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1 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Generating Network Topologies That Obey Power Laws Christopher R. Palmer and J. Gregory Steffan School of Computer Science Carnegie Mellon University

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2 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon What is a Power Law? Faloutsos et al. define four power laws: –they found laws in multiple Internet graphs –others found similar laws, also for the Web y = βx α Log the Internet obeys power laws

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3 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon What is a Topology Generator? Artificial network generation algorithm: –often used to evaluate new network schemes Do artificial networks obey power laws? –artificial networks may not be realistic –conclusions could be inaccurate can we generate these topologies? does it matter?

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4 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Outline Do existing generators obey power laws? Can we generate graphs that obey power laws? Do power law graphs impact results? Related work Conclusions

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5 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Existing Topology Generators Waxman: –place nodes randomly in 2-space –add edges with probability P(u,v)=αe -d/(βL) N-level hierarchical: –connect random graphs in an N-level hierarchy

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6 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Power Laws 1 and 2 PL #1: Out-degree vs. Rank –compute the out-degree of all nodes –sort in descending order PL #2: Frequency vs. Out-degree –compute the out-degree of all nodes –compute the frequency of each out-degree Internet graphs obey

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7 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #1: Out-degree vs. Rank 2-Level and Waxman do not obey Waxman: ρ= Level: ρ=0.81

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8 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #2: Frequency vs. Out-degree 2-Level & Waxman REALLY do not obey! Waxman: ρ= Level: ρ=0.23

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9 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Power Laws 3 and 4 PL #3: Hopcounts –number of pairs of nodes within i hops PL #4: Eigenvalues –compute the largest 10 eigenvalues λ i Internet graphs obey [A][v i ] = λ i [v i ]

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10 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #3: Hopcounts 2-Level and Waxman obey Waxman: ρ= Level: ρ=0.98

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11 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #4: Eigenvalues 2-Level and Waxman obey Waxman: ρ= Level: ρ=0.65

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12 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Outline Do existing generators obey power laws? Can we generate graphs that obey power laws? –Power-Law Out-Degree (PLOD) –Recursive Do power law graphs impact results? Related work Conclusions

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13 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Power-Law Out-Degree Algorithm (PLOD) FOR i:1..N x = uniform_random(1,N) out_degree i = βx -α FOR i:1..M WHILE 1 r = uniform_random(1,N), c = uniform_random(1,N) IF r != c AND out_degree r AND out_degree c AND !A r,c out_degree r --, out_degree c -- A r,c = 1, A c,r = 1 BREAK Assign exponential out-degree credits Place an edge in the adjacency matrix

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14 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PLOD: Example Topology 32 nodes, 48 links

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15 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Recursive Topology Generator β γ α Our Recursive Distribution: 80/20 Distribution: 80%20% generalize to a 2D adjacency matrix

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16 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Recursive Topology: Generation Link Probabilities10 Generated links darker means higher probability / weight

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17 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Recursive Topology: Example 32 nodes, 50 low latency, 10 high latency (red) links

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18 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #1: Out-degree vs. Rank Recursive: good power-law tail, non-power-law start PLOD: EXCELLENT power-law Recursive: ρ=0.89 PLOD: ρ=0.97

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19 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #2: Frequency vs. Degree both GOOD power-laws Recursive: ρ=0.92 PLOD: ρ=0.93

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20 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #3: Hopcounts both EXCELLENT power-laws Recursive: ρ=0.94 PLOD: ρ=0.98

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21 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon PL #4: Eigenvalues both EXCELLENT power-laws Recursive: ρ=0.93 PLOD: ρ=0.98

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22 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Power-Law Summary: Correlations PL #1: Degree PL #2: Deg. Freq PL #3: Hops PL #4: Eigenval 2-Level Waxman PLOD Recursive GREEN cells obey power-laws, RED cells do not our generators have better Internet characteristics!

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23 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Outline Do existing generators obey power laws? Can we generate graphs that obey power laws? Do power law graphs impact results? Related work Conclusions

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24 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon STORM Multicast Algorithm client requests repair from parent with a nack source client (parent) client nackrepair

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25 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Simulation Methodology Original STORM study: –used 2-level random topology –source and clients connected to second-level Generating comparable topologies: – equalize graph size and average out-degree –selection of high and low latency links What impact do we expect of PL topologies? –average results will be similar –distributions will differ

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26 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon STORM Average Overhead STORM overhead averages scale for all topologies

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27 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon STORM Overhead Distribution overhead distribution varies significantly by topology 2-Level

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28 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Loss Distribution loss distribution also varies significantly by topology

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29 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Related Work Barabási et al. (Notre Dame) BRITE (Boston University) What causes power laws in the Internet? –incremental growth –preferential connectivity BRITE uses these factors to generate graphs

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30 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Conclusions Existing generators do not obey all power-laws Our two topology generators do –PLOD: use power-law to generate node degrees –recursive: use 80/20 law to generate links Do power-law topologies have any impact? –maybe: changed distributions for STORM –maybe not: averages unchanged for STORM moral: simulate with different generators!

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31 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Backup Slides

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32 Generating Network Topologies That Obey Power LawsPalmer/Steffan Carnegie Mellon Generating Comparable Topologies Equalize graph characteristics: –number of nodes –average out-degree Ensure connectedness: –randomly connect disconnected components Assign high/low-latency links: –Recursive algorithm provides a distinction –method for putting low-lat. links near clients

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