1. Christos H. Papadimitriou UC Berkeley christos
Networks and Games
Christos H. Papadimitriou
UC Berkeley
christos

2. (Mathematical tools: combinatorics, logic)
Goal of TCS ( ):
Develop a mathematical understanding of the capabilities and limitations of the von Neumann computer and its software –the dominant and most novel computational artifacts of that time
(Mathematical tools: combinatorics, logic)
What should Theory’s goals be today?
cmu, nov

3. cmu, nov

4. The Internet Huge, growing, open, end-to-end
Built and operated by companies in various (and varying) degrees of competition
The first computational artefact that must be studied by the scientific method
Theoretical understanding urgently needed
Tools: math economics and game theory, probability, graph theory, spectral theory
cmu, nov

5. Today: Nash equilibria and the price of anarchy Vickrey shortest paths
Power Laws
Collaborators: Alex Fabrikant, Joan Feigenbaum, Elias Koutsoupias, Eli Maneva, Milena Mihail, Amin Saberi, Rahul Sami, Scott Shenker
cmu, nov

6. (NB: also, many players)
Game Theory
strategies
strategies
3,-2
payoffs
(NB: also, many players)
cmu, nov

7. e.g. 1,-1 -1,1 3,3 0,4 4,0 1,1 0,0 0,1 1,0 -1,-1 matching pennies
prisoner’s dilemma
1,-1
-1,1
3,3
0,4
4,0
1,1
chicken
0,0
0,1
1,0
-1,-1
cmu, nov

8. Nash equilibrium Definition: Double best response
Problem: may not exist
Idea: Randomized Nash equilibrium
Theorem [Nash 1951]: Always exists.
Problem: but is it in P?
Problem: too many!
cmu, nov

9. Exploring the multiplicity: “The price of anarchy”
cost of worst Nash equilibrium
p. of A =
“socially optimum” cost
cmu, nov

10. Selfishness can hurt you!
delays x 1
Social optimum: 1.5 x 1
Anarchical equilibrium: 2
cmu, nov

11. The price of the Internet architecture?
Worst case?
Price of anarchy
= “2” (4/3 for linear delays)
[Roughgarden and Tardos, 2000,
Roughgarden 2002]
The price of the Internet architecture?
cmu, nov

12. Mechanism design (or inverse game theory)
agents have utilities – but these utilities are known only to them
game designer prefers certain outcomes depending on players’ utilities
designed game (mechanism) has designer’s goals as dominating strategies (or other rational outcomes)
cmu, nov

13. e.g., Vickrey auction
sealed-highest-bid auction encourages gaming and speculation
Vickrey auction: Highest bidder wins,
pays second-highest bid
Participants are incentivized to tell the truth: Truthful mechanism
cmu, nov

14. e.g., shortest path auction3 6 5 s 4 t 6 10 3 11
pay e its declared cost c(e),
plus a bonus equal to dist(s,t)|c(e) = - dist(s,t)
cmu, nov

15. Problem: s t Theorem [Elkind, Sahai, Steiglitz, 03]: This is1 1 1 1 1 s 10 t
Theorem [Elkind, Sahai, Steiglitz, 03]: This is
inherent in any truthful mechanism.
cmu, nov

16. But…
…in the Internet (the graph of autonomous systems) VCG overcharge would be only about 30% on the average [FPSS 2002]
Could this be the manifestation of rational behavior at network creation?
cmu, nov

17. Naaah…
Theorem [with Mihail and Saberi, 2003]: In a random graph with average degree d, the expected VCG overcharge is constant (conjectured: ~1/d)
cmu, nov

18. The monster’s tail
[Faloutsos3 1999] the degrees of the Internet are power law distributed
Both autonomous systems graph and router graph
Eigenvalues: ditto!??!
Model?
cmu, nov

19. The world according to Zipf
Power laws, Zipf’s law, heavy tails,…
“the signature of human activity” vs
cmu, nov

20. Models
Size-independent growth (“the rich get richer,” or random walk in log paper)
Carlson and Doyle 1999: Highly optimized tolerance (HOT)
cmu, nov

21. Our model [with Fabrikant and Koutsoupias, 2002]:
minj < i [  dij + hopj]
cmu, nov

22. Theorem: if  < const, then graph is a star degree = n -1
if  > n, then there is exponential concentration of degrees
prob(degree > x) < exp(-ax)
otherwise, if const <  < n, heavy tail:
prob(degree > x) > x -b
cmu, nov

23. Heuristically optimized tradeoffs
Power law distributions seem to come from tradeoffs between conflicting objectives (a signature of human activity?)
cf HOT, file sizes, [Mandelbrot 1954]
Other examples?
General theorem?
cmu, nov

24. Also: eigenvalues
Theorem [with Mihail, 2002]: If the di’s obey a power law, then the nb largest eigenvalues are almost surely very close to
d1,  d2, d3, …
Corollary: Spectral data-mining methods are of dubious value in the presence of large features
cmu, nov

25. PS: How does traffic grow
PS: How does traffic grow? (or: why is the Internet backbone overprovisioned?)
Trees: n2
Expanders (and most degree-balanced sparse graphs): ~ n
The Internet?
Theorem (with Mihail and Saberi, 2003):
“Scale-free graph models” are almost certainly expanders
cmu, nov