1. 1 Milena Mihail Georgia Tech. Algorithmic Performance in Complex Networks

2. 2 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.

3. 3 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.

4. 4 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

5. 5 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? Congestion Graph on nodes. Congestion = flow on most loaded link under optimal routing. Route 1 unit of flow between each pair of nodes. Total flow. 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?

6. 6 Relevant metric: “bottlenecks”Conductance Alon 85 Jerrum & Sinclair 88 Leighton & Rao 95

7. 7 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.

8. 8 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).

9. 9 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.

10. 10 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

11. 11 Given Choose random perfect matching over CONFIGURATIONAL MODEL minivertices edge multiplicity O(log n), a.s. connected, a.s.

12. 12 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

13. 13 Worst case is when all vertices have degree 3. Structural Model, Proof Idea:Difficulty: Non homogeneity in state-space But all vertices do not have the same degree.

14. 14 Growth with Preferential Connectivity Model, Proof Idea: 1st 2nd 3rd 4th 5th 6th 7th Key Observation : To bound conductance of S, suffices to study combinatorics of how these two sequences interleave. Difficulty : whether there is edge depends upon arrival order of all vertices!

15. 15 Growth with Preferential Connectivity Model, Proof Idea: Shifting Argument 1st 2nd 3rd 4th 5th 6th 7th

16. 16 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 ?

17. 17 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 ?

18. 18 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).

19. 19 Conductance, Mixing and Cover Time For Cover Time “mixing” in Rapid Mixing of Random Walk Alon 85 Jerrum & Sinclair 88

20. 20 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.

21. 21 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

22. 22 Proof In the configurational model with Cover Time with Look-Ahead Two Theorem [MM,Saberi,Tetali 05]: can discover vertices in steps.

23. 23 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 ?

24. 24 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

25. 25 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 per client. 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.

26. 26 Theorem [ Feder, Guetz, M, Saberi 06 ]: Rapid 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 general 2-link switch on d-regular graphs is rapidly mixing. LOCALITY: In reality, network can only switch links that are within constant distance.

27. 27 Space of d-regular graphs general 2-link switch Markov chain Space of connected d-regular graphs local Flip Markov chain The proof is a Markov chain comparison argument Map the transitions of S to the transitions of SC, with small load. Load = max # transitions of S mapped to single transition of SC.

28. 28 Space of d-regular graphs general 2-link switch Markov chain Space of connected d-regular graphs local Flip Markov chain The proof is a Markov chain comparison argument Natural mapping from S to SC: map switch between u, v to path of local flip switches. Problem: length of path unbounded. Key Construction: mapping such that each edge in S maps to a constant number of edges in SC

29. 29 P2P Dynamic Network Construction Problem: ? ? ?

30. 30 P2P Dynamic Network Topology Construction by Random Walk Theorem [Law & Siu ‘03]: Construct a constant expander on n vertices with overhead O( log n) per node addition. ? ? ?

31. 31 P2P Dynamic Network Topology Construction by Random Walk

32. 32 P2P Dynamic Network Topology Construction by Random Walk

33. 33 P2P Dynamic Network Topology Construction by Random Walk Gkantsidis, MM, Saberi ’04 Heuristic reminiscent of saving random bits in simulation of BPP [AKS87,ZI89,G95] Overhead O(1) per new node addition.

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36. 36 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

37. 37 How can we maintain an expander in a distributed way under dynamic settings or arriving and departing nodes? Can we develop efficient distributed algorithms that discover critical links in the network? Networking is growing at an unprecedented rate and it is rich with algorithmic questions. In particular, it raises novel new questions related to expander graphs.