Niloy Ganguly Complex Networks Research Group Department of Computer Science & Engineering

1. Maximizing Network Coverage under Resource Constraints Niloy Ganguly Complex Networks Research Group Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Kharagpur 721302 Collaborators : Sudipta Saha, Subrata Nandi, Lutz Brusch, Andreas Deutsch

2. Information Dissemination/Searching • Large scale system • Wireless sensor network • Mobile network • P2P networks • Essential requirements • Dissemination • A node has an information • Wants to spread it to all other nodes in the network • Searching • A node wants to get some information/data • The data is somewhere in the network • Gathering (collection) ? ? ? • Data/Query packets need to cover/visit many nodes in the network

3. Information Dissemination/Searching • Challenge • Unstructured network • No centralized control, fully distributed • Very largescale network • Dynamic network structure • No end to end connectivity • Constraint of Time • Constraint on Energy • Main Goal : Maximize the node coverage within a given constraint of time as well as energy

4. Information Dissemination/Searching Flooding, Time =2 • Existing algorithms • Basic flooding • Wastes a lot of resource • Optimal in time • Single random walk • Wastes a lot of time • Optimal in resource usage Mutual overlap (wastage) Single RW, Time =9 • Flooding and random Walk both are optimal under a single constraint General optimal algorithm for any pair of resource and time constraint

5. Overlap 3 • Wastage of resource • Visiting the same node more than once • Overlap with own trail • Mutual overlap 2 4 7 1 8 Start 5 9 6 Overlap with own trail 10 Trail of single walker Node / site

6. Overlap • Wastage of resource • Visiting the same node more than once • Overlap with own trail • Mutual overlap Start Mutual Overlap Trail of walker 1 Trail of walker 2 Start Node / site

7. Understanding the Problem Space • Three broad regions Bandwidth Time

8. Building Proliferation Strategy • Walkers originate from a single point like flooding and random walk • Walkers multiply at certain rate (say) P- proliferation rate • For each point in the graph, a P would be needed – determining best P for each point

9. Optimal Proliferation Rate • Proliferation Rate which enhances speed but does not cause mutual overlap like single random walker • Each walker has its own area (although new walkers are produced from old walkers)

10. K-RW - The statistical mechanics perspective Results on d – dimensional grid Euclidean dimension (Larralde et. al. ‚02) Increase in coverage Regime I Regime II Regime III Ref: H. Larralde, P. Turnfio, S. Havlin, H. E. Stanley and G. H. Weiss, Nature, 355:423 - 426, 2002.

11. Observations and Inference • In regime I, coverage rate is similar to flooding. • In regime II, walkers move far apart each other and less walkers • co-occupy nodes. However, still some amount of mutual overlap persists. • In regime III, each walker behaves independently like a single • random walker with non-overlapping exploration space, covers with peak efficiency Emax=E1-RW. Regime I Regime II Regime III

12. Proposed Algorithm Start with a small number of random walkers at t = 1 Proliferating each walker at a suitable rate P*(t) at each time step, Aim :- System always remains at the regime boundary (II- III) as desired.

13. Speed-up vis-à-vis 1-RW • Bandwidth consumed • Lower bound of time to achieve Cmax, • Speed-up

14. Contribution • We develop a coverage algorithm P*(t)-RW with proliferating message packets and temporally modulated proliferation rate. • Proliferation -- a walker self-replicates at its current node with rate P*(t)Є R+ at time t such that on average each walker produces one offspring walker every 1/P*(t) time steps • The algorithm performs as efficiently as 1-RW, covers Cmax but (B (d−2)/d) times faster, resulting in significant service speed-up on a regular grid of dimension d. Coverage maximization in networks under resource constraints,Subrata Nandi, Lutz Brusch, Andreas Deutsch and Niloy Ganguly, Phys. Rev. E 81, 061124 (2010)

16. Contribution Resource (B) Time (T) Coverage maximization in networks under resource constraints,Subrata Nandi, Lutz Brusch, Andreas Deutsch and Niloy Ganguly, Phys. Rev. E 81, 061124 (2010)

17. The Problem Space • Phase diagram – explains the problem space Resource (B) Time (t) • We need optimal strategy for Zone 2 and in general for any given (B,T) pair

18. Our Study Average mutual overlap in proliferating random walk Standard deviation of mutual overlap in proliferating random Walk Is heterogeneity better from the perspective of coverage under bounded resource? *\alpha = 1 optimal strategy

19. Our Study Average mutual overlap in K - random walk Standard deviation of mutual overlap in K-random walk

20. Strategy Reduce Heterogeneity or Increase Heterogeneity

21. Our Study Relationship between concentration of the walkers and their mutual overlap • Two distinct phases • Phase I - low mutual overlap, very short, highly sensitive to the concentration • Phase II - High mutual overlap, Insensitive to concentration

22. Main Challenge Hence, the more walkers an algorithm can keep in Phase I, the more utilization of the resource and time, it can make

23. Main Challenge • More walkers in Phase 1 • (low density) Proliferate only when a walker is in a very sparse region

24. Our Strategy • Calculate local density of an walker • A difficult task without centralized control • Our solution - replaces the spatial measurement of the density with the temporal measurement • Walkers record how many of its previous visits are mutual overlap • This approximation of density correlates with actual density Spatial measure Temporal measure • Correlation is better for higher proliferation rates

25. Our Strategy • We made proliferation rate proportional to this temporal density e.g., p(t) is the per walker proliferation rate, we replace α by αh • Rate of proliferation is now inversely proportional to the number of mutual overlap a walker has faced • γ=0 ; The strategy is equivalent to base strategy • γ>0 ; As γ increases, we proliferate more those walkers which face lesser mutual overlap in last H visits • Observation: γ=20 and above gives maximum improvement • Only zero mutual overlaps are proliferated • Proliferating only at zero mutual overlap – produced maximum efficiency Coverage maximization under resource constraints using non-uniform proliferating random walk, Sudipta Saha and Niloy Ganguly, Phys. Rev. E 87, 022807 (2013)

26. Our Strategy • We proliferated only those walkers which faced zero mutual overlap in the last H node visits (identified as Phase-I walkers) • This strategy is denoted by P(t,h)-RW-e and performed extremely well in comparison with other existing strategies • Improvements – • In 2D regular grid 250% • In 3D regular grid 133% • In 4D regular grid 80% • In 5D regular grid 20% • In 2D random geometric graph 233% Coverage maximization under resource constraints using non-uniform proliferating random walk, Sudipta Saha and Niloy Ganguly, Phys. Rev. E 87, 022807 (2013)

27. Conclusion Solution for the entire space Significant Performance Improvement Resource (B) Z Coverage (C) Time (T) Time (T) X Resource (B) New Problem Definition Optimize with Knowledge Y Knowledge Maximize the function C=f(B,T, K)

28. Thank You Complex Network Research group (CNeRG) Coverage maximization in networks under resource constraints,Subrata Nandi, Lutz Brusch, Andreas Deutsch and Niloy Ganguly, Phys. Rev. E 81, 061124 (2010) maximization under resource constraints using non-uniform proliferating random walk, Sudipta Saha and Niloy Ganguly, Phys. Rev. E 87, 022807 (2013)

29. Conclusion Resource (B) Time (t)

30. Effect of History Size • History size has significant effect on the performance of the strategy also • Lower history size may not identify the proper phase 1 walkers • Higher history size may proliferate less frequently than what is required • If both 0 and 1 overlaps are proliferated • For higher history size they behave almost equally • Optimal history size depends on topology as well as walker forwarding policy “Coverage maximization under resource constraints using non-uniform proliferating random walk” Sudipta Saha and Niloy Ganguly, Phys. Rev. E 87, 022807 (2013)

31. A Different Perspective • Key observations • In regular grid – as dimension increases, the random walk strategy becomes more efficient • A random walker travels more distance on average from the start node as dimension increases • A walker in the developed history based strategy travels the maximum distance from the start node in comparison to other existing algorithms The average distance a walker can travel from the start node Efficiency of the strategy Is Correlated ?

32. A Different Perspective Forwarded with higher bias to this node • New strategy • Walker should move in such a way that they can travel maximum distance from the start node on average • Should move along a direction • They should follow unique direction to minimize mutual overlap • Challenges • Finding out unique and optimal direction • For static network – it is outwards the center node • But needs the information of the position of the center • For dynamic networks it is more difficult • Finding out the balancing proliferation rate θ A

33. A Different Perspective • Alternative way (a possible bio-inspired strategy) • We can learn from the spreading mechanism of cancer cells • Density biased proliferating random walk • We approximate spatial density by temporal density • If we an approximate spatial density itself • How exactly the cancer cells estimates density in a distributed fashion • Can be implemented using artificial pheromone ? • How cancer cells migrate from one place to another? • How they sense density ? • At what rate they proliferate ? • Do they optimize food resource and time?

34. Thank You Niloy Ganguly niloy@cse.iitkgp.ernet.in http://cse.iitkgp.ac.in/~niloy