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

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Presentation on theme: "Tight bounds on sparse perturbations of Markov Chains Romain Hollanders Giacomo Como Jean-Charles Delvenne Raphaël Jungers UCLouvain University of Lund."— Presentation transcript:

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 ?

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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

17 probability 1

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19 ??

20 A counter example

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

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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

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