Topologies of Complex Networks Functions vs. Structures Lun Li Advisor: John C. Doyle Co-advisor:...
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Transcript of Topologies of Complex Networks Functions vs. Structures Lun Li Advisor: John C. Doyle Co-advisor:...
Topologies of Complex NetworksFunctions vs. Structures
Lun LiAdvisor: John C. Doyle
Co-advisor: Steven H. LowCollaborators: David Alderson (NPS)
Walter Willinger (AT&T-labs Research)
• Incredibly large size• Dramatic growth, rapid and ongoing evolution• Multitude of measurement data (some are
biased and incomplete)
Challenges
• Structures always affect functions• Evaluate the performances of new regulations
that run on top of the structure.• A better design of complex networks
Importance
Trends
• Identify unifying large-scale properties– Power-law degree distribution
• Build universal models to match properties– Scale-free Networks
• Derive “Emergent properties” from models– “Achilles’ heel”, self-similar, generic…
Trends
• Identify unifying large-scale properties– Power-law degree distribution
• Build universal models to match properties– Scale-free Networks
• Derive “Emergent properties” from models– “Achilles’ heel”, Self-similar, generic
“New Science of Networks!”
Power Laws
Call D={d1, d2, …, dn } degree sequence of graph
Let di denote the degree of node i
ckdk where c>0 and α>0
k: rank of a degree
α: power-law tail index, 0<α≤2 in complex networks
10000
1000
100
10
1
Source: Faloutsos et al. (1999)
Degree
Ran
k
Power Laws
10000
1000
100
10
1
Source: Faloutsos et al. (1999)Most nodes have few connections
Call D={d1, d2, …, dn } degree sequence of graph
Let di denote the degree of node i A few nodes have lots of connections
ckdk where c>0 and α>0
)log()log()log( cdk k
Degree
Ran
k
Power-law and High Variability• For a sequence D,
– CV characterizes the variability of a degree sequence– Regular Graph (di=c), CV(D) = 0– ER random graph (Poisson), CV(D) = c– Some other random graphs (Exponential), CV(D) = c
• If D is Power-law (n∞), <2 , CV(D) = ∞• High variability significantly deviates from classical graphs• Power-law is discovered in many complex networks• Lead to pursue of universal theories to explain it.
Scale-Free (SF) Models
• Preferential Attachment (PA)Barabasi & Albert (1999)
– Growth by sequentially adding new nodes– New nodes connect preferentially to nodes
having more connections
Scale-Free (SF) Networks
• Reproduce power-law degree sequence• Generated by random process (PA, GRG,…)• Highly connected central “hubs”, which are
crucial to the system, “hold network together”– Achilles’ heel: fragile to specific attack
• Self-similar and fractal, Small-world properties… • SF networks have been suggested as
representative models of complex networks
PA GRG
However…
• Scale-free network theories are incomplete and in need of corrective actions. – Power laws are “more normal than Normal”…– Power laws are popular but not universal…– And not a “signature” of specific mechanisms
• Focus on network functions and structures
Functions Internet Router-level topology
s-metricStructures
Our Approach for Internet Topology• Consider the explicit design of the Internet
– Annotated network graphs (bandwidth)– Network Functions
• Carry expected traffic demand
– Constraints• Technological constraints• Economic limitations
– Heuristic optimized tradeoffs (HOT)• Maximize network function subject to constraints
Our Approach for Internet Topology• Consider the explicit design of the Internet
– Annotated network graphs (bandwidth)– Network Functions
• Carry expected traffic demand
– Constraints• Technological constraints• Economic limitations
– Heuristic optimized tradeoffs (HOT)
10K
100K
1M
10M
100M
1G
10G
100G
1000G
1
10 100 1000 10000
Total Router Degree (physical connections)
To
tal
Ro
ute
r B
and
wid
th (
bit
s/se
c)
Shared media at network edge (LAN, DSL, Cable, Wireless, Dial-up)
Corebackbone
High-end gateways
Older/cheapertechnology
Abstracted Technologically Feasible Region
Flow conservation in routers:Routers can either have a few
high-bandwidth connections, or many low bandwidth
connections.
Individual router models specialize in different bandwidth-degree combinations and therefore tend to used in different
regions of the network.
Rank (number of users)
Con
nect
ion
Spee
d (M
bps)
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 (2003)
High cost of links drives traffic
aggregation at network edge
Hosts
Edges
Core
Heuristically Optimal Topology
High degree nodes are at the edges.
Sparse, mesh-like core of fast, low-degree routers.
Relatively uniform low connectivity within core: carry high
bandwidth
high variability in connectivity at
edge:aggregate end
users
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
Optimization-based models• Core: Mesh-like, low
degree • Edge: High degree• From engineering design• Tradeoffs in constraints • Match the real Internet
SF models• Core: Hub-like, high
degree • Edge: Low degree• From random process• Ignore engineering details• Match aggregate
statistics
SF HOT
How to reconcile these two perspectives?
SF
PLRG/GRG
HOT
Abilene-inspired Sub-optimal
What are the key differences among these graphs?
•Functions
•Structures
Network PerformanceGiven 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
BBxgPerf
ijrkjikij
ji jijiij
,..
maxmax)(
:,
, ,
Step 3: Compute max flow
Bi
Bj
xij
Step 2: Compute traffic demand
jiij BBx
SF HOT
Perf(g) = 1.19 x 1010
(bps)Perf(g) = 1.13 x 1012 (bps)
Engineering-based models• Core: Mesh-like, low
degree • Edge: High degree• From explicit design• Tradeoffs in constraints • High throughput• High router utilization• No Achilles’ Heel
Degree-based models• Core: Hub-like, high
degree • Edge: Low degree• From random process• Ignore engineering details• Low throughput• Low router utilization• Achilles’ Heel
SF HOT
Engineering-based models• Core: Mesh-like, low
degree • Edge: High degree• From explicit design• Tradeoffs in constraints • High throughput• High router utilization• No Achilles’ Heel
Degree-based models• Core: Hub-like, high
degree • Edge: Low degree• From random process• Ignore engineering details• Low throughput• Low router utilization• Achilles’ Heel
PA HOT
A Structural Approach
• s-metric
– Structural metric, depending only on the connectivity of a given graph not on the generation mechanism
– Not for a specific network
• High s(g) is achieved by connecting high degree nodes to each other
• Measures how “hub-like” the network core is
jji
iddgs
),(
)(
s and Joint Degree Distribution
• Joint Degree Distribution (JDD): p(k,k’) correlation between the degrees k, k’ of connected nodes– Degree distribution is a first order statistic– JDD is a second order statistic
• For a graph having degree sequence D, s is the aggregation of JDD
– Corollary: If two graphs have the same JDD, define a metric as the aggregation of the third order correlation.
Dkk
kkpkks',
)',('2
1
P(g) Perfomance (bps)
SFHOT
0 0.2 0.4 0.6 0.8 1
1010
1011
1012
S(g)
minmax
min)()(
ss
sgsgS
s-metric
• Structural metric, depending only on the connectivity of a given graph not on the generation mechanism
• Define the extent to which the graph is scale-free
• Differentiate graphs with the same highly variable degree sequence, among them:– smax graph is the one with the highest s value– smin graph is the one with the lowest s value
• smax – smin defines the graph diversity of a given degree sequence in the simple and connected graph space.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
smax
s-va
lue
smin
Variability of a degree sequence
Graph diversity
cv
Variability vs. graph diversity of a degree sequence
A tree with 100 nodes, 99 links
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
smax (SF is closed to smax)
s-va
lue
smin (HOT is closed to smin)
Variability vs. graph diversity of a degree sequence
A tree with 100 nodes, 99 links
Variability of a degree sequence
Graph diversity
cv
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
s-va
lue
chain
star
Variability of a degree sequence
Graph diversity
Low variability graphs
cv
Variability vs. graph diversity of a degree sequence
A tree with 100 nodes, 99 links
s and Assortativity r(g)
• For a given graph, assortativity is:
– r>0, assortative, high degree nodes connect to high degree nodes
– r<0, dissortative, high degree nodes connect to low degree nodes
– A popular metric to measure the degree correlation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
CV(D)
Almost all the simple, connected trees with high variable connectivity have negative assortativity
Ass
ort
ativ
ity
rmax
rmin
Variability vs. assortivity of a degree sequence
A tree with 100 nodes, 99 links
r(SF) = -0.42
r(HOT) = -0.46
Assortativity r(g)
• For a given graph, assortativity is:
Assortativity r(g)
• For a given graph, assortativity is:
• Normalization termsmax of unconstrained graphs:all the nodes connect themselves
Assortativity r(g)
• For a given graph, assortativity is:
• Normalization term
• Centering term
smax of unconstrained graph
Center of unconstrained graph
Assortativity r(g)
• For a given graph, assortativity is:
• Normalization term
• Centering term
• Background set is the unconstrained graph!
smax of unconstrained graph
Center of unconstrained graph
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
s-metric assortativity
smax
smin
rmax
rmin
CV(D) CV(D)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
s/smax assortativity
smin/smax
rmax
rmin
•Assortativity is a metric directed borrowed from classic graph theory
•It works well for the low variability case
•Extremely misleading for the high variability complex network
CV(D) CV(D)
smax (SF) and graph metrics
With s, we can quantitatively characterize the properties claimed in SF literature.
• Node Centrality– In smax graph, node centrality increases with degree
• Small-world phenomena– smax has lowest average shortest path
• Self similarity– smax graph remains smax by trimming, coarse graining,
highest connect motif• Generic
– smax graph is most likely to appear by GRG
Conclusions
FunctionsStructures
• The Internet• Functions vs.
Constraints• HOT vs. SF
•s-metric can highlight the difference (HOT vs SF)•s-metric measures graph diversity•s-metric has a rich connection to self-similarity, assortativity