Understanding the Internet Graph. Christos H. Papadimitriou

Transcription

1 Understanding the Internet Graph Christos H. Papadimitriou

2 Goals To understand the Internet topology using tools from graph and game theory To contribute to the rigorous foundations of Internet science To prove good theorems To meet people (Fabrikant, Koutsoupias, Mihail, Feigenbaum, Sami, Shenker, Saberi, Luthra, Maneva, V. Vazirani, Talwar) Microsoft CITRIS day, March 20 2

3 Today BGP + VCG The price of anarchy The Internet s heavy tail and Internet traffic Microsoft CITRIS day, March 20 3

4 Vickrey auction sealed highest bid auction encourages gaming and speculation Vickrey auction: Highest bidder wins, pays second highest bid Theorem: Vickrey auction is a truthful mechanism. (Theorem: It maximizes social benefit and auctioneer expected revenue.) Microsoft CITRIS day, March 20 4

5 Vickrey Clarke Groves (VCG) shortest paths s t pay e VCG(e) = its declared cost c(e), plus a bonus equal to dist(s,t) c(e) = dist(s,t) Microsoft CITRIS day, March 20 5

6 Problem: s 10 t Microsoft CITRIS day, March 20 6

7 But in the Internet VCG overcharge would be only about 30% on the average [FPSS 2002] Could this be the manifestation of rational behavior/competition at network creation? Nash equilibria of network creation? Microsoft CITRIS day, March 20 7

8 In Internet routing, VCG[e] depends on the origin and destination. Can be computed with little overhead on top of BGP (the standard protocol for interdomain routing). Theorem (with Mihail and Saberi,2003): In a random graph with average degree d, the expected VCG overcharge is ~1/d Microsoft CITRIS day, March 20 8

9 The price of anarchy cost of worst Nash equilibrium socially optimum cost [Koutsoupias and P, 1998] in network routing = 2 [Roughgarden and Tardos, 2000, Roughgargen 2002] Microsoft CITRIS day, March 20 9

10 Simple net creation game Players are the nodes (autonomous systems) The strategies of node i are all possible sets of links (peering agreements) out of it. Result is undirected graph G = (s 1,,s n ) Cost to node i: cost of links, plus sum of hop distances to all other nodes Microsoft CITRIS day, March 20 10

11 Nash equlibria? If links are cheap, then the only Nash equilibrium is the clique If links are not too cheap, not too expensive, then social optimum is clique, Nash equilibrium is the star (price of anarchy = 4/3) If links are expensive, price of anarchy is somewhere between 3 and 5 Microsoft CITRIS day, March 20 11

12 The monster s tail [Faloutsos ] the degrees of the Internet are power law distributed Both autonomous systems graph and router graph Eigenvalues: ditto!??! Model? Microsoft CITRIS day, March 20 12

13 The world according to Zipf Power laws, Zipf s law, heavy tails, i th largest is ~ i a (cities, words: a = 1, Zipf s Law ) Equivalently: prob[greater than x] ~ x b (compare with law of large numbers) the signature of human activity Microsoft CITRIS day, March 20 13

14 Models Size independent growth ( the rich get richer, or random walk in log paper) Carlson and Doyle 1999: Highly optimized tolerance (HOT) Microsoft CITRIS day, March 20 14

15 Our model [with Fabrikant and Koutsoupias, 2002]: min j < i [α d ij + hop j ] Microsoft CITRIS day, March 20 15

16 Theorem: if α < const, then graph is a star degree = n 1 if α > n, then there is exponential concentration of degrees prob(degree > x) < exp( ax) otherwise, if const < α < n, heavy tail: prob(degree > x) > x b Microsoft CITRIS day, March 20 16

17 Heuristically optimized tradeoffs Power law distributions seem to come from tradeoffs between conflicting objectives (a signature of human activity?) cf HOT, [Mandelbrot 1954] Other examples? General theorem? Microsoft CITRIS day, March 20 17

18 Also: eigenvalues Theorem [with Mihail, 2002]: If the d i s obey a power law, then the n b largest eigenvalues are almost surely very close to d 1, d 2, d 3, Corollary: Spectral data mining methods are of dubious value in the presence of large features Microsoft CITRIS day, March 20 18

19 PS: How does traffic grow? Trees: n 2 Expanders (and most degree balanced sparse graphs): ~ n The Internet? Theorem (with Mihail and Saberi, 2003): Scale free graph models with degree > 2 are almost certainly expanders Microsoft CITRIS day, March 20 19