Challenges for search in wireless networks

1. Efficiency of search methods in dynamic wireless networks Gerhard Haßlinger, T-Systems, Germany, Thomas Kunz, Carleton University, Ottawa, Canada, E-Mail: gerhard.hasslinger@telekom.de, tkunz@sce.carleton.ca • Challenges for search in wireless networks • Random walks and flooding for search with partial info. • Evaluation using transient analysis and bounds

2. Experience with wireless routing: Inaccuracy of node state info. for OLSR routing (RFC 3626) Traffic Rate OLSR information enhanced by probing

3. No full support for routing and search can be given in Wireless and mobile networks Sensor networks Overlay, peer-to-peer networks Advantage: More flexibility to set up and expand networks, less overhead in managing networks with high churn Disadvantage: More expensive search by exploration of the network The Internet itself exhibits unstructured growth and churn, but the IETF (Cisco, Juniper ...) established reliable routing and search engines (Google, Yahoo, …) provide content exploration New IETF WG on “Routing for low power & lossy Networks” Challenges for search in wireless networks

4. Network exploration by flooding & random walks • Flooding is exhaustive for all neighbors up to a distance d or time to live (TTL); Parallel search; large amount of messages spread in all directions • Random walks follow some probabilistic winding route

5. Network exploration by flooding & random walks • Control of message overhead for flooding is difficult: Unknown network coverage as a function of the distance d Coverage may rise e.g. from 3% to 30% in one step dd + 1 • A random walk of predefined length L has fixed expense Randoms walks can proceed in parallel, with forking or can be enhanced by flooding with small d from some points  Many ways to combine random walks & flooding schemes (see Gkantsidis et al., IEEE Infocom 2004 & ’05) Network coverage is partial, but random walk searches are efficient e.g. for replicated data in P2P networks Random network growth is also efficiently supported by r. walks

6. Transient analysis of basic random walks Performance studies on random walks prefer simulation, although transient analysis offers a simple and scalable alternative Bounds e.g. based on second largest eigenvalue of the transition matrix prove convergence but often are not tight Performance criteria: Number of steps required - to reach a target node - or for network coverage with predefined probability (e.g. 99%) Transient analysis - computes the probability distribution for the random walks’ sojourn node step by step - starting from a node or an arbitrary initial distribution

7. Basic transient analysis of a random walks An absorbing state („black hole“) is introduced at a considered network node  The probability to enter the absorbing state from some starting conditions, e.g. from steady state, equals the probability to discover the network node during the random walk For networks with heterogeneous nodes, coverage can be studied depending on different types or degrees of nodes

8. 1.hop Example of a transient random walk analysis ... 10.11%  10% 12.hop: Start 2.hop 14.89%  15% 20.23%  20% Absorbing state: Modified graph for considering a target

9. Performance Results for random walk & flooding Random walks: Number of hops required for 90% and 99% success for different levels of available routing info and for different distances to destination (up to 20 hops on a grid diagonal)

10. Extensions of the transient analysis for random walks • The following cases are tractable by extended analysis: • Several random walks in parallel • product formula for the probability that independent trials miss a node • Random walk without step back (except for nodes of degree 1) •  increased state space for analysis: network edges instead of nodes, • but the run time complexity is unchanged • Random walk followed by flooding on distance d after the last step •  extend absorbing state to the set of all neighbors up to distance d • Random walk search for replicated data on n nodes •  use a set of n absorbing states or • assume a binomial distributed hit count based on single node search

11. D:0 D-1 D+1 D-1 S:D D+1 p q Convergence bound for biased random walks Dm: Distance to the destination after m random walk steps (with changes) Dm has a binomial distribution: for unrestricted range of distances in both directions with mean E(Dm) = D –m (q–p)/(q+p); variance s2(Dm) = mpq/(q+p) if m(q–p)/(q+p) > D E(Dm) < 0  destination is reached with > 50% Example: 2-dim. Grid Start S:D 3 2 1 D:0 -1 -2 -3 Destination p q

12. Bound (thin lines) compared to transient analysis

13. Conclusions • Random walks are useful for search and routing in networks with unreliable or incomplete information on target nodes • Random walks outperform flooding for large hop distances • Many options for combining routing, flooding & random walks for improved performance in different scenarios • Transient analysis yields accurate evaluation of random walks - for the basic case and many variants - is scalable for networks of large size - bounds can be given on convergence behaviour