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Topologies of Complex Networks Functions vs. Structures Lun Li Advisor: John C. Doyle Co-advisor: Steven H. Low Collaborators: David Alderson (NPS) Walter Willinger (AT&T-labs Research)

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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

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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…

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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!

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Power Laws Call D={d 1, d 2, …, d n } degree sequence of graph Let d i denote the degree of node i where c>0 and α>0 k : rank of a degree α: power-law tail index, 0<α2 in complex networks Source: Faloutsos et al. (1999) Degree Rank

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Power Laws Source: Faloutsos et al. (1999) Most nodes have few connections Call D={d 1, d 2, …, d n } degree sequence of graph Let d i denote the degree of node i A few nodes have lots of connections where c>0 and α>0 Degree Rank

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Power-law and High Variability For a sequence D, –CV characterizes the variability of a degree sequence –Regular Graph ( d i =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.

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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

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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

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PA GRG

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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-metric Structures

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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

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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)

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10K 100K 1M 10M 100M 1G 10G 100G 1000G Total Router Degree (physical connections) Total Router Bandwidth (bits/sec) Shared media at network edge (LAN, DSL, Cable, Wireless, Dial-up) Core backbone High-end gateways Older/cheaper technology 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.

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Rank (number of users) Connection Speed (Mbps) 1e-1 1e-2 1 1e1 1e2 1e3 1e4 1e211e4 1e6 1e8 Dial-up ~56Kbps Broadband Cable/DSL ~500Kbps Ethernet Mbps Ethernet 1-10Gbps most users have low speed connections a few users have very high speed connections high performance computing academic and corporate residential and small business Variability in End-User Bandwidths (2003) High cost of links drives traffic aggregation at network edge

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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

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SOX SFGP/ AMPATH U. Florida U. So. Florida Miss State GigaPoP WiscREN SURFNet Rutgers U. MANLAN Northern Crossroads Mid-Atlantic Crossroads Drexel U.U. Delaware PSC NCNI/MCNCMAGPI UMD NGIX DARPA BossNet GEANT Seattle Sunnyvale Los Angeles Houston Denver Kansas City Indian- apolis Atlanta Wash D.C. Chicago New York OARNET Northern Lights Indiana GigaPoP Merit U. Louisville NYSERNet U. Memphis Great Plains OneNet Arizona St. U. Arizona Qwest LabsUNM Oregon GigaPoP Front Range GigaPoP Texas TechTulane U. North Texas GigaPoP Texas GigaPoP LaNet UT Austin CENICUniNet WIDE AMES NGIX Pacific Northwest GigaPoP U. Hawaii Pacific Wave ESnet TransPAC/APAN Iowa St. Florida A&M UT-SW Med Ctr. NCSA MREN SINet WPI StarLight Intermountain GigaPoP Abilene Backbone Physical Connectivity (as of December 16, 2003) Gbps Gbps Gbps Gbps

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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 SFHOT How to reconcile these two perspectives?

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SF PLRG/GRG HOT Abilene-inspiredSub-optimal What are the key differences among these graphs? Functions Structures

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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 degree Bandwidth (Mbps) Step 3: Compute max flow BiBi BjBj x ij Step 2: Compute traffic demand

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SF HOT Perf(g) = 1.19 x (bps) Perf(g) = 1.13 x (bps)

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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

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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

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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

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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.

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P(g) Perfomance (bps) SFHOT S(g)

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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: –s max graph is the one with the highest s value –s min graph is the one with the lowest s value s max – s min defines the graph diversity of a given degree sequence in the simple and connected graph space.

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s max s-value s min Variability of a degree sequence Graph diversity cv Variability vs. graph diversity of a degree sequence A tree with 100 nodes, 99 links

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s max (SF is closed to s max ) s-value s min (HOT is closed to s min ) Variability vs. graph diversity of a degree sequence A tree with 100 nodes, 99 links Variability of a degree sequence Graph diversity cv

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s-value 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

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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

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CV(D) Almost all the simple, connected trees with high variable connectivity have negative assortativity Assortativity r max r min Variability vs. assortivity of a degree sequence A tree with 100 nodes, 99 links r(SF) = r(HOT) = -0.46

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Assortativity r(g) For a given graph, assortativity is:

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Assortativity r(g) For a given graph, assortativity is: Normalization term s max of unconstrained graphs: all the nodes connect themselves

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Assortativity r(g) For a given graph, assortativity is: Normalization term Centering term s max of unconstrained graph Center of unconstrained graph

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Assortativity r(g) For a given graph, assortativity is: Normalization term Centering term Background set is the unconstrained graph! s max of unconstrained graph Center of unconstrained graph

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s-metric assortativity s max s min r max r min CV(D)

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s/s max assortativity s min /s max r max r min 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)

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s max (SF) and graph metrics With s, we can quantitatively characterize the properties claimed in SF literature. Node Centrality –In s max graph, node centrality increases with degree Small-world phenomena –s max has lowest average shortest path Self similarity –s max graph remains s max by trimming, coarse graining, highest connect motif Generic –s max graph is most likely to appear by GRG

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Conclusions Functions Structures 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

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