1. Tight bounds on sparse perturbations of Markov Chains Romain Hollanders Giacomo Como Jean-Charles Delvenne Raphaël Jungers UCLouvain University of Lund MTNS’2014

2. PageRank is the average portion of time spent in a node During an infinite random walk

3. PageRank is the average portion of time spent in a node During an infinite random walk

4. PageRank : PageRank is the average portion of time spent in a node During an infinite random walk

5. PageRank : How much can a few nodes affect the PageRank values ?

6. PageRank : How much can a few nodes affect the PageRank values ?

7. PageRank : How much can a few nodes affect the PageRank values ?

8. PageRank : How much can a few nodes affect the PageRank values ?

9. Consensus : How much can a few nodes affect a consensus ?

10. Consensus : the weight of each agent in the final decision How much can a few nodes affect a consensus ?

11. Consensus : How much can a few nodes affect a consensus ?

14. Typically blows up when the network size grows Sensitive mainly to the magnitude of the perturbation We need better, tighter bounds, adapted to local perturbations ! Weak bounds already exist They depend more on the size than the structure of the network / perturbation

15. Captures local perturbationsProvides physical insight Difficult (impossible?) to extend to other norms No reason to believe that it is tight Como & Fagnani proposed a bound for the 1-norm mixing time a nice increasing function

16. Exactly and in polynomial time 1. 2. 3.

17. probability 1

19. ??

20. A counter example

22. We need to loop through every candidate “worst-node”…

23. 1. 2. 3.

24. Perspectives Extend the approach to other norms Compare the results with Como & Fagnani’s bound especially the 1-norm to establish its quality

25. Thank you