best fits the
observed data
fk = fraction of nodes with degree k =
probability of a randomly selected node
to have degree k
zk - z
p(k)  P(k; z)  e
k!
With the random graph model the node
degree distribution is Poisson of mean
z=Np
Highly concentrated around the mean z (average degree) ->
the probability of very high degree nodes is exponentially small
Random Graph Model
Erdös-Renyi (ER) Model
Source: http://www.caida.org (ISMA 2006)
Waxman Models and Generators
(Waxman 1988)
Waxman model:
•
•
Router-level model capturing locality
Idea based on the observation that long-range links are
expensive
–
–
•
Variant of random graphs (ER model) where the probability
that any two vertices are connected decreases with the
increasing distance between them
Intuitively: the farther apart the two nodes are, the less likely
they will be connected)
Successful only in representing small networks
Waxman Models and Generators
(Waxman 1988)
Algorithm:
•
•
Place N nodes randomly on a 2-dim space (plane) with
dimension L (diameter)
Add a link between node pair (u,v) with edge probability (=
probability that two nodes u and v separated by Euclidian
distance d are connected)
P(u,v) =  e-d/( L)
u
d(u,v)
where:
d is Euclidean distance (u,v)
 (0 < ) degree of distance sensitivity
 (  1) edge density
•
Waxman topology has an exponential degree distribution
v
Waxman Models and Generators
(Waxman 1988)
Example of network topology generated by the
Waxman model
Random Graph Model
Erdös-Renyi (ER) Model
Are E-R graphs realistic?

They have small world properties (diameter is
logarithmic in the size of the graph) but low clustering
coefficient:
•
Example for autonomous internet systems, compare 0.30
with 0.0004 [Pastor-Satorras and Vespignani]

Unrealistic degree distributions: degrees not
concentrated around mean

Exponential tails: instead heavy tailed degree
distribution

Result: departure from ER model and variants
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Structural Models
Based on the observation that real networks are not
structured randomly (engineering dimension) but
exhibit
•
•
•
•
Hierarchical structure (two or three tier-level structure)
Specialized domains (transit domains, stub domains)
Connectivity requirements
Redundancy (resiliency)
Characteristics incorporated into
•
•
Tiers generator
GT-ITM: Georgia Tech Internetwork Topology Models
Structural Generators: Tiers
(Doar 1996)
 Router
level model capturing hierarchy
 Three
level hierarchy model (3-level Tiers) comprising
LAN, MAN, and WAN
•
•
•
•
•
•
Can specify numbers of LAN, and MAN partitions, number
of WANs = 1 (connected)
LANs are modeled as star with router at center and host
around (star-shaped LANs)
Produces connected sub-graphs by joining all nodes in one
domain using a spanning tree: WAN and MANs are first
connected using a spanning tree (guarantees connectivity)
Remaining links connected in order of increasing distance
Closer connections preferred, like a mesh/grid
Adding edges for intra-domain and inter-domain
redundancy
Structural Generators: Tiers
(Doar 1996)
Single level Tiers
Three level Tiers
Structural Generators: Transit-Stub
(Calvert/Zegura 1997)

Router level model capturing hierarchy

Basic idea
•
•

Combination of number of simple topologies in a hierarchical structure:
transit domains and stub domains modelled as random graphs
Supports only two-level hierarchy
Transit-Stub generator
•
•
•
•
•
Generate a connected random graph where each node is replace by
connected random graph (transit domain)
For each node in transit domain, a number of connected random
graphs (stub domains) are generated and attached to that transit node
by transit-stub links
Add some “extra” connectivity: transit-stub links, and stub-stub links
Possibility to assign edge weights
Several user-specified parameters to control the structure of the graph:
N transit domains and M stub domains with n and m average routers,
ratio stub to transit ( need for experimental estimates to set values to
the parameters of the model)
Structural Generators: Transit-Stub
(Calvert/Zegura 1997)
Router level model modeling hierarchical structure
N Transit domains
•
•
•
placed in 2-d space
populated with n routers
connected to each other
M Stub domains
•
•
•
placed in 2-d space
populated with m routers
connected to transit domains
Use edge weights so that e.g.
•
•
•
Paths between two nodes in a domain
remains in that domain
Paths between two nodes in two
different domains traverses zero or
more transit domains
Four weights (in order)
–
–
–
–
intra-domain edges
Transit-Transit edges
Stub-Transit edges
Stub-Stub edges
Structural Generators: Transit-Stub
GT-ITM
GT‐ITM (Georgia Tech Internetwork Topology Models)
•
Router-level model simulator capturing the actual Internet
hierarchy by differentiating structural elements: the transit
and the stub domains
Transit domain
•
•
Number of transit domains placed in 2-dim space
Populated with routers and connected to each other
Stub domain
•
•
•
Number of stub domains placed in 2-dim space
Populated with routers
Connected to transit domains
Transit-stub
•
Connectivity
Structural Generators: Transit-Stub
GT-ITM

Characteristics
•
•
More probabilistic than Tiers
More control parameters than Tiers
–
•

Can guarantee large graphs with small average node degrees.
Follows standard Internet routing policy
Note: GT-ITM (as other Transit-stub) have been abandoned
as they did not give power law degree graphs (new topology
generators and explanation for power law degrees were
sought)
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Topology measurement
1. Router-level topologies reflect physical connectivity
between nodes
Inferred from tools like traceroute or well known
public measurement projects like Mercator and
Skitter
2. AS-level Internet topology where AS graph reflects
a peering relationship between two providers/clients
Inferred from inter-domain routers that run BGP and
public projects like Oregon Route Views
Topology measurement
Router-level Internet topology

Construction
•
•
Probing techniques (e.g. traceroute) discovers
compliant IP routers along path between selected
network host computers
Coupled with source-routing (requires sourcerouting enabled routers, which are only about 8%
of the total routers) and removing aliases (which
IP addresses belong to same router) Interface
collapsing algorithms
Topology measurement:
Router-level Internet topology
Data: traceroute, returns sequence
(list) of IP addresses (hops) along the
path from source to given destinations
Collection challenges:
•
•
•
obtaining sufficient traceroute origin points
deciding set of destination IP addresses
(for coverage)
limiting traceroute load
source 0
S1
D1
Post-processing challenges:
•
resolving aliases (which IP addresses
belong to same router)
destination 0
Topology measurement:
Router-level Internet topology

Examples of Large-scale traceroute experiments
•
•
•
•
Pansiot and Grad: router-level map from around 1995
Cheswick and Burch: mapping project started in 1997
Mercator (single source with source-routing): router-level
maps from around 1999 by R. Govindan and H.
Tangmunarunkit)
CAIDA Skitter: multiple sources (now replaced by
“Archipelagos measurement infrastructure” or Ark (*)
–
–
–
–
•
•
Traceroute based tool
Continuous measurement since 1998
25 monitors distributed over the Internet
Monitor destination list of 1 million IPv4 addresses
Rocketfuel: router-level maps of individual ISPs by Univ. of
Washington (http://www.cs.washington.edu/research/networking/rocketfuel/)
Dimes (EU project)
(*) http://www.caida.org/publications/presentations/2006/young_wide0611_ark/young_wide0611_ark.pdf
Problems with Traceroute-based
measurements

Ambiguity
•
•

Inaccuracy
•

traceroute is strictly about IP-level connectivity
traceroute cannot distinguish between high connectivity
nodes that are for real and that are fake and due to
underlying Layer 2 (e.g., Ethernet, ATM) or MPLS
Requires some guesswork in deciding which IP
addresses/interface cards refer to the same router (“alias
resolution” problem)
Incomplete/biased
•
•
IP-level connectivity is more easily/accurately inferred the
closer the routers are to the traceroute source(s)
Node degree distribution is inferred to be of the power-law
type even when the actual distribution is not
Topology measurement:
AS-level Internet topology

Construction
•
AS-Path-based: BGP routing tables and/or update
messages
–
–
•
•

RouteViews (University of Oregon) collects BGP
routing tables
RIPE (Europe): BGP dump
Traceroute-based
Synthetic: power laws
Advantage:
•
•
Coarse-grained
Relatively easy to generate
Problems

BGP-based measurements are incomplete
•
•

Contains most nodes (ASes)
Might miss up to 40-50% of existing links
BGP-based measurements are ambiguous
•
•
•
•
Dynamics of AS-level Internet
Requires some guesswork in deciding whether a
“new” node or link is genuine
BGP-based measurements are inaccurate
Use of heuristics for inferring peering
relationships
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Initial observation
Faloutsos et al. (1999) identify power law in node degree
distribution at both router-level graph and AS-level graph:
A random variable X is said to follow a power law with
scaling index g > 0, if P[X > x] ≈ c x -g , as x → ∞
Rank plots: log-log plot of the out-degree of the nodes (# of edges incident) vs rank of
the nodes (index in the order of decreasing out-degree)
A few nodes have lots of connections
Most nodes have few connections
Source: Faloutsos et al. (1999)
Power Law Relationships of the Internet
Topology
Internet Instances: (1997-1998)
 Int-11-97: inter-domain topology
of the Internet with 3015 nodes,
5156 edges and 3.42 avg. outdegree
 Int-04-98: inter-domain topology
of the Internet with 3530 nodes,
6432 edges, and 3.65 avg. outdegree
 Int-12-98: inter-domain topology
of the Internet with 4389 nodes,
8236 edges, and 3.76 avg. outdegree
 Rout-95: The routers of the
Internet with 3888 nodes, 5012
edges, and an avg. out-degree of
2.57
Inter-domain level
Internet Growth : 45%
[Faloutsos 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos.
Power Laws




Power Law 1: Out-degree of nodes vs. rank
Power Law 2: Frequency of out-degree
Power Law 3: Pairs of nodes within h hops
Power Law 4: Eigenvalues of adjacency matrix
Power Laws
Rank exponent (R)
Outdegree exponent (O)
Hop-plot exponent (H)
Eigenvalue exponent (e)
Expression
dv  r
R
v
fd  d
H
P(h)  h
O
li  i
e
Value
R ~ -0,8
O ~ -2,2
H ~ 4,7
e ~ -0,48
Power Law 1: Rank exponent R
Definition: The rank rv of a node v is its index in the order of
decreasing out-degree
Power Law 1: the out-degree (dv) of a node v, is proportional to
the rank of the node (rv) to the power of a constant, R: dv  rvR
Rank exponent (R) = slope of the plot of the out-degrees of the
nodes versus the rank of the nodes in log-log scale
log(dv)
Rank exponent (slope) R = -0.74
R
log(rv)
Power Law 1: Rank exponent R
Log-Log scale graph
•
X axis is rank r, Y axis is out-degree d
Plot approximated well by linear regression - the correlation coefficient is higher than 0.974 !
Power Law 1: Rank exponent R
Observations:
•
•
For the 3 interdomain-level instances, the rank exponent is 0.81, -0.82 and -0.74
For the router-level graph, the rank exponent is -0.48
 This difference suggests that the rank exponent can
distinguish graphs of different nature (e.g. inter-domain vs.
intra-domain)
Interpretation:
•
•
This power-law most likely reflects a principle of the way
domains and routers connect
Captures equilibrium of the trade-off between the gain and
the cost of adding an edge from a financial and functional
point of view
Power Law 1: Application
dv estimation

The out degree dv, of a node v is a function of the
rank of the node rv and the rank exponent, R, as
follows:
1 R
d v  R rv
N

Proof:
•
•
•
Power law 1: dv  rvR
So there’s a proportional constant C such that dv = C rvR
If we require that the outdegree of the Nth node is 1: dN = 1
then can estimate C
1
d N  CN  C  R
N
R
1 R
d v  R rv
N
Power Law 2: Out-degree exponent O
Definition: the frequency of an out-degree d, fd , is the number of
nodes with out-degree d
Power Law 2: the frequency fd, of an out-degree d, is proportional
to the out-degree to the power of a constant, O: fd  dO
Out-degree exponent (O) = slope of the plot of the frequency of
the out-degrees versus the out-degrees in log-log scale
Log(fd)
Out-degree Exponent (slope) O = -2.15
Log(d)
Power Law 2: Out-degree exponent O
Log-log scale graph: Frequency f vs Out-degree d
Plots
•
X axis is degree, Y axis is frequency
The plots are approximated well by linear regression - the correlation coefficient is
higher than 0.966
Power Law 2: Out-degree exponent O
Observation
•
•
•
The value of the out-degree exponent is practically constant
For the inter-domain topology, the out-degree exponent O
is: -2.15, -2.16 and -2.2
For the router-level, the out-degree exponent O router-level
topology is -2.48
Interpretation
•
Suggest that the out-degree exponent O describes a
fundamental property of the network: lower degrees are
more frequent
Power Law 3: Hop-Plot Exponent H
Definition: the neighborhood size P(h) of distance h,
is the total number of pairs of nodes within less or
equal to h hops, including self-pairs, and counting all
other pairs twice
Examples
•
•
For h = 0, only self-pairs: P(0) = N
For h = δ (graph diameter) self-pairs and all other possible
pairs: P(δ) = N2 (maximum possible number of pairs)
Idea
•
•
For a ring topology we have P(h)  h1
For a 2-dimensional grid, we have P(h)  h2 (for h « δ)
Question: Will the number of pairs P(h) for the Internet
follows a similar power-law ?
Power Law 3: Hop-Plot Exponent H
Power Law 3 (approx.): the total number of nodes pairs P(h)
within h hops, is proportional to the number of hops h to the
power of a constant, H: P(h)  hH, h « δ (diameter)
Hop-plot Exponent (H):
for h « δ: H defined as the slope of the plot of the number of
pairs P(h) within h hops vs the number of hops h, in log-log
scale
for h ≥ δ: P(h) = N2
•
•
Log (P(h))
H = 2.83
Horizontal line represents the
maximum number of pairs (N2)
log(h)
Power Law 3: Hop-Plot Exponent H

Observation
•
•

The 3 domain-level datasets have practically equal hopplot exponents: H ~ 4,7
The router-level dataset has a hop-plot H exponent of 2,8
Interpretation
•
The hop-plot exponent can distinguish families of graphs
efficiently
Power Law 3: Application
Effective Diameter

Effective diameter is useful for protocol improvements such as
broadcast extent selection e.g. Automated Time-to-live (TTL)
for broadcast packets in advanced protocols

Given a graph with N, nodes, E edges, and H hop-plot
exponent, we define the effective diameter def of a graph as
 N


 
 N  2E 
2
δ ef

1
H
Applications:
Any two nodes are within def hops of each other with high
probability
For the internet: def is slightly higher than 4
•
•
Rounding to 4, approximately 80% of the pairs of nodes are within
this distance
– If we take the ceiling, 5, more than 95%
–
Power Law 3: Application
Neighborhood Size

The average neighborhood is commonly used to estimate the
message complexity of protocols

Definition: Neighborhood size, NN(h), is the average number
of nodes in a neighborhood of h hops
P(h) - N P(h)
NN(h) 

-1
N
N
where P(h) - N is the number of pairs without the self-pairs

Using hop-plot estimation P(h)
ch
h  δ
P(h)   2
hδ
N
where c  N  2 E
H
c H
NN'(h) 
h -1
N
Power Law 4: Eigen Exponent e
Let A be the adjacency matrix of graph:
2
A=
1
3
0
1
1
1
0
0
1
0
0
The eigenvalue l is a real number such that A v = l v
where v some vector (eigenvalues are related to
topological properties)
 The eigenvalues of a graph are defined as the
eigenvalues for the adjacency matrix of this graph
Problem: find the eigenvalues of the adjacency matrix
ranked in decreasing order (first 20)
Power Law 4: Eigen Exponent e
Power Law 4: the eigenvalues, li,of a graph are proportional to
the order i,to the power of a constant,e
li  i e
Plot the eigenvalue li versus i in log-log scale (the first 20
eigenvalues) where i is in the order of li in l1 ≥ l2 ≥ … ≥ lN
Eigenvalue
log(li)
Exponent (slope) e = -0.48
Correlation coefficient = 0.99
log(i)
Rank of decreasing eigenvalue
Power Law 4: Eigen Exponent e
Log-log scale graph
•
•
X axis is the order of eigenvalue
Y axis is the eigenvalue
Power Law 4: Eigen Exponent e

Eigen exponents value
•
•
for the interdomain-level graphs: e ~ -0,48
for the router-level graph: e = -,0177
 e can also distinguish families of graphs efficiently

Rich literature proves that eigenvalues of a graph are
closely related to topological properties of graphs
•
•
Graph diameter, number of edges, number of spanning
trees
Number of connected components, and more
Power Laws - Summary



Rank Exponent R: The out-degree, dv, of a node v, is
proportional to the rank of the node, rv, to the power a
constant, R (≈ -0.8): dv  r R
v
Out-degree Exponent O: The frequency, fd, of an out-degree d
is proportional to the out-degree to the power of a constant, O
(≈ -2.2): f d  d o
Hop-plot Exponent H: The total number of pairs of nodes, P(h),
within h hops, is proportional to the number of hops to the
power of a constant, H (≈ 4.7):
Effective Diameter: Given a graph with N nodes and1E
edges, define the effective diameter as:
 N 2 H

d ef  
 N  2E 

Eigen Exponent e: The eigenvalues, λi, of a graph are
proportional to the order, i, to the power of a constant, ε (≈ 0.48): l  i e
i
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Degree-based Network Topology Models

Faloutsos et al. (1999) find power law in node
degree distribution at router-level graph &
Autonomous System (AS) graph

Basic Idea: traditional random graphs [Erdös &
Renyí, 1959] do not produce power laws, so develop
new models that explicitly attempt to match the
observed (power law) distribution in node degree
 Led to active research in degree-based network
models: focus on generators that match degree
distribution of observed graph (descriptive methods)
Degree-based Network Topology
Generators

Two methods for generating random networks having
power law distributions in node degree
•
Growth modelling (evolutionary)
–
–
–
–
–
•
Distribution modelling (non-evolutionary)
–

Inet 3.0: enforced power law degree distribution and Preferential
Attachment
BRITE: model based on Incremental growth and Preferential
Attachment
Barabasi-Albert (BA) model: scale free networks characterized by
Incremental growth and Preferential Attachment
Albert-Barabasi (AB) model: variant of BA model
Generalized Linear Preference (GLP) model
Expected Degree Sequence (Given Expected Degree): Power Law
Random Graph (PLRG) and Generalized Random Graph (GRG)
Common features:
•
•
Ignore all system-specific details
Central core of high-degree, hub-like nodes
Inet model
(Jin 2000)

AS level Internet topology generator
•

Number of links generated depends on
•
•

Originally designed to match the measurements of the
original-maps of the AS graph
Total number of nodes
Percentage of nodes with degree 1
Typically generates 26% less links than extended-AS
graph
Inet model
(Jin 2000)
Input



Total number of nodes
Percentage of degree-one nodes
Random seeds
Construction


Generate degree sequence (power law
distribution)
Step 1: build spanning tree over nodes
with degree larger than 1, using
preferential attachment
•
•


Randomly select node u not in tree
Join u to existing node v with probability
d(v)/d(w)
Step 2: Connect degree 1 nodes using
preferential attachment
Step 3: Add remaining edges using
preferential attachment
BRITE
(Medina 2000)

Capture properties
•
•

Key Ideas
•
•
•

Power law relationship
Network evolution (incremental growth)
Preferential attachment of a new node to existing nodes
Incremental growth of the network
Connection locality
Input
•
•
•
•
Size of plane (to assign the node)
Number of links added per new node
Incremental growth
Preferential attachment
BRITE
(Medina 2000)

Incremental growth
•
Inactive
–
–
•
Active
–
–

Places all nodes at once before adding links
New node randomly selects the node to connect from all nodes
Place nodes in the plane gradually one at a time
New node only considers as candidate neighbors from existing
node
Preferential Attachment
•
NONE
–
–
•
ONLY
–
–
•
Preferential attachment is turned off
Using Waxman’s probability function (locality)
Using preferential attachment
d(i)/d(j)
BOTH
–
–
Combine preferential attachment and connection locality
w(i)d(i) /  w(j)d(j) where w(i) is the Waxman’s probability
BRITE
(Medina 2000)
Construction
•
Step 1: Generate small backbone,
with nodes placed:
–
–
•
•
randomly or
concentrated (skewed)
Step 2: Add nodes one at a time
(incremental growth)
Step 3: New node has constant
number of edges connected using:
–
–
preferential attachment and/or
locality
Barabasi-Albert (BA) model (Barabasi, 1999)
Albert-Barabasi (AB) model (Albert, 2000)

Power-law degree distribution can arise from two
mechanisms :
•
•
Incremental growth: continuous addition of new nodes
and edges to the system
Preferential attachment: new nodes are preferentially
attached to nodes that are already well connected
Probability of attachment : Pi(t) = ki(t) / jkj(t)

It is estimated that BA model
generates networks with degree
distribution P(k) ~ k-3
Barabasi-Albert (BA) model (Barabasi, 1999)
Albert-Barabasi (AB) model (Albert, 2000)

Method
•
•
Start with a small number (m0) of nodes
At every step, add a new node with m  m0 edges that link
the new node to m different nodes already present in the
graph
Probability Pi(t) that a new node will be connected to a
an existing node i depends on the connectivity (degree)
ki of that node
-> at each step: Pi(t) = ki(t) / jkj(t)
•
After t steps the model leads to a random network with t
+ m0 nodes and m x t links
Barabasi-Albert (BA) model (Barabasi, 1999)
Albert-Barabasi (AB) model (Albert, 2000)
Method (extended model):
•
•
Start with m0 isolated (unconnected) nodes
At each time t step, perform one of following three operations
– Add new m links with probability p
–
–
–
Randomly select starting points (one of the link endpoint)
Preferentially select the other link endpoint with probability
BA Model: Pi(t) = ki(t) / jkj(t)
AB Model: Pi(t) = [ki(t) + 1] / j [kj(t) + 1]
Rewire m links with probability q  ki 
m
ki  1
   -q  qm
N
 t 
 (kj  1)
j
–
Add new node with m links with probability 1-p-q
–
Preferentially select m link endpoints to connect with
ki  1
 ki 
   (1 - p - q)m
 t 
 (kj  1)
j
Barabasi-Albert (BA) model (Barabasi, 1999)
Albert-Barabasi (AB) model (Albert, 2000)

Incremental growth: starting from initial graph G(t=t0): G0
•
•
•
Add new nodes to graph G
Add new links to graph G
Rewire links: re-arrangement of already existing links
0.5
0.5
0.25
existing node
new node
0.5
G(t-1)

0.25
G(t)
G(t+1)
Linear preferential attachment: new nodes prefer existing
nodes with large-degree
Pi(t) probability of selecting an existing node i of degree ki at
time t
BA Model: Pi(t) = ki(t) / j kj(t)
AB Model: Pi(t) = [ki(t) + 1] / j [kj(t) + 1]
•
Generalized Linear Preference (GLP)
(Bu, 2002)


A modification of the BA model
Evolution of AS graph mostly due to two reasons
•
•

Addition of new nodes
Addition of new links between existing nodes
Method
•
•
It starts with m0 nodes connected through m0-1 links
At each time step (t), perform one of the following two
operations:
–
–
With probability p, add m < m0 new links between m pairs of
nodes chosen from existing nodes (for each end of each link,
node i is chosen with probability Pi(t)
With probability 1-p, add new node with m new links and
connected to m existing nodes (each link is connected to node i
already present in the system with probability Pi(t)
 Growth of new nodes and new links are independent (the
probability that node i increases its degree ki is a function
of that degree)
Generalized Linear Preference (GLP)
(Bu, 2002)

Probability of selecting existing node i at time t is
proportional to its degree ki−
Pi(t) = [ki(t) - ] / j [kj(t) - ]

 ∈ ]-∞, 1[ is a tunable parameter that
•
•
•

indicates the preference for a new node (edge)
connecting to more popular nodes
can be adjusted such that nodes have a stronger
preference of high degree nodes than BA model
Note: the smaller the value of  is, the less preference
gives to high degree nodes
Matches AS graph (original-maps) in terms of two
characteristics of small-world networks
•
•
Characteristic path length
Clustering coefficient: quantifies how likely the
neighbors of a node are to be connected
Expected Degree Sequence

Non-evolutionary models

Based on random graph models (inspired by graph
theory) that skew probability distribution to produce
power laws in expectation (expected degree
sequence)

Examples:
•
•
Power Law Random Graph (PLRG) [Aiello00]: enforce
power law degree distribution and random matching of
nodes
Generalized Random Graph (GRG) [Chung03]
Power Law Random Graph (PLRG)
(Aiello, 2000)

Suppose n vertices of degree k where k and n satisfy
Log n =  -  Log k

What can be calculated
•
•
•
The maximum degree of the graph
The number of vertices
The number of edges

Input: , 

Construction:
•
•
•
•
Assign to nodes v degree k (kv) drawn from power law distribution
Pool creation: form a set L (pool) containing kv distinct copies of each
node v of degree k
Pairing: choose a random matching of the elements of L to form actual
links
For two nodes u and v, the number of links joining u and v is equal to the
number of links in the matching of L joining copies of u to copies of v
Generalized random graph (GRG)
(Chung, 2003)

Generalized random graph (GRG) with a given
expected degree sequence K = {k1,…,kn} for vertices
1,…,n

Construction:
•
•
Step 1: assign each node its (expected) degree
Step 2: insert links between the nodes i and j chosen
independently according to a probability that is
proportional to the product of their degrees: pij = c ki kj
(where c is small constant)
If the assigned expected node degree sequence follows a
power-law, the generated graph’s node degree distribution
will exhibit the same power law
Properties of Degree-based Models
Preferential Attachment



Expected Degree Sequence (PLRG)
Degree sequence follows a power law (by construction)
High-degree nodes correspond to highly connected
central “hubs”, which are crucial to the system
Achilles’ heel: robust to random failure, fragile to
specific attack
Power Laws Relationships of the Internet
Topology: Revisited
Main Findings
•
•
•
•
•
•
BGP AS paths might not cover the complete AS Topology.
Distribution of node degrees not exactly a power law but
definitely a heavy tailed distribution.
A vast majority of new ASes are born with vertex degree 1
or 2
ASs can die also!! (deaths not included in the BA Model)
ASs have much stronger preference to connect to high
vertex degree ASs than predicted by the linear preferential
model
Rewiring not a significant factor in the evolution of the
Internet
Scale Free networks
Scale free networks (term introduced by Barabasi)
 Idea: universal model of network topologies that
exhibit power law distributions in the network node
connectivity
 Definition of scale free: any function f(x) that remains
unchanged to within a multiplicative factor under a
rescaling of the independent variable x
 Power law function since only solutions to f(a x) =
g(a) f(x)

New Node
Scale Free networks
1. Continuous (incremental) growth
•
•
Existing models of networks did not include the addition of
nodes over time. The graphs remained static.
Scale free networks are in a state of continuous growth by
incremental addition of new nodes and links to the system
2. Preferential attachment
•
•
New nodes tend to connect to nodes that are already well
connected. New nodes have higher probability of
connecting to the existing nodes with high connectivity,
i.e., a “rich-gets-richer”
“Rich club” phenomenon - power laws in asymptotic limit:
new nodes attach preferentially to high-degree nodes
(well-connected nodes) in linear proportion to their degree
Note: Role of Rewiring process: Re-arrangement of
the already existing links
Rich Club Phenomenon
Rich nodes
•
Power-law technologies have small number of nodes having
large number of links
AS graph shows this phenomenon
•
•
Rich nodes are well connected to each other
Rich nodes are connected preferentially to the other rich
nodes
Measured in the
•
•
Original-maps of the AS graph (BGP Routing tables by
University of Oregon Route Views Project)
Extended-maps of the AS graph (BGP Routing tables +
Looking Glass (LG) data + Internet Routing Registry data)
Scale Free networks - Controversy
Scientists spot Achilles heel of the Internet
Fact: scale-free networks have approximately power law
degree distributions
Claim:
If the Internet has power law degree distribution
Then, the Internet must be scale-free
⇒ The Internet has the properties of a scale-free network
•
•
Implications of “scale free” network structure
Few centrally located and highly connected hubs (highdegree nodes correspond to highly connected central
“hubs”, critical to the system)
⇒ Achilles’ heel: robust to random attack/node failures
(probability of targeting hub very low) but vulnerable to
targeted attacks
•
"The reason this is so is because there are a couple of very big nodes
and all messages are going through them. But if someone maliciously
takes down the biggest nodes you can harm the system in incredible
ways. You can very easily destroy the function of the Internet,..."
-- “Achilles heel of the Internet” Albert, Jeong, Barabasi, Nature 2000
Real network vs Preferential attachment
Networks with the same statistical features can be
OPPOSITES in terms of engineering
~ Real network
Meshed, low-degree core
Result of design
High performance and robustness
Preferential Attachment
High degree central “hubs”
From random construction
Poor performance and robustness
Problems and Controversy

Scale-free claims: based critically on the implied
relationship between power laws and a network
structure that has highly connected “central hubs”
•
•

The scale-free models ignore all system-specific
details in making their claims
•
•
•

Not all networks with power law degree distributions have
properties of scale free networks (The Internet is just one
example!)
Building a model to replicate power law data is no more
than curve fitting (descriptive, not explanatory)
Ignore architecture e.g. hardware, protocol stack
Ignore objectives e.g. performance
Ignore constraints e.g. geography, economics
Conclusion
•
•
The scale-free claims of the Internet are not merely wrong,
they suggest properties that are opposite to the real thing
Fundamental difference: random vs. designed
Outline
1. Organization of the Internet
2. Evolution of the Internet
3. Internet Topology modelling
•
•
•
•
•
•
•
Network properties
Random Graphs models and generators
Structural models and generators
Topology measurements
Power Law relationships
Degree-based models and generators
Internet topology metrics
Internet Topology Metrics
A network topology is characterized by topological
parameters, or topology metrics like:
•
•
•
•
•
•
•
•
Average degree
Degree Distribution (DD)
Joint Degree Distribution (JDD) a.k.a Degree correlation
Clustering
Rich club coefficient (RCC)
Distance
Betweenness
Spectrum
Average degree

Definition: average node degree k
k = 2m/n
where m = number of links
n = number of nodes (a.k.a graph size)

Interpretation:
•
•
•
Coarsest connectivity characteristic of the topology
Networks with higher k are “better-connected” on average
and, consequently, are likely to be more robust
Detailed topology characterization based only on the
average degree is limited
Reason: graphs with the same average node degree can
have very different structure
Degree Distribution (DD)
Definition:
•
Node degree distribution (DD) P(k) is the probability that a
randomly selected node is k-degree:
n( k )
P(k ) 
n
where n(k) = number of nodes of degree k (k-degree nodes)
•
DD contains more information about connectivity than the
average degree
Reason: given a specific form of P(k) we can always restore
the average degree by
k = k=1
kmax
k P(k)
where kmax is the maximum node degree in the graph
Degree Distribution

Interpretation
•
•
Most frequently used topology characteristic
[Faloutsos99] observation that Internet’s degree distribution (both
router and AS-level) follows a power law had significant impact
on network topology research
–
–
•
Smooth power law degree distribution indicates
–

Structural Internet models before failed to exhibit power laws 
organized hierarchy existence among ASes
[Tangmunarunkit02]: topologies derived from structural generators
that incorporated hierarchies of AS tiers did not have much in
common with topologies obtained from real observed data
Indicates no organized tiers among ASes. The power law
distribution also implies substantial variability associated with
degrees of individual nodes
Note:
•
Node DD tells how many nodes of a given degree are in the
network but it does NOT provide information on the
interconnection between these nodes
Reason: given P(k), structure of the neighborhood of the average
node of a given degree is still unknown
Degree Distribution
Approximated by long tail power law distribution of node degree
k: P(k) ∼ k-g, where the power-law exponent g = 2.254
log(dv)

log(rv)

In practice, the distribution is not a strict power law
•
•

The Internet contains more 2-degree nodes than 1-degree nodes
The distribution has a longer tail, i.e. the maximum degree is much
larger large than expected by the power-law
The Internet is characterized by a fewer nodes with a large
degree a large number of nodes with a low degree
Source: Faloutsos et al (1999)
Degree Correlation
Definition:
•
The joint degree distribution (JDD) P(k1,k2), or the node
degree correlation matrix: probability that a randomly
selected edge connects k1- and k2-degree nodes:
P(k1,k2) = μ(k1, k2) × m(k1, k2)/(2m)
where
μ(k1, k2) = 1 if k1 = k2 and 2 otherwise
m(k1, k2) is the total number of edges connecting
nodes of degrees k1 and k2
•
JDD contains more information about the graph connectivity
than the degree distribution
Reason: given a specific form of P(k1, k2) we can always
restore both the degree distribution P(k) and k
Degree Correlation
Summary statistic of JDD:
•
Average neighbor connectivity:
Average neighbor degree of the average k-degree node
•
•
•
JDD shows whether AS of a given degree preferentially
connect to high- or low-degree AS
JDD provides more information than DD (information about
1-hop neighborhoods around a node) but JDD does not tell
us how neighbors interconnect
Note: in a full mesh graph, knn(k) reaches its maximal
possible value: n − 1. Therefore, for uniform graph
comparison plot normalized values knn(k)/(n − 1)
JDD and Assortativity coefficient r
Summary statistic of JDD: assortativity coefficient r
where −1 ≤ r ≤ 1
Interpretation of r :
•
Disassortative networks (r < 0) have an excess of radial links
(links connecting high-degree nodes to low-degree nodes) i.e.
links connecting nodes of dissimilar degrees
–
–
•
Cons: more vulnerable to both random failures and targeted attacks
Pros: vertex covers in disassortative graphs are smaller, which is
important for applications such as traffic monitoring and prevention of
DoS attack
Assortative networks (r > 0) have an excess of tangential links
i.e links connecting nodes of similar degrees
Assortative coefficient r

The Internet exhibits a negative correlation between a node’s
degree k and its nearest-neighbors average degree
Diassortative mixing
r<0
Assortative mixing
r>0
 Disassortative mixing (r = -0.236 < 0): high-degree nodes
tend to connect with low-degree nodes and visa versa
Likelihood S

[Li04] introduces likelihood S (structural metric)
•
Definition: sum of products of degrees of adjacent nodes
S(g) = i,j ki kj
•
S measures randomness to differentiate between multiple
graphs with the same DD (measures how “hub-like” the
network core is)
–
•
•
(ki = degree of node i)
(where node i, j are connected)
S depends on graph structure, not the generation mechanism
S is linearly related to the assortativity coefficient
S is used as measure of graph randomness: a topology
with low likelihood is not random; it results from some
sophisticated evolution processes involving specific design
purposes
–
–
S provides a measure of the amount of order, e.g., engineering
design constraints, present in a given topology
Router-level topologies are not “very random” but instead the
result of sophisticated engineering design
Example: Five networks with the same
node DD
(b) Network resulting from preferential
attachment
(c) Network resulting from the general
model of random graphs (GRG) method
with a given expected degree sequence
(a) Node degree distribution (degree
versus rank on log-log scale)
(d) Heuristically optimal topology (HOT)
using Power Law Random Graph (PLRG)
(e) Abilene-inspired topology
(f) Sub-optimally designed topology
Structure determines performance
HOT
P(g) = 1.13 x 1012
PA
P(g) = 1.19 x 1010
PLRG/GRG
P(g) = 1.64 x 1010
Clustering

Quantifies how close node’s neighbors are to forming
a clique (complete graph i.e. every pair of distinct
vertices is connected by an edge)

Definition:
Local clustering C(k): C(k) = 2mnn(k) / [k(k-1)]
•
where mnn(k) is the average number of links between the
neighbors of k-degree nodes
k(k-1)/2 is the maximum possible number of links
between neighbors of k-degree nodes
If two neighbors of a node are connected, then these three
nodes together form a triangle (3-cycle)
 Local clustering measure of average number of 3cycles involving k-degree nodes
Clustering
Associated statistics
•
Mean local clustering (average value of C(k)):
Cm = k C(k)P(k)
•
Clustering coefficient Ccoeff percentage of 3-cycles among all
connected node triplets in the entire graph
Interpretation
•
•
Clustering is a measure of local robustness in the graph
Implications:
–
–
–
The higher the local clustering of a node, the more interconnected are
its neighbors, thus increasing path diversity locally around the node
Networks with strong clustering are likely to be chordal or of low
chordality, 4 which makes certain routing strategies perform better
Clustering used as litmus test for verifying the accuracy of a topology
model or generator
Rich-Club Coefficient (RCC)
Rich club coefficient f(r/n)
In a graph of size n, r = 1 . . . n are the first r nodes ordered
by their non-increasing degrees
Definition: ratio of the number of links in the subgraph
induced by the r largest-degree nodes to the maximum
possible number of such links
Interpretation: RCC is a measure of how close r-induced
subgraphs are to cliques
Rich-Club Coefficient (RCC)

In the Internet, the high-degree nodes (a.k.a rich
nodes) are tightly interconnected with themselves,
forming a rich-club
•

Club membership: the richest r nodes (= nodes with
degree larger than k)
The interrelationship between a set of rich nodes is
quantified by the rich-club connectivity:
•
Ratio of actual number of links between club members to
maximum possible links between club members
Distance

Definition:
•

The distance distribution d(x) is the number of pairs of
nodes at a distance x, divided by the total number of pairs
n2 (self-pairs included)
Associated statistics with distance distribution of a
graph
•
•
Average distance dm
Standard deviation s (a.k.a distance distribution width
since distance distributions in Internet graphs have a
characteristic Gaussian-like shape)
Distance
Interpretation
•
Distance distribution is important for routing:
–
–
•
Distance distribution plays a vital role in robustness of the
network to worms
–
–
•
Distance-based locality-sensitive approach root of most modern
routing algorithms: performance of routing algorithms depend mostly
on the distance distribution
Short average distance and narrow distance distribution width break
the efficiency of traditional hierarchical routing: root causes of interdomain routing scalability issues in the Internet
Worms can quickly contaminate a network that has small distances
between nodes
Topology models (that accurately reproduce observed distance
distributions) will benefit development of techniques to quarantine
the network from worms
Expansion metric: renormalized version of distance
distribution d(x)
–
Critical metric for topology comparison analysis
Betweeness

Betweenness
•
•

Measures the number of shortest paths traversing a
vertex(node) or edge(link) if each individuals send a
message to all other individuals
Estimation of the potential traffic load (flow of information)
on this node/link assuming uniformly distributed traffic
following shortest paths
Definition
•
•
•
sij : number of shortest paths between nodes i and j
y : either a node or link
sij(y) : number of shortest paths between i and j going
through y
Betweeness By = ij sij(y)/sij
•
The maximum possible value for node and link
betweenness is n(n − 1)  to compare betweenness in
graphs of different sizes, normalization by n(n − 1)
Betweeness
Interpretation
•
•
Important metric for traffic engineering applications that try to
estimate potential traffic load on nodes/links and potential
congestion points in a given topology
Critical for evaluating the accuracy of topology sampling by
tree-like probes (e.g. skitter and BGP)
–
–
•
The broader the betweenness distribution, the higher the statistical
accuracy of the sampled graph
Note: exploration process statistically focuses on nodes/links with
high betweenness thus providing an accurate sampling of the
distribution tail and capturing relevant statistical information
Note: link betweenness is not a measure of centrality but a
measure of a certain combination of link centrality and
radiality
Spectrum

Definition
•
A : n × n adjacency matrix of a graph constructed by
setting the value of its element as
–
–
•
•

aij = aji = 1 if there is a link between nodes i and j
all other elements have value 0
Scalar l are the eigenvalue and vector v the eigenvector
of A if A v = l v
Spectrum of a graph is the set of eigenvalues l of its
adjacency matrix A
Interpretation: (one of the) most important global
characteristics of the topology
•
•
Provides bounds for critical graph characteristics such as
distance-related parameters, expansion properties, and
values related to separator problems estimating graph
resilience under node/link removal
Most networks with high values as eigenvalues have small
diameter, expand faster, and are more robust
Spectrum

Example of spectrum-related metrics
•
Robustness of network
–
–
–
•
Performance: Max. traffic throughput of network
–

Relation to spectrum: network conductance can be tightly estimated
by the gap between the first and second largest eigenvalues
Application to Traffic engineering
•

Critical metric for topology comparison analysis
Measure of network robustness under link removal (equals
minimum balanced cut size of a graph)
Relation to spectrum: graph’s largest eigenvalues provide bounds
on network robustness with respect to both link and node removals
Graphs with larger eigenvalues have, in general, more
node- and link-disjoint paths to choose from
Spectral analysis
•
•
Powerful tool for detailed investigation of network structure
Example: discovering clusters of highly interconnected
nodes and revealing AS hierarchy
Scaling dependency on Topology
Internet topological properties characterized by
•
•
•
•
Node degree distribution: approximated by long tail power law
distribution P(k) ∼ k-γ, γ = 2.254 (scaling index)
The Internet is characterized by a fewer nodes with a large degree a
large number of nodes with a low degree
Node degree correlation: negative correlation between a node’s
degree k and its nearest-neighbors average degree
Disassortative mixing (r = -0.236 < 0): high-degree nodes tend to
connect with low-degree nodes and visa versa
Clustering: large numbers of short subgraphs (3-/4-cycles) ><
regular tree structures basic units for routing redundancy and
community clustering
Shortest Path Length: The average length of shortest paths
between all pairs of nodes on the Internet is just over 3 hops
Average AS-path length ~constant (avg. 3,4) >< hierarchical routing
(performs well for graphs with large distances between nodes)
Backup Material
Power laws

Power-laws are laws of the form: P(k) = C k-g
where

•
g : scale index (power law exponent, typically 2 ≤ g ≤ 3)
•
C : constant
Properties of power laws
P(k) = C k-g
 log(P(k))
= -g log(k) + log C
Power-law distribution gives a line in log-log plot
log frequency
frequency
degree
α
log degree
Power-law distributions: Examples

Heavy-tail distribution
•
•
non-negligible fraction of nodes has very high degree (hubs)
scale-free: no characteristic scale, average is not informative
Source [Newman 2003]
Graphs: Examples
Ring graph
Power Law Graph
Fully Connected graph
Random graph
References
References
[Aiello00]
[Aiello01]
[Aiello02]
[Albert00a]
[Albert00b]
[Albert02]
[Alderson05a]
[Alderson05b]
[Barabasi99a]
[Barabasi99b]
[Barabasi01]
[Barabasi03a]
[Barabasi03b]
[Bollobas01]
[Bollobas03a]
[Bollobas03b]
W. Aiello, F. Chung, and L. Lu. A Random Graph Model for Massive Graphs. In Proceedings
of ACM Symposium on Theory of Computing, pp.171—180, 2000.
W. Aiello, F. Chung, and L. Lu. A Random Graph Model for Power Law Graphs.
Experimental Math. 10, pp.53—66, 2001.
W. Aiello, F. Chung, and L. Lu. Random Evolution in MassiveGraphs. In Handbook on
Massive Data Sets, vol.2, pp. 97–122, Spinger, 2002.
Reka Albert, Hawoong Jeong, and Albert-Laszlo Barabasi, Attack and error tolerance of
complex networks, Nature 406, pp.378-382, 2000.
R. Albert and A. L. Barabasi, Topology of Evolving Networks: Local Events and Universality,
Physical Review Letters, 85(24):5234-5237, 2000.
R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Reviews of Modern
Physics, vol.74, pp.47-97, 2002.
D. Alderson, L. Li, W. Willinger, J.C. Doyle, Understanding Internet Topology: Principles,
Models, and Validation. ACM/IEEE Trans. on Networking 13(6), 2005.
D. Alderson and W. Willinger, A contrasting look at self-organization in the Internet and nextgeneration communication networks. IEEE Comm. Magazine. July 2005.
Albert-Laszlo Barabasi and Reka Albert, Emergence of scaling in random networks,
Science, vol.286(5439), pp.509-512, 1999.
A.-L. Barabasi, R. Albert, and H. Jeong, Mean-field theory for scale-free random networks,
Physica A 272, pp.173-187, 1999.
A.-L. Barabasi, E. Ravasz, and T. Vicsek. Deterministic scale- free networks, Physica A,
299:599, 2001.
Albert-Laszlo Barabasi, Linked: How Everything Is Connected to Everything Else and What
It Means (Plume, 2003).
A.-L.Barabasi, Z. Dezso, E. Ravasz, S.H.Yook, and Z. Oltvai. Scale-free and Hierarchical
Structures in Complex Networks, AIP Conference Proceedings, 2003.
B. Bollobas, O. Riordan, J. Spencer, and G. Tusnady. The Degree Sequence of a ScaleFree Random Process. Random Structures and Algorithms, 18(3):279—290, 2001.
Bela Bollobas and Oliver Riordan, Robustness and Vulnerability of Scale-Free Random
Graphs, Internet Mathematics, vol.1, no.1, pp.1-35, 2003.
Bela Bollobas and Oliver Riordan, Coupling Scale-Free and Classical Random Graphs,
Internet Mathematics, vol.1, no.2, pp. 215-225, 2003.
References
[Bollobas04]
B. Bollobas and O. Riordan. The Diameter of a Scale-Free Random Graph, Combinatorica,
vol.24, no.1, pp.5–34, 2004.
[Bu02]
T. Bu and D. Towsley, "On distinguishing between Internet power law topology generators,"
in Proc. of IEEE INFOCOM, Dec. 2002.
[Carlson99]
J. M. Carlson and J. Doyle, "Highly optimized tolerance: A mechanism for power laws in
designed systems," Physical Review E, vol.60, pp.1412-1427, 1999.
[Chang03]
H. Chang, S. Jamin, and W. Willinger, Internet connectivity at the AS-level: An optimizationdriven modeling approach, in MoMeTools, 2003.
[Chang04]
H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Willinger, Towards capturing
representative AS-level Internet topologies, Computer Networks Journal, vol.44, pp.737-755,
April 2004.
[Chang06]
H. Chang, S. Jamin, and W. Willinger, To peer or not to peer: Modeling the evolution of the
Internet's AS-level topology, in Proc. IEEE INFOCOM, 2006.
[Chen02]
Q. Chen, H. Chang, R. Govindan, S. Jamin, S. J. Shenker, and W. Willinger, The origin of
power laws in Internet topologies revisited, in Proc. IEEE INFOCOM, June 2002.
[Chung03a]
F. Chung and L. Lu. The Average Distance in a Random Graph with Given Expected Degrees.
Internet Mathematics, vol.1, no.1, pp.91–114, 2003.
[Chung03b]
Fan Chung, Linyuan Lu, and Van Vu. The Spectra of Random Graphs with Given Expected
Degrees. Internet Mathematics, vol.1, no.3, pp.257-275, 2003.
[Chung04]
Fan Chung and Linyuan Lu. Coupling Online and Offline Analyses for Random Power Law
Graphs. Internet Mathematics, vol.1, no.4, pp.409-461, 2004.
[Cooper04]
Colin Cooper, Alan Frieze, and Juan Vera. Random Deletion in a Scale-Free Random Graph
Process. Internet Mathematics, vol.1, no.4, pp.463-483, 2004.
[DallAsta05]
L. Dall'Asta, I. Alvarez-Hamelin, A. Barrat, A. Vazquez, and A. Vespignani, "Exploring networks
with traceroute-like probes: Theory and simulations," Theoretical Computer Science, Special
Issue on Complex Networks, 2005.
[Dimitropoulos06] X. Dimitropoulos, D. Krioukov, G. Riley, and K. Claffy, Revealing the Autonomous System
taxonomy: The machine learning approach, in Proc. PAM, March 2006.
[Dimitropoulos07] X. Dimitropoulos, D. Krioukov, M. Fomenkov, B. Huffaker, Y. Hyun, K. Claffy, and G. Riley,
AS relationships: Inference and validation, Computer Communication Review, vol.37, no.1, 2007
[Dorogovstev03] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks. Oxford, England:Oxford
University Press, 2003.
References
[Doyle05]
J.C. Doyle, D. Alderson, L. Li, S. Low, M. Roughan, S. Shalunov, R. Tanaka, and W. Willinger,
The "robust yet fragile" nature of the Internet. PNAS 102(41), 2005.
[Erdos60a]
P. Erdos, and A. Rényi. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad.
Sci.5, pp.17-60, 1960.
[Erdos60b]
P. Erdos and T. Gallai. Graphs with prescribed degrees of vertices. Mat. Lapok (Hungarian),
11, pp.264-274, 1960.
[Erdos59]
P. Erdos and A. Renyi. On random graphs I Publ. Math. (Debrecen) 9, pp.290-297, 1959.
[Fabrikant02]
A. Fabrikant, E. Koutsoupias, and C. H. Papadimitriou. “Heuristically Optimized Tradeoffs: A
New Paradigm for Power Laws in the Internet. In Proceedings of ICALP, Lecture Notes in
Computer Science 2380, pp.110—122, Springer—Verlag, 2002.
[Faloutsos99]
M. Faloutsos, P. Faloutsos, and C. Faloutsos, On power-law relationships of the Internet
topology, in Proc. ACM SIGCOMM 1999 and in ACM Computer Communication Review, 29,
pp.251-263, 1999.
[Jin00]
C. Jin, Q. Chen, and S. Jamin, "INET topology generator", Technical Report, EECS Department,
University of Michigan, 2000.
[Jin02]
S. Jin and A. Bestavros, "Small-world internet topologies: Possible causes and implications
on scalability of end-system multicast", Technical Report, Boston University, 2002.
[Krapivsky01]
P. L. Krapivsky and S. Redner, "Organization of growing random networks," Physical Review
E, vol.63, no.6, May 2001.
[Krioukov04]
D. Krioukov, K. Fall, and X. Yang. Compact routing on Internet-like graphs. in Proc. IEEE
INFOCOM, 2004.
[Li04]
L. Li, D. Alderson, W. Willinger, and J. Doyle, A first-principles approach to understanding
the Internet’s router-level topology, in Proc. ACM SIGCOMM, 2004.
[Li05]
L. Li, D. Alderson, J.C. Doyle, and W. Willinger. Toward a Theory of Scale-Free Networks:
Definition, Properties, and Implications. Internet Mathematics, vol.2, no.4, pp.431-523, 2005.
[Mahadevan06a] P. Mahadevan, D. Krioukov, M. Fomenkov, B. Huffaker, X. Dimitropoulos, kc claffy, and A.
Vahdat. The Internet AS-level topology: Three data sources and one definitive metric. Computer
Communication Review, vol.36, no.1, 2006.
[Mahadevan06b] P. Mahadevan, D. Krioukov, K. Fall, and A. Vahdat, Systematic topology analysis and generation
using degree correlations, in Proc. ACM SIGCOMM, 2006.
[Medina00]
A. Medina, I. Matta, and J. Byers, "On the origin of power laws in Internet topologies", ACM
Computer Communication Review, vol.30, no.2, pp.18-28, April 2000.
[Mihail02]
M. Mihail, C.H. Papadimitriou: On the Eigenvalue Power Law. RANDOM, pp.254-262, 2002.
References
[Newman02]
[Newman03]
M. E. J. Newman, Assortative Mixing in Networks, Phys. Rev. Lett. 89, 208701 (2002).
M. E. J. Newman, The Structure and Function of Complex Networks, SIAM Reviews, vol.45,
no.2, pp.167-256, 2003.
[Newman05]
M. E. J. Newman. Power laws, Pareto distributions and Zipfs law, Contemporary Physics,
vol.46, no.5, pp.323–351, January 2005.
[Oliveira05]
R. Oliveira and J. Spencer, Connectivity Transitions in Networks with Super-Linear
Preferential Attachment, Internet Mathematics, vol.2, no.2, pp.121-163, 2002.
[Palmer01]
C. Palmer, G. Siganos, M. Faloutsos, C. Faloutsos and P. Gibbons, The connectivity and
fault-tolerance of the Internet topology (NRDM 2001), Santa Barbara, CA, May 2001.
[Radoslavov00] P. Radoslavov, H. Tangmunarunkit, H. Yu, R. Govindan, S. Shenker, D. Estrin, On
characterizing network topologies and analyzing their impact on protocol design, USC-CSTR-00-731, March 2000.
[Serrano06]
M. A. Serrano, M. Boguna, and A. Daz-Guilera, Modeling the Internet, The European Physical
Journal B, vol.50, pp.249-254, 2006.
[Siganos03]
G. Siganos, M. Faloutsos, P. Faloutsos, and C. Faloutsos, Power Laws and the AS-level
Internet topology, Transactions on Networking, vol.11, no.4, pp.514-524, August 2003.
[Tangmunarunkit02] H. Tangmunarunkit, R. Govindan, S. Jamin, S. Shenker, W. Willinger, Network
Topology Generators: Degree-Based vs. Structural, in Proc. of ACM SIGCOMM 2002.
[Vukadinovic02] D. Vukadinovic, P. Huang, T. Erlebach. On the Spectrum and Structure of Internet
Topology Graphs. in Proc. I2CS 2002.
[Waxman98]
Waxman, Routing of multipoint connections, IEEE JSAC, 1988.
[Willinger04a]
W. Willinger, D Alderson, J.C. Doyle, and L. Li, More "normal" than Normal: scaling
distributions in complex systems. Proc. Winter Simulation Conference 2004.
[Willinger04b]
W. Willinger, D Alderson, and L. Li, A pragmatic approach to dealing with high-variability in
network measurements, in Proc. ACM SIGCOMM, IMC 2004.
[Xang06]
X. Wang and D. Loguinov, "Wealth-based evolution model for the Internet AS-level
topology," in Proc. IEEE INFOCOM, 2006.
[Yook02]
S.-H. Yook, H. Jeong, and A.-L. Barabasi. Modeling the Internets large-scale topology, PNAS
99, 13382-86 (2002).
[Zegura97]
E. W. Zegura, K.L. Calvert and M.J. Donahoo. A Quantitative Comparison of Graph-based
Models for Internet Topology. IEEE/ACM Transactions on Networking, vol.5, no.6,Dec.1997.
[Zhou04]
S. Zhou and R. J. Mondragon, Accurately modeling the Internet topology, Physical Review
E, vol.70, p.066108, 2004.