1. 1 “Erdos and the Internet” Milena Mihail Georgia Tech. The Internet is a remarkable phenomenon that involves graph theory in a natural way and gives rise to new questions and models. E.g. the Internet at the level of Autonomous Systems supports the critical BGP routing protocol.

2. 2 Search and routing networks, like the WWW, the internet, P2P networks, ad-hoc (mobile, wireless, sensor) networks are pervasive and scale at an unprecedented rate. Performance analysis/evaluation in networking: measure parameters hopefully predictive of performance. Important in network simulation and design.

3. 3 Sparse small-world graphs with large degree-variance. Want metrics predictive or explanatory of network function. 4102 100 degree frequency, but no sharp concentration Erdos-Renyi

4. 4 Networking questions Routing Does packet drop (blocking) scale? Does the network evolve towards monopolies? Are network resources used efficiently? How does delay scale in routing? Is there load balancing? Is it or ? Searching Design How fast can you crawl the WWW? Can you search a P2P network with low overhead? How can you maintain a well connected topology? How about distributed and dynamic networks? Are there strategies to improve crawling and searching? Is it or ? Congestion Congestion = flow on most loaded link under optimal routing. Route 1 unit of flow between each pair of nodes. Graph on nodes. Total flow.

5. 5 Relevant metric: “bottlenecks”Conductance Alon 85 Jerrum & Sinclair 88 Leighton & Rao 95

6. 6 Second eigenvalue of the lazy random walk associated with the adjacency matrix closely approximates conductance: computationally soft Matlab does 1-2M node sparse graphs Random Graph Internet This is also another point of view of the small-world phenomenon This also says that congestion under link capacities, search time and sampling time scale smoothly Plots at 700 nodes, 3000 nodes, and 15000 nodes. 100 largest eigenvalues + - + + + - - - Eigenvectors associated with large eigenvalues are “shadows” of sets with bad conductance.

7. 7 Beyond today, we need network models to predict future behavior. What are suitable network models? The Internet grows anarchically, so random graphs are good canditates. Current network models are random graphs which produce power law degree sequences (thus also matching this important observed data).

8. 8 One vertex at a time New vertex attaches to existing vertices EVOLUTIONARY:Growth & Preferential Attachment Simon 55, Barabasi & Albert 99, Kumar et al 00, Bollobas & Riordan 01, Bollobas, Riordan, Spencer & Tusnady 01.

9. 9 CONFIGURATIONAL aka structural MODEL Given choose random perfect matching over minivertices “Random” graph with given “power law” degree sequence. Bollobas 80s, Molloy & Reed 90s, Aiello, Chung & Lu 00s, Sigcomm/Infocom 00s

10. 10 Given Choose random perfect matching over CONFIGURATIONAL MODEL minivertices edge multiplicity O(log n), a.s. connected, a.s.

11. 11 Theorem [M, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with,, a.s. Theorem [Gkantsidis, M, Saberi 03]: For a random graph in the configurational model arising from degree sequence,, a.s. Bounds on Conductance Previously: Cooper & Frieze 02 Independent: Chung,Lu&Vu 03 Technique: Probabilistic Counting Arguments & Combinatorics. Difficulty: Non homogeneity in state-space, Dependencies. for a different structural random graph model and

12. 12 Worst case is when all vertices have degree 3. Structural Model, Proof Idea:Difficulty: Non homogeneity in state-space

13. 13 Growth with Preferential Connectivity Model, Proof Idea: Difficulty: Arrival Time Dependencies Shifting Argument

14. 14 firstlast first last

15. 15 Theorem [Gkantsidis,MM, Saberi 03]: For a random graph in the structural model arising from degree sequence there is a poly time computable flow that routes demand between all vertices and with max link congestion a.s. Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with there is a poly time computable flow that routes demand between all vertices and with max link congestion, a.s. Each vertex with degree in the network core serves customers from the network periphery. Note: Why is demand ?

16. 16 Networking questions Routing Congestion Searching Design Does packet drop (blocking) scale? How fast can you crawl the WWW? Does the network evolve towards monopolies? Can you search a P2P network with low overhead? How can you maintain a well connected topology? Are network resources used efficiently? How does delay scale in routing? Is there load balancing? How about distributed and dynamic networks? Are there strategies to improve crawling and searching? It is Is it or ?

17. 17 Searching, Cover Time and Mixing Time Cover time = expected time to visit all nodes. Search the graph by random walk. Graph on nodes. Mixing time = time to reach stationary distribution (arbitrarily close).

18. 18 Conductance, Mixing and Cover Time For Cover Time “mixing” in Rapid Mixing of Random Walk Alon 85 Jerrum & Sinclair 88

19. 19 Extensions of Cover Time In practice, when crawling the WWW or searching a P2P network, when a node is visited, all nodes incident to the node are also visited. This can be implemented by one-step local replication of information.

20. 20 can discover vertices in steps. Cover Time with Look-Ahead OneIn the configurational model with Theorem [MM,Saberi,Tetali 05]: Proof Adamic et al 02 Chawathe et al 03 Gkanstidis, MM, Saberi 05

21. 21 Proof In the configurational model with Cover Time with Look-Ahead Two Theorem [MM,Saberi,Tetali 05]: can discover vertices in steps.

22. 22 Networking questions Searching Cover time Does packet drop (blocking) scale? How fast can you crawl the WWW? Does the network evolve towards monopolies? Can you search a P2P network with low overhead? How can you maintain a well connected topology? Are network resources used efficiently? How does delay scale in routing? Is there load balancing? How about distributed and dynamic networks? Are there strategies to improve crawling and searching? It is It is and local replication offers substantial improvement Routing Congestion Design Is it or ?

23. 23 The case of Peer-to-Peer Networks n nodes, d-regular graph Each node has resources O(polylogn) and knows a very small size neighborhood around itself Distributed, decentralized Search for content, e.g. by flooding or random walk ? Must maintain well connected topology, e.g. a graph with good concuctance, a random graph

24. 24 Gnutella: constantly drops existing connections and replaces them with new connections P2P networks are constantly randomizing their links There are between 5 and 30 requests for new connections per second. About 1% of these requests are satisfied and existing links are dropped. The network is working “in panic” trying to randomize thus avoiding network configurations with bottlenecks and trying to maintain high conductance.

25. 25 Theorem [ Feder, Guetz, M, Saberi 06 ]: The Markov chain on d-regular graphs is rapidly mixing, even under local 2-link switches or flips. P2P Network Topology Maintenance by Constant Randomization Theorem [ Cooper, Frieze & Greenhill 04 ]: The Markov chain corresponding to a 2-link switch on d-regular graphs is rapidly mixing. Question: How does the network “pick” a random 2-link switch? In reality, the links involved in a switch are within constant distance.

26. 26 Space of d-regular graphs general 2-link switch Markov chain Space of connected d-regular graphs local Flip Markov chain Define a mapping from to such that (a) (b) each edge in maps to a path of constant length in The proof is a Markov chain comparison argument

27. 27 Networking questions Congestion Cover time Mixing time Does packet drop (blocking) scale? How fast can you crawl the WWW? Does the network evolve towards monopolies? Can you search a P2P network with low overhead? How can you maintain a well connected topology? Are network resources used efficiently? How does delay scale in routing? Is there load balancing? How about distributed and dynamic networks? Are there strategies to improve crawling and searching? It is Conductance

28. 28 The Internet topology has constant second eigenvalue, but larger than the second eigenvalue of random graphs. Can we develop random graph models (with powerlaw degree distributions) and with varying values of the second eigenvalue ? Preliminary work by Flaxman, Frieze & Vera Routing on the Internet is done according to shortest paths. Can we characterize congestion under shortest path routing? How can we maintain a P2P topology with good connectivity under dynamic settings or arriving and departing nodes? Can we develop efficient distributed algorithms that discover critical links in the network? Preliminary work by Boyd, Diaconis & Xiao. Open Problems