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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 MTNS’2014

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

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

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

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

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

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

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

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

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

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

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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

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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

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Exactly and in polynomial time 1. 2. 3.

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probability 1

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??

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A counter example

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

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1. 2. 3.

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

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Thank you

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