15-744: computer networking
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15-744: Computer Networking
L-14 Network Topology
Sensor Networks
• Structural generators
• Power laws
• HOT graphs
• Graph generators
• Assigned reading• On Power-Law Relationships of the Internet
Topology• A First Principles Approach to Understanding
the Internet’s Router-level Topology
2
Outline
• Motivation/Background
• Power Laws
• Optimization Models
• Graph Generation
3
Why study topology?
• Correctness of network protocols typically independent of topology
• Performance of networks critically dependent on topology• e.g., convergence of route information
• Internet impossible to replicate
• Modeling of topology needed to generate test topologies
4
Internet topologies
AT&T
SPRINTMCI
AT&T
MCI SPRINT
Router level Autonomous System (AS) level
5
More on topologies..
• Router level topologies reflect physical connectivity between nodes• Inferred from tools like traceroute or well known public
measurement projects like Mercator and Skitter
• AS graph reflects a peering relationship between two providers/clients• Inferred from inter-domain routers that run BGP and publlic
projects like Oregon Route Views
• Inferring both is difficult, and often inaccurate
6
Hub-and-Spoke Topology
• Single hub node• Common in enterprise networks• Main location and satellite sites• Simple design and trivial routing
• Problems• Single point of failure• Bandwidth limitations• High delay between sites• Costs to backhaul to hub
7
Simple Alternatives to Hub-and-Spoke
• Dual hub-and-spoke• Higher reliability• Higher cost• Good building block
• Levels of hierarchy• Reduce backhaul cost• Aggregate the
bandwidth• Shorter site-to-site
delay…
8
Abilene Internet2 Backbone
9
SOX
SFGP/AMPATH
U. Florida
U. So. Florida
Miss StateGigaPoP
WiscREN
SURFNet
Rutgers U.
MANLAN
NorthernCrossroads
Mid-AtlanticCrossroads
Drexel U.
U. Delaware
PSC
NCNI/MCNC
MAGPI
UMD NGIX
DARPABossNet
GEANT
Seattle
Sunnyvale
Los Angeles
Houston
Denver
KansasCity
Indian-apolis
Atlanta
Wash D.C.
Chicago
New York
OARNET
Northern LightsIndiana GigaPoP
MeritU. Louisville
NYSERNet
U. Memphis
Great Plains
OneNetArizona St.
U. Arizona
Qwest Labs
UNM
OregonGigaPoP
Front RangeGigaPoP
Texas Tech
Tulane U.
North TexasGigaPoP
TexasGigaPoP
LaNet
UT Austin
CENIC
UniNet
WIDE
AMES NGIX
PacificNorthwestGigaPoP
U. Hawaii
PacificWave
ESnet
TransPAC/APAN
Iowa St.
Florida A&MUT-SWMed Ctr.
NCSA
MREN
SINet
WPI
StarLight
IntermountainGigaPoP
Abilene BackbonePhysical Connectivity(as of December 16, 2003)
0.1-0.5 Gbps0.5-1.0 Gbps1.0-5.0 Gbps5.0-10.0 Gbps
10
Points-of-Presence (PoPs)
• Inter-PoP links• Long distances• High bandwidth
• Intra-PoP links• Short cables between
racks or floors• Aggregated bandwidth
• Links to other networks• Wide range of media
and bandwidth
Intra-PoP
Other networks
Inter-PoP
11
Deciding Where to Locate Nodes and Links
• Placing Points-of-Presence (PoPs)• Large population of potential customers• Other providers or exchange points• Cost and availability of real-estate• Mostly in major metropolitan areas
• Placing links between PoPs• Already fiber in the ground• Needed to limit propagation delay• Needed to handle the traffic load
12
Trends in Topology Modeling
Observation• Long-range links are expensive
• Real networks are not random, but have obvious hierarchy
• Internet topologies exhibit power law degree distributions (Faloutsos et al., 1999)
• Physical networks have hard technological (and economic) constraints.
Modeling Approach• Random graph (Waxman88)
• Structural models (GT-ITM Calvert/Zegura, 1996)
• Degree-based models replicate power-law degree sequences
• Optimization-driven models topologies consistent with design tradeoffs of network engineers
13
Waxman model (Waxman 1988)
• Router level model
• Nodes placed at random in 2-d space with dimension L
• Probability of edge (u,v):• ae^{-d/(bL)}, where d is
Euclidean distance (u,v), a and b are constants
• Models locality
14
v
u d(u,v)
Real world topologies
• Real networks exhibit• Hierarchical structure• Specialized nodes (transit, stub..)• Connectivity requirements• Redundancy
• Characteristics incorporated into the Georgia Tech Internetwork Topology Models (GT-ITM) simulator (E. Zegura, K.Calvert and M.J. Donahoo, 1995)
15
Transit-stub model (Zegura 1997)
• Router level model
• Transit domains • placed in 2-d space
• populated with routers
• connected to each other
• Stub domains • placed in 2-d space
• populated with routers
• connected to transit domains
• Models hierarchy
16
So…are we done?
• No!
• In 1999, Faloutsos, Faloutsos and Faloutsos published a paper, demonstrating power law relationships in Internet graphs
• Specifically, the node degree distribution exhibited power laws
That Changed Everything…..
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Outline
• Motivation/Background
• Power Laws
• Optimization Models
• Graph Generation
18
Power laws in AS level topology
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A few nodes have lots of connections
Ran
k R
(d)
Degree d
Source: Faloutsos et al. (1999)Power Laws and Internet Topology
• Router-level graph & Autonomous System (AS) graph• Led to active research in degree-based network models
Most nodes have few connections
R(d
) =
P (
D>
d) x
#no
des
GT-ITM abandoned..
• GT-ITM did not give power law degree graphs
• New topology generators and explanation for power law degrees were sought
• Focus of generators to match degree distribution of observed graph
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Inet (Jin 2000)
• Generate degree sequence • Build spanning tree over nodes
with degree larger than 1, using preferential connectivity• randomly select node u not in
tree• join u to existing node v with
probability d(v)/d(w)
• Connect degree 1 nodes using preferential connectivity
• Add remaining edges using preferential connectivity
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Power law random graph (PLRG)• Operations
• assign degrees to nodes drawn from power law distribution• create kv copies of node v; kv degree of v.• randomly match nodes in pool• aggregate edges
may be disconnected, contain multiple edges, self-loops• contains unique giant component for right choice of
parameters
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2
11
Barabasi model: fixed exponent
• incremental growth• initially, m0 nodes• step: add new node i with m edges
• linear preferential attachment• connect to node i with probability ki / ∑ kj
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0.5
0.5 0.25
0.5 0.25
new nodeexisting node
may contain multi-edges, self-loops
Features of Degree-Based Models
• 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
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Preferential Attachment Expected Degree Sequence
Does Internet graph have these properties?
• No…(There is no Memphis!)
• Emphasis on degree distribution - structure ignored
• Real Internet very structured
• Evolution of graph is highly constrained
26
Problem With Power Law
• ... but they're descriptive models!
• No correct physical explanation, need an understanding of:• the driving force behind deployment• the driving force behind growth
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Outline
• Motivation/Background
• Power Laws
• Optimization Models
• Graph Generation
28
Li et al.
• Consider the explicit design of the Internet• Annotated network graphs (capacity, bandwidth)• Technological and economic limitations• Network performance
• Seek a theory for Internet topology that is explanatory and not merely descriptive.• Explain high variability in network connectivity• Ability to match large scale statistics (e.g. power
laws) is only secondary evidence
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100
101
102
Degree10
-1
100
101
102
103
Ban
dwid
th (
Gbp
s)
15 x 10 GE
15 x 3 x 1 GE
15 x 4 x OC12
15 x 8 FE
Technology constraint
Total Bandwidth
Bandwidth per Degree
Router Technology Constraint
30
Cisco 12416 GSR, circa 2002high BW low degree
high degree low BW
0.01
0.1
1
10
100
1000
10000
100000
1000000
1 10 100 1000 10000degree
To
tal R
ou
ter
BW
(M
bp
s)
cisco 12416
cisco 12410
cisco 12406
cisco 12404
cisco 7500
cisco 7200
linksys 4-port router
uBR7246 cmts(cable)
cisco 6260 dslam(DSL)
cisco AS5850(dialup)
approximateaggregate
feasible region
Aggregate Router Feasibility
core technologies
edge technologies
older/cheaper technologies
Source: Cisco Product Catalog, June 2002 31
Rank (number of users)
Con
nect
ion
Spe
ed (
Mbp
s)
1e-1
1e-2
1
1e1
1e2
1e3
1e4
1e21 1e4 1e6 1e8
Dial-up~56Kbps
BroadbandCable/DSL~500Kbps
Ethernet10-100Mbps
Ethernet1-10Gbps
most users have low speed
connections
a few users have very high speed connections
high performancecomputing
academic and corporate
residential and small business
Variability in End-User Bandwidths
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Heuristically Optimal Topology
Hosts
Edges
Cores
Mesh-like core of fast, low degree routers
High degree nodes are at the edges.
33
Comparison Metric: Network Performance
Given realistic technology constraints on routers, how well is the network able to carry traffic?
Step 1: Constrain to be feasible
Abstracted Technologically Feasible Region
1
10
100
1000
10000
100000
1000000
10 100 1000
degree
Ban
dw
idth
(M
bp
s)
kBxts
BBx
ijrkjikij
ji jijiij
,..
maxmax
:,
, ,
Step 3: Compute max flow
Bi
Bj
xij
Step 2: Compute traffic demand
jiij BBx
34
Likelihood-Related Metric
• Easily computed for any graph• Depends on the structure of the graph, not the generation
mechanism• Measures how “hub-like” the network core is
• For graphs resulting from probabilistic construction (e.g. PLRG/GRG),
LogLikelihood (LLH) L(g)
• Interpretation: How likely is a particular graph (having given node degree distribution) to be constructed?
j
connectedji
iddgL ,
)(Define the metric (di = degree of node i)
35
Lmax
l(g) = 1P(g) = 1.08 x 1010
P(g) Perfomance (bps)
PA PLRG/GRGHOT Abilene-inspired Sub-optimal
0 0.2 0.4 0.6 0.8 1
1010
1011
1012
l(g) = Relative Likelihood
36
PA PLRG/GRGHOT
Structure Determines Performance
P(g) = 1.19 x 1010 P(g) = 1.64 x 1010 P(g) = 1.13 x 1012
37
Summary Network Topology• Faloutsos3 [SIGCOMM99] on Internet topology
• Observed many “power laws” in the Internet structure• Router level connections, AS-level connections, neighborhood sizes
• Power law observation refuted later, Lakhina [INFOCOM00]
• Inspired many degree-based topology generators• Compared properties of generated graphs with those of measured
graphs to validate generator• What is wrong with these topologies? Li et al [SIGCOMM04]
• Many graphs with similar distribution have different properties• Random graph generation models don’t have network-intrinsic
meaning• Should look at fundamental trade-offs to understand topology
• Technology constraints and economic trade-offs• Graphs arising out of such generation better explain topology and its
properties, but are unlikely to be generated by random processes!
38
Outline
• Motivation/Background
• Power Laws
• Optimization Models
• Graph Generation
39
Graph Generation
• Many important topology metrics• Spectrum• Distance distribution• Degree distribution• Clustering…
• No way to reproduce most of the important metrics
• No guarantee there will not be any other/new metric found important
40
dK-series approach
• Look at inter-dependencies among topology characteristics
• See if by reproducing most basic, simple, but not necessarily practically relevant characteristics, we can also reproduce (capture) all other characteristics, including practically important
• Try to find the one(s) defining all others
0K
Average degree <k>
1K
Degree distribution P(k)
2K
Joint degree distribution P(k1,k2)
3K
“Joint edge degree” distribution P(k1,k2,k3)
3K, more exactly
4K
Definition of dK-distributions
dK-distributions are degree correlations within simple connected graphs of size d
Nice properties of properties Pd
• Constructability: we can construct graphs having properties Pd (dK-graphs)
• Inclusion: if a graph has property Pd, then it also has all properties Pi, with i < d (dK-graphs are also iK-graphs)
• Convergence: the set of graphs having property Pn consists only of one element, G itself (dK-graphs converge to G)
Rewiring
50
Graph Reproduction
51
Other Useful Tools
• Rocketfuel• Biases?
• RouteViews
52
53
• Faloutsos3 (Sigcomm’99)
• frequency vs. degree
Power Laws
topology from BGP tables of 18 routers54
• Faloutsos3 (Sigcomm’99)
• frequency vs. degree
Power Laws
topology from BGP tables of 18 routers55
• Faloutsos3 (Sigcomm’99)
• frequency vs. degree
Power Laws
topology from BGP tables of 18 routers56
• Faloutsos• frequency vs.
degree• empirical ccdf
P(d>x) ~ x-
Power Laws
57
Power Laws
• Faloutsos3 (Sigcomm’99)
• frequency vs. degree
• empirical ccdf P(d>x) ~ x-
α ≈1.15
58
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