“Erdos and the Internet”

1. E.g. the Internet at the level of Autonomous Systems supports the critical BGP routing protocol. “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.

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. Want metrics predictive or explanatory of network function. Sparse small-world graphs with large degree-variance. , but no sharp concentration frequency Erdos-Renyi 100 2 4 10 degree

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

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

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

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

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

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

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

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

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

14. first last first last

15. 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. 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. Note: Why is demand ? Each vertex with degree in the network core serves customers from the network periphery.

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

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

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

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

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

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

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

24. P2P networks are constantly randomizing their links Gnutella: constantly drops existing connections and replaces them with new connections 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. 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. Theorem[Feder, Guetz, M, Saberi 06]: The Markov chain on d-regular graphs is rapidly mixing, even under local 2-link switches or flips. Question: How does the network “pick” a random 2-link switch? In reality, the links involved in a switch are within constant distance.

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

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

28. Open Problems 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.