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Tight bounds on sparse perturbations of Markov Chains Romain Hollanders Giacomo Como Jean-Charles Delvenne Raphal Jungers UCLouvain University of Lund MTNS2014 Slide 2 PageRank is the average portion of time spent in a node During an infinite random walk Slide 3 PageRank is the average portion of time spent in a node During an infinite random walk Slide 4 PageRank : PageRank is the average portion of time spent in a node During an infinite random walk Slide 5 PageRank : How much can a few nodes affect the PageRank values ? Slide 6 PageRank : How much can a few nodes affect the PageRank values ? Slide 7 PageRank : How much can a few nodes affect the PageRank values ? Slide 8 PageRank : How much can a few nodes affect the PageRank values ? Slide 9 Consensus : How much can a few nodes affect a consensus ? Slide 10 Consensus : the weight of each agent in the final decision How much can a few nodes affect a consensus ? Slide 11 Consensus : How much can a few nodes affect a consensus ? Slide 12 Slide 13 Slide 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 Slide 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 Slide 16 Exactly and in polynomial time 1. 2. 3. Slide 17 probability 1 Slide 18 Slide 19 ?? Slide 20 A counter example Slide 21 Slide 22 We need to loop through every candidate worst-node Slide 23 1. 2. 3. Slide 24 Perspectives Extend the approach to other norms Compare the results with Como & Fagnanis bound especially the 1-norm to establish its quality Slide 25 Thank you Slide 26