Complex Networks: Connectivity and Functionality

1. Complex Networks:Connectivity and Functionality Milena Mihail Georgia Tech.

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. Which are criticalnetwork parameters/metrics that determinealgorithmic performance? Important in network simulation and design. Predictive of routing and searching performance is conductance, expansion, spectral gap.

3. This talk: The case of internet routing topology How can network models capture the parameters/metrics that are critical in network performance? This talk: The case of P2P networks Can we design network algorithms/protocols that optimize these critical network parameters?

4. Current Models for Internet Routing Topologies focus on large degree-variance Erdos-Renyi-like, Configurational : Chung&Lu A random graph with given degrees Evolutionary, macroscopic and microscopic : The graph grows one vertex at a time and attaches preferentially to degrees or according to some optimization criterion Fabrikant,Koutsoupias,Papadimitriou Barabasi&Albert Bollobas&Riordan The case of modeling the internet routing topology Nodes are routers or Autonomous Systems Two nodes connected by a link if they are involved in direct exchange of traffic Sparse small-world graphs with large degree-variance But are degrees the “right” parameter to measure?

5. computationally soft Matlab does 1-2M node sparse graphs An important metric: Conductance and the second eigenvalue of the stochastic normalization of the adjacency matrix characterize routing congestion under link capacities, mixing rate, cover time. Leighton-Rao Jerrum-Sinclair Broder-Karlin How does the second eigenvalue (more generally the principal eigenvalues) scale as the size of the network increases?

6. This is also another point of view of the small-world phenomenon This also says that congestion under link capacities scales smoothly Second eigenvalue of internet is larger than that of random graphs but spectral gap remains constant as number of nodes increases. Gkantsidis,M,Zegura M,Papadimitriou,Saberi random graph configurational model Gkantsidis,M,Saberi Open problem: Erdos-Renyi like, configurational models which include spectral gap parameter?

7. Some evolutionary random graph models may capture clustering Growth & Preferential Attachment One vertex at a time New vertex attaches to existing vertices

8. Open Problem: characterize clustering as a function of model parameter Flaxman,Frieze,Vera plots as number of nodes increases ? M,Saberi,Papadimitriou ie, indicate which parameter ranges are important in simulations

9. Other discrepancies of random graph models from real internet topologies: Li, Alderson, Willinger, Doyle high degree nodes away from “network core” high degrees mostly connected to low degrees “core” of low degrees what do internet topologies “optimize” ? real network random graph, evolutionary model random graph, configurational model

10. Open Problem: Research direction: Algorithms improving congestion conductance and spectral gap Boyd&Saberi Rao&Vazirani Given total length l and n random points in a metric space construct a graph with total link length l that - maximizes spectral gap, conductance - minimizes congestion under node capacities

11. Algorithms optimizing connectivity How do you maintain a P2P network with good search performance ?

12. Lv&al Chawathe&al Gkantsidis&al The case of Peer-to-Peer Networks Distributed, decentralized nnodes,d-regular graph each node has resources O(polylogn) and knows a constant size neighborhood ? Search for content, e.g. by flooding or random walk Must maintain well connected topology, e.g. a random graph, an expander Jerrum-Sinclair Broder-Karlin

13. P2P Network Topology Maintenance by Constant Randomization Gnutella: constantly drops existing connections and replaces them with new connections random graph, expander 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.

14. 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

15. ? ? ? Padurangan,Raghavan,Upfal Law,Siu Gkantsidis,M,Saberi Ajtai,Komlos,Szemeredi Impagliazzo,Zuckerman Question: How do we add new nodes with low network overhead? Question: How do we delete nodes with low network overhead?

16. Algorithms developing topology awareness Link Criticality Boyd,Diaconis,Xiao

17. 3$ local information 1$ 7$ local information local information 2S Generalized Search: Link Criticality A node has a query and a budget Subtract 1 from budget Arbitrarily partition the remaining budget and forward the parts to the neighbors Boyd,Diaconis,Xiao Gkantsidis,M,Saberi

18. Let be a graph. Assign symmetric transition probabilities to links in (and self loops) so that the resulting matrix is stochastic and the second in absolute value largest eigenvalue is minimized. s.t. Fastest Mixing Markov Chain Boyd,Diaconis,Xiao SDP formalization

19. Fastest Mixing Markov Chain Subgradient Algorithm is some vector on of initial transition probabilities is the eigenvector corresponding to second in absolute value largest eigenvalue is a vector on with repeat subgradient step projection to feasible subspace Open Question: Is there a decentralized implementation or algorithm?

20. The Case of Ad-Hoc Wireless Networks How does Capacity/Throughput/Delay Scale? Capacity of Wireless Networks, Gupta & Kumar, 2000 Is there a connection with Lipton & Tarjan’s separators for planar graphs? Mobility Increases Capacity, Grossgaluser & Tse, 2001 Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003 Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004